Twisted Forms of Differential Lie Algebras over ${\mathbb {C}}(t)$ Associated with Complex Simple Lie Algebras

Abstract

Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over ${\mathbb {C}}(t)$ associated with complex simple Lie algebras. Hopf–Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors.

1 Introduction: Problem and Answer

Rings and algebras are supposed to be associative and containing 1, and their morphisms are supposed to send 1 to 1. Moreover, rings, algebras and Hopf algebras are assumed to be commutative, unless otherwise stated.

We let $\delta$ mean “differential” and use the symbol $\delta$ to indicate differential operators in general. A $\delta$-ring is thus a (commutative) ring R equipped with an additive operator $\delta : R \rightarrow R$ satisfying the Leibniz rule

$$\begin{aligned} \delta (xy)=(\delta x)y+ x(\delta y) \end{aligned}$$

for all $x, y \in R$. It is called a $\delta$-field if the ring is a field. The rational function field ${\mathbb {C}}(t)$ in one variable is regarded as a $\delta$-field with respect to the standard operator such that $\delta t=1$ and $\delta c=0$ for every $c \in {\mathbb {C}}$. The field

$$\begin{aligned} C_{{\mathbb {C}}(t)}= \{ \, a \in {\mathbb {C}}(t)\mid \delta a=0 \, \} \end{aligned}$$

of constants is ${\mathbb {C}}$. A $\delta$-${\mathbb {C}}(t)$-Lie algebra is a Lie algebra $\mathfrak {g}$ over ${\mathbb {C}}(t)$ which is equipped with an additive operator $\delta : \mathfrak {g}\rightarrow \mathfrak {g}$ such that

$$\begin{aligned} \delta (a X)= (\delta a)X+ a(\delta X),\quad \delta [X,Y]=[\delta X, Y]+ [X,\delta Y] \end{aligned}$$

for all $a \in {\mathbb {C}}(t)$ and $X,Y \in \mathfrak {g}$. In the same way a $\delta$-R-Lie algebra is defined for any $\delta$-ring R. Let $\mathfrak {g}$ be a $\delta$-${\mathbb {C}}(t)$-Lie algebra. Given a $\delta$-ring map ${\mathbb {C}}(t)\rightarrow R$ (that is, a ring map preserving the $\delta$-operator), the base extension $\mathfrak {g} \otimes _{{\mathbb {C}}(t)} R$ of $\mathfrak {g}$ is naturally a $\delta$-R-Lie algebra. A twisted form of $\mathfrak {g}$ is a $\delta$-${\mathbb {C}}(t)$-Lie algebra $\mathfrak {f}$ such that

$$\begin{aligned} \mathfrak {g} \otimes _{{\mathbb {C}}(t)} R \simeq \mathfrak {f} \otimes _{{\mathbb {C}}(t)} R \quad \text {as}\ \delta \text {-}R\text {-Lie algebras} \end{aligned}$$

for some $\delta$-ring map ${\mathbb {C}}(t)\rightarrow R$ with $R\ne 0$.

Let $n \ge 2$. We can and do regard the ${\mathbb {C}}(t)$-Lie algebra $\mathfrak {sl}_n({\mathbb {C}}(t))$ which consists of all traceless matrices $X=\big (x_{ij}\big )$ with $x_{ij}\in {\mathbb {C}}(t)$, as a $\delta$-${\mathbb {C}}(t)$-Lie algebra with respect to the entry-wise $\delta$-operator $\delta \big (x_{ij}\big ) :=\big (\delta x_{ij}\big )$. A. Pianzola posed personally to the first-named author, a problem which is specialized to the following.

Problem 1.1

Describe all twisted forms of the $\delta$-${\mathbb {C}}(t)$-Lie algebra $\mathfrak {sl}_n({\mathbb {C}}(t))$.

Clearly, to generalize the problem, one can replace $\mathfrak {sl}_n({\mathbb {C}})$ with a complex simple Lie algebra ${\mathfrak {g}}_0$ (of finite dimension), and regard ${\mathbb {C}}(t)$-Lie algebra

$$\begin{aligned} {\mathfrak {g}}_0({\mathbb {C}}(t))={\mathfrak {g}}_0 \otimes _{\mathbb {C}}{\mathbb {C}}(t)\end{aligned}$$

(1.1)

as a $\delta$-${\mathbb {C}}(t)$-Lie algebra with the $\delta$ operating on the tensor factor ${\mathbb {C}}(t)$. The notation (1.1) is used since ${\mathfrak {g}}_0$ is seen to give the functor $R\mapsto {\mathfrak {g}}_0 \otimes _{\mathbb {C}}R$, and ${\mathfrak {g}}_0 \otimes _{\mathbb {C}}{\mathbb {C}}(t)$ is then its value. Pianzola’s problem is precisely the following one, which we are going to solve.

Problem 1.2

Given a complex simple Lie algebra ${\mathfrak {g}}_0$, describe all twisted forms of ${\mathfrak {g}}_0({\mathbb {C}}(t))$.

Our interest in this problem or in studying twisted forms of Lie algebras in the differential context comes from differential Galois Theory, in which examples of such twisted forms naturally arise. In fact, given a homogeneous linear differential equation with coefficients in a differential field, say ${\mathbb {C}}(t)$, we have the Galois group of the equation, $\mathsf {G}_0$, which is an affine algebraic ${\mathbb {C}}$-group. In addition we naturally have an affine algebraic “differential” ${\mathbb {C}}(t)$-group $\mathsf {G}$, called the intrinsic Galois group (or Katz group); see [11]. One sees that the Lie algebra $\mathrm{Lie}(\mathsf {G})$ of $\mathsf {G}$ is a $\delta$-${\mathbb {C}}(t)$-Lie algebra which is a twisted form of the $\delta$-${\mathbb {C}}(t)$-Lie algebra $\mathrm{Lie}(\mathsf {G}_0)\otimes _{{\mathbb {C}}}{\mathbb {C}}(t)$, where $\mathrm{Lie}(\mathsf {G}_0)$ is the (complex) Lie algebra of $\mathsf {G}_0$.

Returning to the situation before, recall that the complex simple Lie algebras are classified, labeled by their root systems

$$\begin{aligned} \mathrm {A}_{\ell }~(\ell \ge 1),\ \mathrm {B}_{\ell } ~(\ell \ge 2),\ \mathrm {C}_{\ell }~(\ell \ge 3), \ \mathrm {D}_{\ell }~(\ell \ge 4),\ \mathrm {E}_6, \ \mathrm {E}_7,\ \mathrm {E}_8,\ \mathrm {F}_4,\ \mathrm {G}_2. \end{aligned}$$

See [9, Chapter IV], for example. Let ${\mathfrak {g}}_0$ be a complex simple Lie algebra, and let $\varGamma$ denote the automorphism group of the associated Dynkin diagram. Explicitly, the group is

$$\begin{aligned} \varGamma ={\left\{ \begin{array}{ll} \{ 1 \} &{} \text {type}\ \mathrm {A}_1,\ \mathrm {B}_{\ell }~(\ell \ge 2), \ \mathrm {C}_{\ell }~(\ell \ge 3),\ \mathrm {E}_7,\ \mathrm {E}_8, \ \mathrm {F}_4\ \text {or}\ \mathrm {G}_2;\\ {\mathbb {Z}}_2 &{} \text {type}\ \mathrm {A}_{\ell }\ (\ell \ge 2), \ \mathrm {D}_{\ell }\ (\ell \ge 5)\ \, \text {or}\ \, \mathrm {E}_6; \\ \mathfrak {S}_3 &{} \text {type}\ \mathrm {D}_4 \end{array}\right. } \end{aligned}$$

(1.2)

according to the type of ${\mathfrak {g}}_0$; see [15, Table 3 on Page 298]. Here and in what follows ${\mathbb {Z}}_n$ denotes the cyclic group of order n. In addition, $\mathfrak {S}_3$ denotes the symmetric group of degree 3. The action by $\varGamma$ naturally (up to conjugation) gives rise to automorphisms of ${\mathfrak {g}}_0$, which forms a group naturally identified with the group $\mathsf {Out}({\mathfrak {g}}_0)$ of outer-automorphisms of ${\mathfrak {g}}_0$.

Roughly speaking, our answer, Theorem 1.5, to the problem tells that all non-trivial twisted forms are obtained by the Galois descent (see [12, Section 18]) for which $\varGamma$ (and its subgroups for type $\mathrm {D}_4$) act as Galois groups. To make a precise statement we introduce below the notion of being quasi-isomorphic.

Lemma 1.3

If $\mathfrak {g}=(\mathfrak {g}, \delta )$ is a $\delta$-R-Lie algebra, then for any element $D \in \mathfrak {g}$,

$$\begin{aligned} \delta + {\mathsf {ad}}(D) : \mathfrak {g}\rightarrow \mathfrak {g},\ X \mapsto \delta X +[D,X] \end{aligned}$$

is a $\delta$-operator with which $\mathfrak {g}$ is again a $\delta$-R-Lie algebra.

Indeed, one sees, more generally, that for any R-linear derivation $\mathfrak {D} : \mathfrak {g}\rightarrow \mathfrak {g}$, $(\mathfrak {g}, \delta +\mathfrak {D})$ is a $\delta$-R-Lie algebra. Note that the inner derivation $\mathsf {ad}(D)$ above is R-linear.

Definition 1.4

Let R be as above, We say that two $\delta$-R-Lie algebras $\mathfrak {g}=(\mathfrak {g}, \delta )$ and $\mathfrak {g}'=(\mathfrak {g}', \delta ')$ are quasi-isomorphic, if there is an element $D \in \mathfrak {g}$ such that

$$\begin{aligned} (\mathfrak {g}, \delta +\mathsf {ad}(D))\simeq (\mathfrak {g}', \delta ') \quad \text {as}\ \delta \text {-}R\text {-Lie algebras}. \end{aligned}$$

The condition is equivalent to saying that there is an element $D' \in \mathfrak {g}'$ such that $(\mathfrak {g}, \delta )\simeq (\mathfrak {g}', \delta '+\mathsf {ad}(D'))$, as is easily seen. It follows that the quasi-isomorphism gives an equivalence relation among all $\delta$-R-Lie algebras.

Theorem 1.5

Suppose that ${\mathfrak {g}}_0$ is a complex simple Lie algebra, and let $\varGamma$ be the automorphism group of the associated Dynkin diagram. Then a $\delta$-${\mathbb {C}}(t)$-Lie algebra is a twisted form of ${\mathfrak {g}}_0({\mathbb {C}}(t))$ if and only if it is quasi-isomorphic to one of those listed below, according to the case $\varGamma =\{1\}$, ${\mathbb {Z}}_2$ or $\mathfrak {S}_3$; see (1.2).

• (1) Case $\varGamma =\{1\}$: ${\mathfrak {g}}_0({\mathbb {C}}(t))$;

• (2) Case $\varGamma ={\mathbb {Z}}_2$:

  • (i) ${\mathfrak {g}}_0({\mathbb {C}}(t))$;

  • (ii) ${\mathfrak {g}}_0(L)^{\varGamma }$, where $L/{\mathbb {C}}(t)$ is a quadratic field extension.

• (3) Case $\varGamma =\mathfrak {S}_3$:

  • (i) ${\mathfrak {g}}_0({\mathbb {C}}(t))$;

  • (ii) ${\mathfrak {g}}_0(L)^{{\mathbb {Z}}_2}$, where $L/{\mathbb {C}}(t)$ is a quadratic field extension;

  • (iii) ${\mathfrak {g}}_0(L)^{{\mathbb {Z}}_3}$, where $L/{\mathbb {C}}(t)$ is a cubic Galois extension;

  • (iv) ${\mathfrak {g}}_0(L)^{\varGamma }$, where $L/{\mathbb {C}}(t)$ is a Galois extension of fields with Galois group $\varGamma (=\mathfrak {S}_3)$.

We should immediately add some explanations about the statement above. First, any finite field extension $L/{\mathbb {C}}(t)$ uniquely turns into an extension of $\delta$-fields, whence ${\mathfrak {g}}_0(L)$ turns into a $\delta$-L-Lie algebra. Second, in (ii) of (2) and (iv) of (3) above, the group $\varGamma$ is supposed to act diagonally on ${\mathfrak {g}}_0(L)={\mathfrak {g}}_0\otimes _{{\mathbb {C}}} L$, as outer-automorphisms on ${\mathfrak {g}}_0$, and as the Galois group on L. In addition, ${\mathfrak {g}}_0(L)^{\varGamma }$ denotes the $\varGamma$-invariants in ${\mathfrak {g}}_0(L)$, which is in fact a $\delta$-${\mathbb {C}}(t)$-Lie algebra by Galois descent; see Sect. 4.3. Third, in (ii) of (3), we choose arbitrarily an order 2 subgroup ${\mathbb {Z}}_2$ of $\varGamma \, (=\mathfrak {S}_3)$, and let it act on ${\mathfrak {g}}_0$ by restriction. The $\delta$-${\mathbb {C}}(t)$-Lie algebra ${\mathfrak {g}}_0(L)^{{\mathbb {Z}}_2}$ which results in the same way as above does not depend (up to isomorphism) on the choice since the order 2 subgroups are conjugate to each other; on the other hand it may depend on L. Finally, in (iii) of (3), we suppose that ${\mathbb {Z}}_3$ is the unique order 3 subgroup of $\varGamma \, (=\mathfrak {S}_3)$, and let it act on ${\mathfrak {g}}_0$ by restriction, again. We add the following remark: there exist infinitely many quadratic and cubic Galois extensions over ${\mathbb {C}}(t)$, as is easily seen, while the existence of a Galois extension over ${\mathbb {C}}(t)$ with Galois group $\mathfrak {S}_3$ will be ensured by Example 4.9.

The theorem will be proved in the final Sect. 4, which contains as well, explicit descriptions (see Sect. 4.6) of the non-trivial $\delta$-${\mathbb {C}}(t)$-Lie algebras listed in (ii) of (2) and (ii)–(iv) of (3). The preceding two sections provide preliminaries, some of which are beyond what will be needed, but are of interest by themselves. Section 2 presents descent theory in the differential context; Sect. 3 prepares technical tools mainly from Hopf–Galois Theory, which is a ring-theoretic counterpart of theory of torsors for group schemes. In particular, Schauenburg’s bi-Galois Theory [17] will play a role in two stages (see Sects. 3.5, 4.3), when we grasp the twisted forms in question from $\delta$-torsors.

2 Descent Theory in Differential Context

Let R be a $\delta$-ring. A $\delta$-R-module is an R-module M equipped with an additive operator $\delta : M \rightarrow M$ satisfying

$$\begin{aligned} \delta (xm)=(\delta x)m+ x(\delta m),\quad x\in R,\ m \in M. \end{aligned}$$

All $\delta$-R-modules form a symmetric tensor category $\delta$-R-$\mathtt {Modules}$ with respect to the tensor product $M_1 \otimes _R M_2$, the unit object R and the obvious symmetry

$$\begin{aligned} M_1 \otimes _R M_2\rightarrow M_2\otimes _R M_1,\quad m_1\otimes m_2 \mapsto m_2 \otimes m_1. \end{aligned}$$

The $\delta$-operator on $M_1\otimes _R M_2$ is given by

$$\begin{aligned} \delta (m_1\otimes m_2)=\delta m_1\otimes m_2+m_1\otimes \delta m_2. \end{aligned}$$

The notion of $\delta$-R-Lie algebra defined in the previous section is precisely a Lie algebra in the category $\delta$-R-$\mathtt {Modules}$. In general, any linear object, such as algebra or Hopf algebra, in $\delta$-$R\text {-}\mathtt {Modules}$ is called a $\delta$-R-object, so as $\delta$-R-algebra or $\delta$-R-Hopf algebra; important is the fact that the structure is defined by morphisms of $\delta$-R-$\mathtt {Modules}$ between tensor powers of the object. Given a $\delta$-R-algebra S, we have the base-extension functor

$$\begin{aligned} \otimes _R S : \delta \text {-}R\text {-}\mathtt {Modules} \rightarrow \delta \text {-}S\text {-}\mathtt {Modules}, \end{aligned}$$

which induces base-extension functors for linear objects such as above.

We are concerned with descent theory (see [22], for example) in differential context. To make this clearer, let us fix a $\delta$-R-object A. A $\delta$-R-object B is called an S/R-form of A, or a twisted form of A split by S, if S is a $\delta$-R-algebra such that

• (i) S is faithfully flat as an R-algebra, and

• (ii) $A \otimes _R S \simeq B \otimes _R S$ as $\delta$-S-objects.

A $\delta$-R-object B is called a twisted form of A, if there exists a $\delta$-R-algebra S which satisfies (i) and (ii) above.

The $\delta$-automorphism-group functor of A is the functor

$$\begin{aligned} \mathsf {Aut}_{\delta }(A) : \delta \text {-}R\text {-}\mathtt {Algebras} \rightarrow \mathtt {Groups},\quad T \mapsto \mathrm{Aut}_{\delta \text {-}T}(A\otimes _R T) \end{aligned}$$

(2.1)

from the category $\delta \text {-}R\text {-}\mathtt {Algebras}$ of $\delta$-R-algebras to the category $\mathtt {Groups}$ of groups, which associates to each $\delta$-R-algebra T, the automorphism-group ${\mathrm{Aut}_{\delta \text {-}T}(A \otimes _R T)}$ of the $\delta$-T-object $A \otimes _R T$. When constructing the 1st Amitsur cohomology (pointed) set as in [22, Section 17.6], replace faithfully flat homomorphisms of rings and automorphism-group functors with our $R \rightarrow S$ (satisfying (i) above) and $\mathsf {Aut}_{\delta }(A)$, respectively. The resulting differential analogue is denoted by

$$\begin{aligned} H^1_{\delta }(S/R, \mathsf {Aut}_{\delta }(A)). \end{aligned}$$

(2.2)

This is seen to classify the S/R-forms of A; to be more precise, there is a natural bijection from $H^1_{\delta }(S/R, \mathsf {Aut}_{\delta }(A))$ to the set of all $\delta \text {-}R$-isomorphism classes of the S/R-forms. An important consequence is: if $A'$ is another $\delta$-R-object of some distinct kind, which has the $\delta$-automorphism-group functor isomorphic to $\mathsf {Aut}_{\delta }(A)$, then there is a one-to-one correspondence (up to isomorphism) between the S/R-forms of A and S/R-forms of $A'$.

We remark that for any functor ${\mathsf {G}}: \delta \text {-}R\text {-}\mathtt {Algebras}\rightarrow \mathtt {Groups}$, the cohomology set $H^1_{\delta }(S/R, {\mathsf {G}})$ is defined just as the one in (2.2). The set will appear in what follows (see (3.6)) only when ${\mathsf {G}}$ is representable, and turns out, indeed, to be an automorphism-group functor.

Remark 2.1

We have used so far the base-on-right notation $A \otimes _R T$ which denotes the extended base on the right; it seemingly looks nicer than the base-on-left notation $T\otimes _R A$. But we may and do (when it is natural) use the latter notation.

Remark 2.2

In this Remark the differential structure is ignored.

• (1) We would like to clarify our use of the term (twisted) form. Consider a faithfully flat homomorphism $R\rightarrow S$ of rings and an R-object A. If R and S are fields, it is more conventional (as one of the referees mentioned) for experts in representation theory or physicists to speak of R-form of the S-object $A \otimes S$. For example, $\mathfrak {so}_3({\mathbb {R}})$ is a real form of $\mathfrak {sl}_2({\mathbb {C}})$. Our terminology is that $\mathfrak {so}_3({\mathbb {R}})$ is a twisted form, or a ${\mathbb {C}}/{\mathbb {R}}$-form of $\mathfrak {sl}_2({\mathbb {R}})$. The terminology we have chosen, namely an S/R-form or twisted form of the R-object A, is familiar in number theory and algebraic geometry, and it is also the standard terminology in Grothendieck’s descent theory. The base R is fixed, and the S can vary.

• (2) The referee suggested the authors to add the articles [3,4,5,6,7, 18, 19] to the References. Those articles classify real forms of loop algebras, which refer to the complex Lie superalgebras of the form ${\mathfrak {g}}_0({\mathbb {C}}[t,t^{-1}])$, where ${\mathfrak {g}}_0$ is a complex simple Lie superalgebra of finite dimension. Moreover, they associate with those forms affine Kac–Moody superalgebras. Notable is Serganova’s method (see [19]) based on the classification of real forms of the complexified Lie algebra of vector fields on a circle.

• (3) Pianzola introduced to the authors his [16]; it characterizes affine Kac–Moody Lie algebras as twisted forms (in the étale sense) of the untwisted algebras. This paper likely contains ideas which may be used to replace some of the argument of Sect. 4.3 below by appealing to Galois cohomology.

3 Affine $\delta \text {-}K$-Groups, Their Lie Algebras and Torsors

In this section K denotes a $\delta$-field. We assume that the characteristic $\mathrm{char} K$ of K is zero.

3.1 Affine $\delta \text {-}K$-Groups and Their Lie Algebras

An affine $\delta \text {-}K$-group scheme is by definition a representable functor

$$\begin{aligned} {\mathsf {G}}: \delta \text {-}K\text {-}\mathtt {Algebras} \rightarrow \mathtt {Groups} \quad \mathrm{(see}\ (2.1)); \end{aligned}$$

this will be called an affine $\delta \text {-}K$-group for short. Such a functor ${\mathsf {G}}$ is uniquely represented by a $\delta \text {-}K$-Hopf algebra, say H, and is presented so as ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$ or $\mathrm{Spec}_{\delta \text {-}K}(H)$. We say that ${\mathsf {G}}$ is algebraic, or it is an affine algebraic $\delta \text {-}K$-group, if H is finitely generated as a K-algebra. If one forgets $\delta$, then ${\mathsf {G}}=\mathrm{Spec}(H)$ is an affine K-group, which has the Lie algebra

$$\begin{aligned} \mathrm{Lie}({\mathsf {G}})=\mathrm{Der}_{\epsilon }(H, K). \end{aligned}$$

(3.1)

Recall that this consists of all K-linear maps $D : H \rightarrow K$ that satisfy

$$\begin{aligned} D(ab)=D(a)\epsilon (b)+\epsilon (a)D(b),\quad a, b \in H, \end{aligned}$$

where $\epsilon : H \rightarrow K$ is the counit of H. This is in fact a $\delta \text {-}K$-Lie algebra with respect to the operator defined by

$$\begin{aligned} (\delta D)(a):=\delta (Da)-D(\delta a),\quad D \in \mathrm{Lie}({\mathsf {G}}),\ a \in H . \end{aligned}$$

Note that the canonical pairing $H\otimes _K \mathrm{Lie}({\mathsf {G}}) \rightarrow K$ is a morphism in $\delta \text {-}K$-$\mathtt {Modules}$. We have $\dim _K(\mathrm{Lie}({\mathsf {G}}))<\infty$, if ${\mathsf {G}}$ is algebraic.

Remark 3.1

The notion of being “algebraic” defined above would be rather restricted for those who would like to work intensively in differential algebra. It should be distinguished from the more natural (for those above) notion of being “$\delta$-algebraic”, which will be discussed briefly in Sect. 3.3, being less crucial for our purpose though.

3.2 $\delta \text {-}K$-Torsors and Galois $\delta \text {-}K$-Algebras

An affine $\delta \text {-}K$-scheme is by definition a representable set-valued functor

$$\begin{aligned} {\mathsf {X}}: \delta \text {-}K\text {-}\mathtt {Algebras}\rightarrow \mathtt {Sets}. \end{aligned}$$

It is uniquely represented by a $\delta \text {-}K$-algebra, say A, being presented so as ${\mathsf {X}}= \mathrm{Spec}_{\delta }(A)$; it is said to be algebraic if A is finitely generated as a K-algebra. The category of affine $\delta \text {-}K$-schemes, whose morphisms are natural transformations, has direct products. The direct product ${\mathsf {X}}_1 \times {\mathsf {X}}_2$ of two affine $\delta \text {-}K$-schemes ${\mathsf {X}}_i= \mathrm{Spec}_{\delta }(A_i)$, $i=1,2$, is represented by $A_1 \otimes _K A_2$. The notion of group object of the category is naturally defined, and such an object is precisely an affine $\delta \text {-}K$-group. Given an affine $\delta \text {-}K$-group ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$, the notion of right (or left) ${\mathsf {G}}$-equivariant objects is defined, as well. Such an object is called a right (or left) ${\mathsf {G}}$-equivariant $\delta \text {-}K$-scheme. Giving such a $\delta \text {-}K$-scheme ${\mathsf {X}}= \mathrm{Spec}_{\delta }(A)$ is the same as giving a right (or left) H-comodule $\delta \text {-}K$-algebra; it is an object A in $\delta \text {-}K\text {-}\mathtt {Algebras}$ equipped with a morphism $A \rightarrow A\otimes _K H$ (or $A \rightarrow H\otimes _K A$) in the category which satisfy the co-associativity and the counit property. Obviously, ${\mathsf {G}}$ itself is ${\mathsf {G}}$-equivariant on both sides.

Let R be a $\delta \text {-}K$-algebra. An affine $\delta \text {-}K$-group or (equivariant or ordinary) $\delta \text {-}K$-scheme ${\mathsf {X}}= \mathrm{Spec}_{\delta }(A)$ has the base change ${\mathsf {X}}_R=\mathrm{Spec}_{\delta \text {-}R}(A\otimes _K R)$; it is by definition the functor $T \mapsto {\mathsf {X}}(T)$ defined on $\delta \text {-}R\text {-}\mathtt {Algebras}$, where each $T \in \delta \text {-}R\text {-}\mathtt {Algebras}$ is regarded naturally as a $\delta \text {-}K$-algebra. We can discuss twisted forms of ${\mathsf {X}}$; it is the same as discussing twisted forms of A.

Let ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$ be an affine $\delta \text {-}K$-group. A twisted form of the right ${\mathsf {G}}$-equivariant $\delta \text {-}K$-scheme ${\mathsf {G}}$ is called a right $\delta \text {-}K$-torsor for ${\mathsf {G}}$. To be explicit it is a right ${\mathsf {G}}$-equivariant $\delta \text {-}K$-scheme ${\mathsf {X}}$ such that ${\mathsf {X}}_R \simeq {\mathsf {G}}_R$ as right ${\mathsf {G}}$-equivariant $\delta \text {-}R$-schemes for some non-zero $\delta \text {-}K$-algebra R. Such an ${\mathsf {X}}$ is uniquely represented by a right H-comodule $\delta \text {-}K$-algebra B which is a twisted form of H. Such a twisted form B is characterized as a right H-Galois $\delta \text {-}K$-algebra [14, Section 8.1]; it is by definition a non-zero right H-comodule $\delta \text {-}K$-algebra B such that the $\delta \text {-}K$-algebra map

$$\begin{aligned} \tilde{\rho } : B \otimes _K B \rightarrow B \otimes _K H,\quad \tilde{\rho }(b \otimes c)=b \rho (c) \end{aligned}$$

(3.2)

is an isomorphism. Here and in what follows, $\rho : B \rightarrow B \otimes _K H$ denotes the structure map. Note that $\tilde{\rho }$ is a $\delta \text {-}B$-algebra isomorphism (with the base-on-left notation, see Remark 2.1), and B is split by B itself.

The analogous notions of left $\delta \text {-}K$-torsors for ${\mathsf {G}}$ and of left H-Galois $\delta \text {-}K$-algebras are defined in the obvious manner, and those two are in one-to-one correspondence.

3.3 Affine $\delta$-Algebraic $\delta \text {-}K$-Groups

An $\delta \text {-}K$-algebra A is said to be $\delta$-finitely generated if it is generated as a K-algebra by finitely many elements $a_1,\ldots ,a_n$ together with their iterated differentials $\delta ^ra_1,\ldots ,\delta ^ra_n$, $r>0$. An extension L/K of $\delta$-fields said to be $\delta$-finitely generated, if L is the quotient field of some $\delta \text {-}K$-finitely generated $\delta \text {-}K$-subalgebra of L.

An affine $\delta$-group ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$ is said to be $\delta$-algebraic if the $\delta \text {-}K$-Hopf algebra H is $\delta$-finitely generated as a $\delta \text {-}K$-algebra; see Remark 3.1. Obviously, “algebraic” implies “$\delta$-algebraic”.

Lemma 3.2

Every right (or left) $\delta \text {-}K$-torsor for an affine $\delta \text {-}K$-group ${\mathsf {G}}$ is split by some $\delta \text {-}K$-field. It is split by a $\delta$-finitely generated extension L/K of $\delta$-fields, if ${\mathsf {G}}$ is $\delta$-algebraic.

Proof

Suppose that B is a right H-Galois $\delta \text {-}K$-algebra, as above. Choose arbitrarily a maximal $\delta$-stable ideal $\mathfrak {m}$ of B, and construct $R=B/\mathfrak {m}$, a simple $\delta \text {-}K$-ring. Since $\mathrm{char} K=0$, R is an integral domain by [21, Lemma 1.17]. The quotient field $L=Q(R)$ of R uniquely turns into a $\delta \text {-}K$-field. By applying $L \otimes _B$ to $\tilde{\rho }$, it follows that B is split by L, proving the first assertion. If H is $\delta$-finitely generated, then B and R are so. It follows that the L/K above is $\delta$-finitely generated, proving the second. $\square$

Proposition 3.3

Suppose that A is $\delta \text {-}K$-object of finite K-dimension. Then $\mathsf {Aut}_{\delta }(A)$ is an affine $\delta$-algebraic $\delta \text {-}K$-group, and every twisted form of A is split by some $\delta$-finitely generated extension L/K of $\delta$-fields.

Proof

We have only to prove that $\mathsf {Aut}_{\delta }(A)$ is an affine $\delta$-algebraic $\delta \text {-}K$-group, since the rest then follows from the preceding Lemma.

Choose a K-basis $v_1,\ldots ,v_n$ of A. Let

$$\begin{aligned} F=K[x_{ij}, x_{ij}', x_{ij}'', \ldots , x_{ij}^{(r)}, \ldots ] \end{aligned}$$

denote the free $\delta \text {-}K$-algebra in indeterminates $x_{ij}$, where $1\le i,j\le n$. Let

$$\begin{aligned} G=F_{d}\, (=F[1/d]) \end{aligned}$$

denote the localization by the determinant $d=\det X$ of the $n\times n$ matrix $X=\big (x_{ij}\big )_{i,j}$ which has the indeterminates above as entries. This G has the $\delta$-operator uniquely extending the one $\delta x_{ij}^{(r)}=x_{ij}^{(r+1)}$, $r\ge 0$, on F. We have a G-linear bijection $\phi : A\otimes _K G\rightarrow A \otimes _K G$ determined by

$$\begin{aligned} \phi (v_j \otimes 1)= \sum _{i=1}^n v_i\otimes x_{ij},\quad 1 \le j \le n. \end{aligned}$$

This is alternatively expressed as

$$\begin{aligned} \phi (v_1\otimes 1,\ldots ,v_n\otimes 1)=(v_1,\ldots ,v_n)\otimes X \end{aligned}$$

by matrix presentation; such presentation will be used in (3.3), (3.4) and (3.5), as well.

Let

$$\begin{aligned} H=G/\mathfrak {a}, \end{aligned}$$

where $\mathfrak {a}$ is the smallest $\delta$-stable ideal of G such that the base extension $\phi _H : A\otimes _K H \rightarrow A \otimes _K H$ of $\phi$ along $G \rightarrow G/\mathfrak {a}=H$ is an endomorphism of the $\delta$-H-object $A\otimes _K H$; obviously, it is necessarily an automorphism. This $\mathfrak {a}$ is, in fact, given by the relations which ensure that $\phi _H$ commutes with the structure maps of A (see [22, Section 7.6]), and with the $\delta$-operator. Explicitly, the latter relation for commuting with $\delta$-operator is

$$\begin{aligned} XD=DX +\delta X, \end{aligned}$$

(3.3)

where $D \in M_n(K)$ is the matrix determined by

$$\begin{aligned} \delta (v_1,\ldots ,v_n)= (v_1, \ldots , v_n )D. \end{aligned}$$

(3.4)

We see that H represents the functor $\mathsf {Aut}_{\delta }(A)$ regarded to be set-valued. In fact, for every $R \in \delta \text {-}K\text {-}\mathtt {Algebras}$, we have the natural bijection

$$\begin{aligned} \mathrm{Spec}_{\delta }(H)(R) \rightarrow \mathrm {Aut}_{\delta \text {-}R}(A\otimes _K R), \quad f\mapsto ~\text {the base extension of}~\phi _H~\text {along}~f. \end{aligned}$$

By Yoneda’s Lemma, H uniquely turns into a $\delta \text {-}K$-Hopf algebra with respect to the familiar Hopf-algebra structure

$$\begin{aligned} \varDelta X=X \otimes X,\quad \epsilon X=I,\quad {\mathcal {S}}X=X^{-1}, \end{aligned}$$

(3.5)

where $\varDelta ,\ \epsilon$ and ${\mathcal {S}}$ denote the coproduct, the counit and the antipode, respectively, and it represents the group-valued functor $\mathsf {Aut}_{\delta }(A)$. Since H is obviously $\delta$-finitely generated, the desired result follows. $\square$

For K as above, we choose and fix an extension ${\mathcal {U}}/K$ of $\delta$-fields into which every $\delta$-finitely generated extension L/K of $\delta$-fields can be embedded. There exists such an extension; a universal extension [13, Chapter III, Section 7] over K is an example.

For an affine $\delta$-algebraic $\delta \text {-}K$-group ${\mathsf {G}}$, we define $H^1_{\delta }(K,{\mathsf {G}})$ by

$$\begin{aligned} H^1_{\delta }(K,{\mathsf {G}}):= H^1_{\delta }({\mathcal {U}}/K, {\mathsf {G}}). \end{aligned}$$

(3.6)

The $\delta$-automorphism-group functor $\mathsf {Aut}_{\delta }({\mathsf {G}}) : T \mapsto \mathrm{Aut}_{\delta \text {-}T}({\mathsf {G}}_T)$ of the right ${\mathsf {G}}$-equivariant $\delta \text {-}K$-scheme ${\mathsf {G}}$ is naturally isomorphic to ${\mathsf {G}}$ itself; $\mathrm{Aut}_{\delta \text {-}T}({\mathsf {G}}_T)$ consists of the natural automorphisms of the functor ${\mathsf {G}}_T: \delta \text {-}T\text {-}\mathtt {Algebras}\rightarrow \mathtt {Groups}$. This fact, combined with Lemma 3.2, shows that $H^1_{\delta }(K,{\mathsf {G}})$ classifies all right $\delta \text {-}K$-torsors for ${\mathsf {G}}$.

For a $\delta \text {-}K$-object A of finite K-dimension, we define

$$\begin{aligned} H^1_{\delta }(K,\mathsf {Aut}_{\delta }(A)):= H^1_{\delta }({\mathcal {U}}/K, \mathsf {Aut}_{\delta }(A)). \end{aligned}$$

This classifies all twisted forms of A, as is seen from Proposition 3.3.

3.4 $\delta \text {-}K$-Bi-torsors and Bi-Galois $\delta \text {-}K$-Algebras

Let ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$ be an affine $\delta \text {-}K$-group. Suppose that ${\mathsf {X}}=\mathrm{Spec}_{\delta }(B)$ is a right $\delta \text {-}K$-torsor for ${\mathsf {G}}$, or in other words, $B=(B,\rho )$ is a right H-Galois $\delta \text {-}K$-algebra. Tracing the argument of [17] modified into our differential situation, we see that there exists uniquely (up to isomorphism) a pair $(H', \lambda )$ of a $\delta \text {-}K$-Hopf algebra $H'$ and a left $H'$-comodule $\delta \text {-}K$-algebra structure $\lambda : B \rightarrow H'\otimes _K B$ such that (i) $(B, \lambda )$ is a left $H'$-Galois $\delta \text {-}K$-algebra, and (ii) $\lambda$ and $\rho$ commute in the sense that

$$\begin{aligned} (\lambda \otimes \mathrm{id}_H)\circ \rho = (\mathrm{id}_{H'}\otimes \rho )\circ \lambda . \end{aligned}$$

(3.7)

We say that B is an $(H', H)$-bi-Galois $\delta \text {-}K$-algebra. Accordingly, we have uniquely a pair of an affine $\delta \text {-}K$-group ${\mathsf {G}}'$ and its action on ${\mathsf {X}}$ from the left, such that (i) ${\mathsf {X}}$ is a left $\delta \text {-}K$-torsor for ${\mathsf {G}}'$, and (ii) the actions on ${\mathsf {X}}$ by ${\mathsf {G}}'$ and by ${\mathsf {G}}$ commute with each other. We say that ${\mathsf {X}}$ is a $\delta \text {-}K$-bi-torsor. We write

$$\begin{aligned} H^B,\quad {\mathsf {G}}^{{\mathsf {X}}} \end{aligned}$$

(3.8)

for $H'$, ${\mathsf {G}}'$, respectively. If B (or equivalently, ${\mathsf {X}}$) is trivial, or namely if $B=H$ (or ${\mathsf {X}}={\mathsf {G}}$), then $H^B=H$ and ${\mathsf {G}}^{{\mathsf {X}}}={\mathsf {G}}$. This, applied after base extension to B, shows the following; see the proof of Proposition 3.5 below for detailed argument.

Proposition 3.4

$H^B$ and ${\mathsf {G}}^{{\mathsf {X}}}$ are B/K-forms of H and of ${\mathsf {G}}$, respectively.

With K replaced by a non-zero $\delta$-ring R, the results above remain true if the relevant $\delta$-R-Hopf algebra is flat over R. We remark that $\delta$-R-torsors are then required, in addition to the $\tilde{\rho }$ being isomorphic, to be faithfully flat over R.

3.5 Interpretation of $H^1_{\delta }(K, {\mathsf {G}})\rightarrow H^1_{\delta }(K, \mathsf {Aut}_{\delta }({\mathfrak {g}}))$

Let ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$ be an affine algebraic $\delta \text {-}K$-group, and set ${\mathfrak {g}}:=\mathrm{Lie}({\mathsf {G}})$. Then ${\mathfrak {g}}$ is a $\delta \text {-}K$-Lie algebra of finite K-dimension, whence the $\delta$-automorphism-group functor $\mathsf {Aut}_{\delta }({\mathfrak {g}})$ is an affine $\delta$-algebraic $\delta \text {-}K$-group by Proposition 3.3. We see that the left adjoint action by ${\mathsf {G}}$ on ${\mathfrak {g}}$ gives rise to a morphism of affine $\delta$-algebraic $\delta \text {-}K$-groups

$$\begin{aligned} \mathsf {Ad} : {\mathsf {G}}\rightarrow \mathsf {Aut}_{\delta }({\mathfrak {g}}), \end{aligned}$$

which induces naturally a map between the cohomology sets

$$\begin{aligned} \mathsf {Ad}_* : H^1_{\delta }(K, {\mathsf {G}}) \rightarrow H^1_{\delta }(K, \mathsf {Aut}_{\delta }({\mathfrak {g}})). \end{aligned}$$

(3.9)

Given a right $\delta \text {-}K$-torsor ${\mathsf {X}}$ for ${\mathsf {G}}$, we define

$$\begin{aligned} {\mathfrak {g}}^{{\mathsf {X}}}:=\mathrm{Lie}({\mathsf {G}}^{{\mathsf {X}}}). \end{aligned}$$

(3.10)

This is a twisted form of ${\mathfrak {g}}=\mathrm{Lie}({\mathsf {G}})$, since ${\mathsf {G}}^{{\mathsf {X}}}$ is a twisted form of ${\mathsf {G}}$; see Proposition 3.4.

Proposition 3.5

$\mathsf {Ad}_*$ is interpreted in terms of twisted forms so as

$$\begin{aligned} {[}\text {a right}~\delta \text {-}K\text {-torsor}\ {\mathsf {X}}\ \text {for}\ {\mathsf {G}}] \ \mapsto \ [{\mathfrak {g}}^{{\mathsf {X}}}], \end{aligned}$$

(3.11)

where $[\ \, ]$ indicates isomorphism classes.

Proof

In this proof we write $\otimes$ for $\otimes _K$, and use the base-on-left notation for base extensions; see Remark 2.1.

Suppose that ${\mathsf {X}}=\mathrm{Spec}_{\delta }(B)$ is a right $\delta \text {-}K$-torsor for ${\mathsf {G}}$, or in other words, $B=(B, \rho )$ is a right H-Galois $\delta \text {-}K$-algebra.

Let $\gamma \in {\mathsf {G}}(B \otimes B)$. This gives the automorphism

$$\begin{aligned} \ell _{\gamma } : (B\otimes B)\otimes H \overset{\simeq }{\longrightarrow } (B\otimes B) \otimes H \end{aligned}$$

of the right $((B\otimes B) \otimes H)$-Galois $\delta \text {-}(B\otimes B)$-algebra $(B\otimes B)\otimes H$ defined by

$$\begin{aligned} \ell _{\gamma }((b\otimes c) \otimes h)=(b\otimes c)\gamma (h_{(1)})\otimes h_{(2)},\quad b, c \in B,\ h \in H. \end{aligned}$$

Here and in what follows, we let

$$\begin{aligned} \varDelta (h)=h_{(1)}\otimes h_{(2)},\quad (\varDelta \otimes \mathrm{id}) \circ \varDelta (h)=h_{(1)}\otimes h_{(2)}\otimes h_{(3)} \end{aligned}$$

denote the coproduct on H. The right co-adjoint action

$$\begin{aligned} \mathsf {Coad}(\gamma ) : (B\otimes B)\otimes H \overset{\simeq }{\longrightarrow } (B\otimes B) \otimes H \end{aligned}$$

by $\gamma$ is defined by

$$\begin{aligned} \mathsf {Coad}(\gamma )((b\otimes c) \otimes h)=(b\otimes c) \gamma (h_{(1)})\gamma ^{-1}(h_{(3)})\otimes h_{(2)}. \end{aligned}$$

This is an automorphism of the $\delta$-$(B\otimes B)$-Hopf algebra $(B\otimes B)\otimes H$. Note that $\ell _{\gamma }$ turns into an isomorphism of left $((B\otimes B) \otimes H)$-Galois $\delta \text {-}(B\otimes B)$-algebras, if one twists through $\mathsf {Coad}(\gamma )$ the obvious co-action by $(B\otimes B) \otimes H$ on the domain. Explicitly, this means that

$$\begin{aligned} (\mathsf {Coad}(\gamma )\, \otimes _{B\otimes B} \, \ell _{\gamma })\circ \varDelta _{(B\otimes B)\otimes H} = \varDelta _{(B\otimes B)\otimes H} \circ \ell _{\gamma } \ \, \text {on} \ \, (B\otimes B)\otimes H, \end{aligned}$$

(3.12)

where $\varDelta _{(B\otimes B)\otimes H}$ denotes the coproduct on $(B\otimes B)\otimes H$.

Suppose that the $\gamma$ above is a cocycle for computing $H^1_{\delta }(B/K,{\mathsf {G}})$ which gives the B/K-form B through $\tilde{\rho }$. This means that the commutative diagram

of right $((B\otimes B) \otimes H)$-Galois $\delta \text {-}(B\otimes B)$-algebras, where $d_i$, $i=1,2$, denote the base extensions along

$$\begin{aligned} B\rightarrow B\otimes B,\quad b\mapsto 1\otimes b,\ b\otimes 1. \end{aligned}$$

Recall that B is an $(H^B,H)$-bi-Galois $\delta$-K-algebra. By [17, Theorem 3.5], the Hopf algebra $H^B$ consists of the elements $\sum _i b_i \otimes c_i$ in $B\otimes B$ such that

$$\begin{aligned} \sum _i (b_i)_{(0)}\otimes (c_i)_{(0)}\otimes (b_i)_{(1)}(c_i)_{(1)} =\sum _ib_i\otimes c_i \otimes 1\ \, \text {in} \ \, (B \otimes B) \otimes H, \end{aligned}$$

(3.13)

where $\rho (b)=b_{(0)}\otimes b_{(1)}$. Moreover,

$$\begin{aligned} \mu : B \otimes H^B \rightarrow B\otimes B,\quad \mu (b \otimes z)= bz \end{aligned}$$

(3.14)

is an isomorphism of left $(B \otimes H^B)$-Galois $\delta$-B-algebras. Define

$$\begin{aligned} \nu :=\tilde{\rho }\circ \mu : B\otimes H^B \rightarrow B\otimes H. \end{aligned}$$

Recall from Sect. 3.4 uniqueness of the pair $(H', \lambda )$, and apply it first over B, and next over $B \otimes B$. Then one sees the following. First, there uniquely exists an isomorphism $\theta : B\otimes H^B \rightarrow B\otimes H$ of $\delta$-B-Hopf algebras such that

$$\begin{aligned} (\theta \otimes \nu )\circ \varDelta _{B\otimes H^B}= \varDelta _{B\otimes H}\circ \nu , \end{aligned}$$

where $\varDelta _{B\otimes H^B}$ and $\varDelta _{B\otimes H}$ denote the coproducts on the $\delta$-B-Hopf algebras. In fact, this $\theta$ is the unique isomorphism between the two $\delta$-B-Hopf-algebras that is compatible with their co-actions on $B \otimes B$. (Notice that this $\theta$ ensures Proposition 3.4.) Next, the last commutative diagram, with $((B\otimes B) \otimes H)^{(-)}$ applied (see (3.8)), induces the commutative diagram

of $\delta \text {-}(B\otimes B)$-Hopf algebras; notice from (3.12) that $\ell _{\gamma }$ induces $\mathsf {Coad}(\gamma )$.

Notice from (3.1) that ${\mathfrak {g}}^{{\mathsf {X}}}=\mathrm{Der}_{\epsilon }(H^B,K)$. Then one sees that $\theta$ induces an isomorphism

$$\begin{aligned} \theta ^* : B \otimes {\mathfrak {g}}\overset{\simeq }{\longrightarrow } B \otimes {\mathfrak {g}}^{{\mathsf {X}}} \end{aligned}$$

of $\delta$-B-Lie algebras. Moreover, the last commutative diagram induces by duality the commutative diagram

of $\delta \text {-}(B\otimes B)$-Lie algebras, where the horizontal arrow indicates the left adjoint action by $\gamma ^{-1}$. We may reverse the direction of the arrow, changing the label into the left adjoint action $\mathsf {Ad}(\gamma )$ by $\gamma$. The result shows that $\mathsf {Ad}(\gamma )$, regarded as a cocycle for computing $H^1_{\delta }(B/K, \mathsf {Aut}_{\delta }({\mathfrak {g}}))$, gives the twisted form ${\mathfrak {g}}^{{\mathsf {X}}}$ of ${\mathfrak {g}}$ which is split by B, indeed.

Recall from (3.6) the definition $H^1(K,{\mathsf {G}}):=H^1({\mathcal {U}}/K,{\mathsf {G}})$. Let $\psi$ is an element of $H^1(K,{\mathsf {G}})$. Then this arises from a cocycle $\gamma$ such as above, which gives a $\delta$-K-torsor ${\mathsf {X}}=\mathrm{Spec}_{\delta }(B)$ for ${\mathsf {G}}$, through a $\delta$-K-algebra map $j: K \rightarrow {\mathcal {U}}$. Thus, $\psi$ is represented by the cocycle given as the composite

$$\begin{aligned} H \overset{\gamma }{\longrightarrow } B\otimes B\overset{j\otimes j}{\longrightarrow } {\mathcal {U}}\otimes {\mathcal {U}}. \end{aligned}$$

This cocycle is seen to give the ${\mathcal {U}}/K$-form B of the right H-comodule $\delta$-K-algebra H. The argument in the preceding paragraphs shows that $\mathsf {Ad}_*(\psi )$ is represented by the base extension of the automorphism $\mathsf {Ad}(\gamma )$ along the $\delta$-K-algebra map $j\otimes j$. This base extension is seen to be a cocycle which gives the ${\mathcal {U}}/K$-form ${\mathfrak {g}}^{{\mathsf {X}}}$ of ${\mathfrak {g}}$. This completes the proof. $\square$

Let ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$ be an affine algebraic $\delta \text {-}K$-group with ${\mathfrak {g}}=\mathrm{Lie}({\mathsf {G}})$, as above. Recall from (3.1) that ${\mathfrak {g}}=\mathrm{Der}_{\epsilon }(H, K)$.

Proposition 3.6

Regard H merely as the trivial right H-Galois K-algebra, forgetting $\delta$ on it.

• (1) Given an element $D \in {\mathfrak {g}}$, define

  $$\begin{aligned} \delta _D : H \rightarrow H,\quad \delta _D(h)=\delta h+ D(h_{(1)})h_{(2)}, \end{aligned}$$

  (3.15)

  where $\delta$ denotes the original operator on H. Then this is a $\delta$-operator with which H is made into a right H-Galois $\delta \text {-}K$-algebra. Conversely, such a $\delta$-operator uniquely arises in this way.

• (2) Given an element $D\in {\mathfrak {g}}$, let ${\mathsf {X}}_D$ denote the right $\delta \text {-}K$-torsor for ${\mathsf {G}}$ which is represented by the right H-Galois $\delta \text {-}K$-algebra $(H, \delta _D)$ obtained above. Then the twisted form ${\mathfrak {g}}^{{\mathsf {X}}_D}$ of ${\mathfrak {g}}$ is the K-Lie algebra ${\mathfrak {g}}$ equipped with the new $\delta$-operator

  $$\begin{aligned} \delta + {\mathsf {ad}}(D) : {\mathfrak {g}}\rightarrow {\mathfrak {g}},\quad z \mapsto \delta z +[D, z], \end{aligned}$$

  where $\delta$ denotes the original operator on ${\mathfrak {g}}$. Thus ${\mathfrak {g}}^{{\mathsf {X}}_D}$ is quasi-isomorphic to the original ${\mathfrak {g}}$; see Definition 1.4.

Proof

• (1) Suppose that $\delta _1$ is a desired operator, or namely, $(H,\delta _1)$ is a right H-Galois $\delta \text {-}K$-algebra. Then one sees that $\delta _1-\delta : H \rightarrow H$ is a K-linear derivation and is at the same time a right H-comodule map. It follows that $\delta _1$ is necessarily of the form $\delta _D$ with $D\in {\mathfrak {g}}$ uniquely determined. Such $\delta _D$ is seen to be a desired operator for any D, indeed.

• (2) Let $H'=H^{(H, \delta _D)}$. Then ${\mathfrak {g}}^{{\mathsf {X}}_D}=\mathrm{Der}_{\epsilon }(H', K)$. Using the uniqueness of the $\delta \text {-}K$-Hopf algebra $H'$ in general, which was discussed in Sect. 3.4, we see that the present $H'$ is the K-Hopf algebra H equipped with the $\delta$-operator

  $$\begin{aligned} H \rightarrow H,\quad h\mapsto \delta h+ D(h_{(1)})h_{(2)}- h_{(1)}D(h_{(2)}). \end{aligned}$$

  This implies the desired result.

$\square$

A simple consequence of the proposition above is the following.

Corollary 3.7

Let ${\mathfrak {g}}$ be a $\delta \text {-}K$-Lie algebra of finite K-dimension. Once the Lie algebra $\mathrm{Lie}({\mathsf {G}})$ of some affine algebraic $\delta \text {-}K$-group ${\mathsf {G}}$ is shown to be a twisted form of ${\mathfrak {g}}$, then every $\delta \text {-}K$-Lie algebra quasi-isomorphic to $\mathrm{Lie}({\mathsf {G}})$ is a twisted form of ${\mathfrak {g}}$, as well.

3.6 Differential $\delta \text {-}K$-Objects Arising from C-Linear Objects

Let K be a $\delta$-field of characteristic zero. Let

$$\begin{aligned} C=C_K\, (=\{ \, x\in K \mid \delta x=0\, \}) \end{aligned}$$

denote the field of constants in K, which is necessarily of characteristic zero. In this subsection we let $\otimes$ denote the tensor product $\otimes _C$ over C.

Let $A_0$ be a C-linear object. We can and do regard the base extension $A_0\otimes K$ as a $\delta \text {-}K$-object with respect to the operator $\delta _0$ defined by

$$\begin{aligned} \delta _0 : A_0\otimes K \rightarrow A_0\otimes K,\quad a \otimes x \mapsto a \otimes \delta x. \end{aligned}$$

For every $\delta \text {-}K$-algebra R, $A_0 \otimes R$ is similarly a $\delta$-R-object, and is a base extension of the $\delta \text {-}K$-object $A_0\otimes K$ above.

Proposition 3.8

If the automorphism-group functor $\mathsf {Aut}(A_0)$ of $A_0$ happens to be an affine C-group, represented by a C-Hopf algebra $H_0$, then the $\delta$-automorphism-group functor $\mathsf {Aut}_{\delta }(A_0\otimes K)$ of the $\delta \text {-}K$-object $(A_0\otimes K, \delta _0)$ is an affine $\delta \text {-}K$-group, represented by the $\delta \text {-}K$-Hopf algebra $H_0\otimes K$.

Proof

Let $R\in \delta \text {-}K\text {-}\mathtt {Algebras}$. One sees that every automorphism of the $\delta$-R-object $A_0 \otimes R$ restricts to an automorphism of $A_0 \otimes C_R$ over the C-algebra $C_R$ of constants in R, and so it is uniquely presented as the base extension of the restriction. This shows $\mathrm {Aut}_{\delta \text {-}R}(A_0\otimes R)= \mathrm {Aut}_{C_R}(A_0\otimes C_R)$; this last is naturally isomorphic to $\mathrm{Spec}_C(H_0)(C_R)=\mathrm{Spec}_{\delta \text {-}K}(H_0\otimes K)(R)$. This proves the proposition. $\square$

We remark that the proposition follows from the proof of Proposition 3.3 if $(\dim _K(A_0\otimes K)=)\dim _CA_0<\infty$. For the relation (3.3) turns into $\delta X=O$ since $D=O$.

Corollary 3.9

If $\dim _C A_0<\infty$, then $\mathsf {Aut}_{\delta }(A_0\otimes K)$ is an affine algebraic $\delta \text {-}K$-group.

Proof

This follows from the proposition above, since $\mathsf {Aut}(A_0)$ is an affine algebraic C-group under the assumption; see [22, Section 7.6]. $\square$

The following result would be worth presenting, though it is not essentially used in this paper.

Proposition 3.10

Assume $\dim _C A_0<\infty$, and that the field C is algebraically closed. Then every twisted form of the $\delta \text {-}K$-object $(A_0 \otimes K, \delta _0)$ is split by some (finitely generated) Picard-Vessiot extension L over K.

Proof

By the preceding results the first assumption implies $\mathsf {Aut}(A_0)=\mathrm{Spec}_C(H_0)$ and $\mathsf {Aut}_{\delta }(A_0\otimes K)=\mathrm{Spec}_{\delta \text {-}K}(H_0\otimes K)$, where $H_0$ is a finitely generated C-Hopf algebra.

Let B be a right $(H_0\otimes K)$-Galois $\delta \text {-}K$-algebra, and regard it as a twisted form of the right $(H_0\otimes K)$-comodule $\delta \text {-}K$-algebra $H_0\otimes K$. We should prove that this twisted form B is split by some L/K as above. It suffices to prove that there exists a $\delta \text {-}K$-algebra map from B to such an L, since B is split by B, itself.

We have the $\delta$-B-algebra isomorphism

$$\begin{aligned} \tilde{\rho } : B\otimes _K B \overset{\simeq }{\longrightarrow } B\otimes _K(H_0\otimes K)=B\otimes H_0 \end{aligned}$$

as in (3.2). Choose a simple quotient $\delta \text {-}K$-algebra R of B, as in the proof of Lemma 3.2. Then R is an integral domain by [21, Lemma 1.17], as before. This is finitely generated as a K-algebra since B is such. The quotient field $L=Q(R)$ of R uniquely turns into a $\delta$-field, which is necessarily a finitely generated extension over K. The second assumption above, combined with [2, Lemma 4.2], implies that the field $C_L$ of constants in L equals C. This L/K will be proved to be a desired Picard–Vessiot extension by [2, Definition 1.8 and Theorem 3.11], if one sees that the canonical $\delta$-R-algebra map $R \otimes C_{R\otimes _K R} \rightarrow R\otimes _K R$ which arises from the embedding $C_{R\otimes _K R} \hookrightarrow R\otimes _K R$ of the constants into $R\otimes _K R$ is surjective. (By [2, Proposition 6.7] this canonical map is injective, though this fact is not needed here.) Indeed, the desired surjectivity is seen from the commutative diagram

Here the vertical arrow on the left-hand side naturally arises from the composite of $\tilde{\rho }^{-1}|_{H_0}:H_0\rightarrow B \otimes _K B$ with the natural surjection $B\otimes _KB\rightarrow R\otimes _KR$, which clearly takes values in $C_{R\otimes _K R}$. $\square$

4 Proof of the Theorem and Computations

Throughout in this section we let $K:={\mathbb {C}}(t)$, and write $\otimes$ for $\otimes _K$.

Suppose that we are in the situation of Sect. 1. Let ${\mathfrak {g}}_0$ be a complex simple Lie algebra, and let ${\mathfrak {g}}={\mathfrak {g}}_0(K)$ denote the $\delta \text {-}K$-Lie algebra as in (1.1).

4.1 Two Key Facts

One key fact for us is the following description of the automorphism-group scheme $\mathsf {Aut}({\mathfrak {g}}_0)$ of ${\mathfrak {g}}_0$. Recall that the finite group $\varGamma =\mathsf {Out}({\mathfrak {g}}_0)$ of outer-automorphisms of ${\mathfrak {g}}_0$ is explicitly given by (1.2); this $\varGamma$ will be identified with the associated, finite constant group scheme. Let ${\mathsf {G}}_0^{\circ }$ be the adjoint simple ${\mathbb {C}}$-group associated with ${\mathfrak {g}}_0$. A natural action by $\varGamma$ on ${\mathsf {G}}_0^{\circ }$ constitutes an affine algebraic ${\mathbb {C}}$-group

$$\begin{aligned} {\mathsf {G}}_0 = {\mathsf {G}}_0^{\circ }\rtimes \varGamma \end{aligned}$$

of semi-direct product, so that

$$\begin{aligned} \mathrm{Lie}({\mathsf {G}}_0)=\mathrm{Lie}({\mathsf {G}}_0^{\circ })={\mathfrak {g}}_0, \end{aligned}$$

and the adjoint action by ${\mathsf {G}}_0$ on ${\mathfrak {g}}_0$ gives an isomorphism

$$\begin{aligned} \mathsf {Ad} : {\mathsf {G}}_0 \overset{\simeq }{\longrightarrow } \mathsf {Aut}({\mathfrak {g}}_0) \end{aligned}$$

(4.1)

of affine algebraic ${\mathbb {C}}$-groups. By restriction this $\mathsf {Ad}$ induces the identity $\varGamma = \mathsf {Out}({\mathfrak {g}}_0)$. Note that ${\mathsf {G}}_0^{\circ }$ is the connected component of ${\mathsf {G}}_0$ containing the identity element. See [15, Chapter 4, Section 4, $1^{\circ }$].

Suppose ${\mathsf {G}}_0=\mathrm{Spec}_{{\mathbb {C}}}(H_0)$, and define

$$\begin{aligned} {\mathsf {G}}=\mathrm{Spec}_{\delta \text {-}K}(H_0\otimes _{{\mathbb {C}}} K). \end{aligned}$$

Then one sees ${\mathfrak {g}}=\mathrm{Lie}({\mathsf {G}})$. Moreover, it follows from (4.1) and Proposition 3.8 that the adjoint action by ${\mathsf {G}}$ on ${\mathfrak {g}}$ gives an isomorphism

$$\begin{aligned} \mathsf {Ad} : {\mathsf {G}}\overset{\simeq }{\longrightarrow } \mathsf {Aut}_{\delta }({\mathfrak {g}}) \end{aligned}$$

of affine algebraic $\delta$-K-groups. This together with Proposition 3.5 prove the following.

Proposition 4.1

Every form of the $\delta \text {-}K$-Lie algebra ${\mathfrak {g}}$ uniquely arises, as described by (3.11), from a right $\delta \text {-}K$-torsor for ${\mathsf {G}}$.

Another key fact is the cohomology vanishing of the (non-differential) Amitsur 1st cohomology due to Steinberg (see Serre [20, III, 2.3, Theorem 1$'$]),

$$\begin{aligned} H^1(K, \mathsf {F})=0, \end{aligned}$$

(4.2)

where $\mathsf {F}$ is a connected affine algebraic K-group. This is proved more generally when K is replaced by a perfect field, say $K'$, of dimension $\le 1$ [20, Definition on Page 78], and in addition, $\mathsf {F}$ is assumed to be smooth if $\mathrm{char} K'>0$; note that every affine algebraic K-group is necessarily smooth since $\mathrm{char} K=0$. One sees that $K\, (={\mathbb {C}}(t))$ is a $\mathrm {(C_1)}$-field by Tsen’s Theorem, whence K is of dimension $\le 1$ by [20, Corollary on Page 80].

4.2 Proof of Theorem 1.5, Part 1: Case $\varGamma =\{1\}$

In this case, ${\mathsf {G}}$, regarded as an affine K-group, is connected. By (4.2) applied to this ${\mathsf {G}}$, it follows that every right $\delta \text {-}K$-torsor for ${\mathsf {G}}$, regarded as a right K-torsor, is trivial. Propositions 3.6 and  4.1 conclude the proof.

4.3 Galois Descent

To proceed to Parts 2 and 3, suppose that we are now in Case $\varGamma \ne \{1\}$.

Note that $\varGamma$, regarded as a finite constant ${\mathbb {C}}$-group scheme, is represented by the dual $({\mathbb {C}}\varGamma )^*$ of the group algebra ${\mathbb {C}}\varGamma$; this $({\mathbb {C}}\varGamma )^*$ is the separable part $\pi _0(H_0)$ of $H_0$ [22, Page 49], that is, the largest separable subalgebra (in fact, Hopf subalgebra) of the ${\mathbb {C}}$-Hopf algebra $H_0$. Suppose ${\mathsf {G}}_0^{\circ }=\mathrm{Spec}_{{\mathbb {C}}}(J_0)$, and define

$$\begin{aligned} H:=H_0\otimes _{{\mathbb {C}}} K,\quad J:=J_0\otimes _{{\mathbb {C}}}K,\quad Z :=({\mathbb {C}}\varGamma )^* \otimes _{{\mathbb {C}}} K\, (=(K\varGamma )^*), \end{aligned}$$

which are naturally $\delta \text {-}K$-Hopf algebras, such that ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$, in particular. One sees that $Z\subset H$ is a $\delta \text {-}K$-Hopf subalgebra, and

$$\begin{aligned} J=H/(Z^+), \end{aligned}$$

(4.3)

where $(Z^+)$ is the ideal (in fact, $\delta$-stable Hopf ideal) generated by the augmentation ideal $Z^+=\mathrm{Ker}(\epsilon :Z\rightarrow K)$ of Z, that is, the kernel of the counit. Since $\varGamma$ acts innerly on ${\mathsf {G}}_0^{\circ }\, (\subset {\mathsf {G}}_0)$ from the right, it acts from the left on J as $\delta \text {-}K$-Hopf-algebra automorphisms. The action gives rise by adjoint to the co-action $J \rightarrow J \otimes Z$ by $Z=(K\varGamma )^*$, so that the associated smash coproduct $Z\blacktriangleright \!\!<J$ (see [14, Definition 10.6.1]) coincides with H. Here one should recall ${\mathsf {G}}_0=\varGamma < imes {\mathsf {G}}_0^{\circ }\, (={\mathsf {G}}_0^{\circ }\rtimes \varGamma )$.

Choose arbitrarily a right $\delta \text {-}K$-torsor ${\mathsf {X}}=\mathrm{Spec}_{\delta }(B)$ for ${\mathsf {G}}=\mathrm{Spec}_{\delta }(H)$. In view of Proposition 4.1 we wish to describe the $\delta \text {-}K$-Lie algebra ${\mathfrak {g}}^{{\mathsf {X}}}\, (=\mathrm{Lie}({\mathsf {G}}^{{\mathsf {X}}}))$. Let $H':=H^B$, or in other words, suppose ${\mathsf {G}}^{{\mathsf {X}}}=\mathrm{Spec}_{\delta }(H')$, so that B is an $(H',H)$-bi-Galois $\delta \text {-}K$-algebra. We are going to prove the following.

Proposition 4.2

There is a finite-dimensional $\delta \text {-}K$-Hopf subalgebra $Z'$ of $H'$ such that

• (i) $Z'$ is separable as a K-algebra;

• (ii) the associated quotient $\delta \text {-}K$-Hopf algebra

  $$\begin{aligned} J':=H'/(Z'^+)\quad (\text {cf}.~(4.3)) \end{aligned}$$

  (4.4)

  has the trivial separable part, $\pi _0(J')=K$, or in other words, it includes no non-trivial separable K-subalgebra.

This implies that the affine K-group $\mathrm{Spec}(H')$ includes $\mathrm{Spec}(J')$ as the connected component containing the identity element, and thereby concludes

$$\begin{aligned} {\mathfrak {g}}^{{\mathsf {X}}}=~\text {the Lie algebra of the affine}~\delta \text {-}K\text {-group}~\mathrm{Spec}_{\delta }(J') \end{aligned}$$

(4.5)

as $\delta \text {-}K$-Lie algebras. Therefore, we aim first to prove the proposition above, and then to describe the ${\mathfrak {g}}^{{\mathsf {X}}}$ above.

Let $\rho : B \rightarrow B \otimes H$ denote the structure map on B, and define

$$\begin{aligned} R := \rho ^{-1}(B \otimes Z). \end{aligned}$$

Then this R is a right $\delta \text {-}K$-Galois algebra for Z, or in other words, $\mathrm{Spec}_{\delta }(R)$ is a right $\delta \text {-}K$-torsor for the finite constant $\delta \text {-}K$-group scheme $\varGamma _K$ given by $\varGamma$; it arises from the the right $\delta \text {-}K$-torsor ${\mathsf {X}}=\mathrm{Spec}_{\delta }(B)$ for ${\mathsf {G}}$ through the restriction map $H^1_{\delta }(K, {\mathsf {G}})\rightarrow H^1_{\delta }(K,\varGamma _K)$ which is defined in the differential situation, as well, just as in the ordinary situation. Note that R is naturally a $\delta \text {-}K$-algebra of finite K-dimension, and is a Galois K-algebra with Galois group $\varGamma$ in the classical sense that the K-algebra map $R \rtimes \varGamma \rightarrow \mathrm{End}_{K}(R)$ which arises from the natural module-action on R by the semi-direct product $R\rtimes \varGamma$ is an isomorphism. Note that $R\rtimes \varGamma$ is naturally a non-commutative $\delta$-K-algebra with $\varGamma \, (=\{1\}\times \varGamma )$ included in constants. A $\delta$-$(R\rtimes \varGamma )$-module is thus an R-module M equipped with an additive operator $\delta$ and a $\varGamma$-action of K-linear automorphisms, such that

$$\begin{aligned} \delta (\gamma m)=\gamma (\delta m),\quad \delta (am)=(\delta a)m+a(\delta m),\quad \gamma (am)=(\gamma a)(\gamma m), \end{aligned}$$

where $\gamma \in \varGamma$, $a \in R$ and $m \in M$. We call this a $(\delta ,\varGamma )$-R-module, to treat $\delta$ and $\varGamma$ on an equality, and let $(\delta ,\varGamma )\text {-}R\text {-}\mathtt {Modules}$ denote the category of those modules. The classical Galois Descent Theorem (see [12, Section 18]) tells us that the functor $M \mapsto M^{\varGamma }$, $\varGamma$-invariants in M, gives the category equivalence

$$\begin{aligned} (\delta ,\varGamma )\text {-}R\text {-}\mathtt {Modules}\overset{\approx }{\longrightarrow } \delta \text {-}K\text {-}\mathtt {Modules}, \end{aligned}$$

whose quasi-inverse is given by the base-extension functor $\otimes _KR$. In fact, this is a symmetric tensor equivalence, so that there is induced the category equivalence between their (commutative-)algebra objects, or between any other kind of linear objects. The category on the left-hand side has the tensor product $\otimes _R$, the unit object R and the obvious symmetry, while the category on the right-hand side has the tensor product $\otimes _K$, the unit object K and the obvious symmetry. A commutative algebra in $(\delta ,\varGamma )\text {-}R\text {-}\mathtt {Modules}$ will be called a $(\delta ,\varGamma )\text {-}R$-algebra; it descends to a $\delta \text {-}K$-algebra by the category equivalence above. Similarly, a $(\delta ,\varGamma )\text {-}R$-Hopf or Lie algebra is defined, and it descends to a $\delta \text {-}K$-object.

We have the commutative diagram

of $\delta \text {-}K$-algebras, where the upper horizontal arrow indicates the isomorphism $\tilde{\rho }$ (see (3.2)) associated with the structure map $\rho : B \rightarrow B \otimes H$ on B, and the lower one is the analogous isomorphism for the right Z-Galois $\delta \text {-}K$-algebra R. In addition, $\text {mult} : R\otimes R \rightarrow R$ indicates the multiplication $x \otimes y\mapsto xy$. By the base extensions along the two diagonal arrows $\text {mult}: R \otimes R \rightarrow R$ and $\mathrm{id}_R \otimes \, \epsilon : R \otimes H \rightarrow R$, the $\tilde{\rho }$ induces the isomorphism

$$\begin{aligned} B\otimes _R B \overset{\simeq }{\longrightarrow } B\otimes J =B\otimes _R(J\otimes R). \end{aligned}$$

(4.6)

Recall that $\varGamma$ acts on J as $\delta \text {-}K$-Hopf algebra automorphisms. Then one sees that $J\otimes R$ is a $(\delta ,\varGamma )\text {-}R$-Hopf algebra, and hence descends to a $\delta \text {-}K$-Hopf algebra

$$\begin{aligned} {\mathcal {J}}:=(J\otimes R)^{\varGamma }. \end{aligned}$$

The composite

$$\begin{aligned} B \rightarrow B\otimes H\rightarrow B\otimes J=B\otimes _R(J\otimes R) \end{aligned}$$

of the structure map on B with the natural surjection onto $B \otimes _R(J\otimes R)$ is a $(\delta ,\varGamma )$-R-algebra map, and hence descends to a $\delta \text {-}K$-algebra map $B^{\varGamma }\rightarrow B^{\varGamma }\otimes {\mathcal {J}}$, which we call $\varrho$.

Lemma 4.3

$B^{\varGamma }$ is a right ${\mathcal {J}}$-Galois $\delta \text {-}K$-algebra by the $\varrho$ above.

Proof

One sees that $\varrho$ satisfies the co-associativity and the counit property since the last composite does. One sees that (4.6) is an isomorphism of $(\delta ,\varGamma )$-R-algebras, and descends to $\tilde{\varrho }: B^{\varGamma }\otimes B^{\varGamma }\rightarrow B^{\varGamma }\otimes {\mathcal {J}}$, which is, therefore, an isomorphism. $\square$

Recall $H'=H^B$. Define $Z':=Z^R$, so that R is a $(Z',Z)$-bi-Galois $\delta \text {-}K$-algebra.

Lemma 4.4

$Z'$ is a finite-dimensional $\delta \text {-}K$-Hopf subalgebra of $H'$ which has the property (i) of Proposition 4.2, that is, $Z'$ is separable as a K-algebra.

Proof

By (3.13) we have $Z'\subset H'$. This inclusion is compatible with the Hopf-algebra structure maps, as is seen from the construction of $H^B$ given in [17, Theorem 3.5]. To verify this here only for the coproduct, recall from (3.14) that $H'\subset B\otimes B$ gives rise to a left B-linear isomorphism $B \otimes H'=B\otimes B$. Therefore, we have

$$\begin{aligned} H'\otimes H'\subset B\otimes H'\otimes H'=B\otimes B\otimes H'=B\otimes B\otimes B. \end{aligned}$$

The construction cited above tells us that the coproduct on $H'$ is the restriction of

$$\begin{aligned} B\otimes B\rightarrow B\otimes B\otimes B,\quad b\otimes c \mapsto b\otimes 1\otimes c. \end{aligned}$$

This, combined with the analogous restriction of $R\otimes R\rightarrow R\otimes R\otimes R$ to the coproduct $Z'\rightarrow Z\otimes Z'$, shows the desired compatibility, as is verified by a commutative diagram in cube.

The K-algebras Z, R and $Z'$ turn to be mutually isomorphic after base extension to some algebraically closed field. It follows that $Z'$ is finite-dimensional separable, since Z is. $\square$

Define $J':=H'/(Z'^+)$, as in (4.4). The proof of Proposition 4.2 completes by proving the next lemma. The following proposition describes the $\delta \text {-}K$-Lie algebra ${\mathfrak {g}}^{{\mathsf {X}}}$; see (3.10).

Lemma 4.5

$B^{\varGamma }$ is a $(J',{\mathcal {J}})$-bi-Galois $\delta \text {-}K$-algebra, and $J'$ has the property (ii) of Proposition 4.2, that is, $\pi _0(J')=K$.

Proof

The same argument as proving Lemma 4.3 shows that $B^{\varGamma }$ is a left $J'$-Galois $\delta \text {-}K$-algebra. Here one should notice that $\varGamma$ acts (or Z co-acts) trivially on $H'$, and hence on $J'$. Indeed, $B^{\varGamma }$ is bi-Galois, since the structure maps

$$\begin{aligned} H'\otimes B \leftarrow B \rightarrow B\otimes H \end{aligned}$$

on B commute with each other (see (3.7)), and hence those on $B^{\varGamma }$ do.

Note that $\pi _0(J)\, (=\pi _0(J_0)\otimes _{{\mathbb {C}}} K)$ equals K. This is equivalent to saying that the K-algebra J contains no non-trivial idempotent even after base extension to some (or any) algebraically closed field. It follows that ${\mathcal {J}}$ and $J'$ have the same property, since J and ${\mathcal {J}}$, as well as ${\mathcal {J}}$ and $J'$, are mutually isomorphic after base extension such as above. $\square$

Since ${\mathfrak {g}}_0(R)={\mathfrak {g}}_0\otimes _{{\mathbb {C}}} R$, on which $\varGamma$ acts diagonally, is a $(\delta ,\varGamma )\text {-}R$-Lie algebra, it descends to ${\mathfrak {g}}_0(R)^{\varGamma }$, a $\delta \text {-}K$-Lie algebra. Our aim of this subsection is achieved by the following.

Proposition 4.6

The $\delta \text {-}K$-Lie algebra ${\mathfrak {g}}^{{\mathsf {X}}}$ is quasi-isomorphic to ${\mathfrak {g}}_0(R)^{\varGamma }$.

Proof

Recall (4.5) and the result of Proposition 4.2 that $B^{\varGamma }$ is a $(J',{\mathcal {J}})$-bi-Galois $\delta \text {-}K$-algebra. By Steinberg’s Cohomology-Vanishing (4.2) applied to the connected affine K-group ${\text {Spec}}({\mathcal {J}})$, we see that the right ${\mathcal {J}}$-Galois K-algebra $B^{\varGamma }$ is isomorphic to ${\mathcal {J}}$. This together with Proposition 3.6 prove the desired result. $\square$

We add an important consequence.

Corollary 4.7

The twisted forms of ${\mathfrak {g}}_0(K)$ are precisely the $\delta \text {-}K$-Lie algebras quasi-isomorphic to ${\mathfrak {g}}_0(R)^{\varGamma }$, where R ranges over all right $(K\varGamma )^*$-Galois $\delta \text {-}K$-algebras.

Proof

By Propositions 4.1 and 4.6, every twisted form is quasi-isomorphic to some ${\mathfrak {g}}_0(R)^{\varGamma }$. Conversely, any ${\mathfrak {g}}_0(R)^{\varGamma }$ is clearly a twisted form, whence any one that is quasi-isomorphic to ${\mathfrak {g}}_0(R)^{\varGamma }$ is, as well, by (4.5) and Corollary 3.7. $\square$

Before proceeding we make the following remark: given an integer $n \ge 2$, let $\varLambda _n=K^{\times }/(K^{\times })^n$ denote the quotient group of the multiplicative group $K^{\times }$ by the subgroup of all n-th powers. This is an infinite group, as is easily seen. Removing the identity element, let $\varLambda _n^{+}=\varLambda _n{\setminus } \{ 1 \}$. Needed here is the set only in $n=2, 3$. The set $\varLambda _2^{+}$ parametrizes the quadratic field extensions over K, while the set $\varLambda _3^{+}$ modulo the equivalence relation $x\sim x^{\pm 1}$ parametrizes the cubic Galois field extensions over K. Therefore, these two classes of field extensions both consist of infinitely many ones.

4.4 Proof of Theorem 1.5, Part 2: Case $\varGamma ={\mathbb {Z}}_2$

In this case, the right $(K\varGamma )^*$-Galois $\delta \text {-}K$-algebras R are precisely

• (i) the trivial one $(K\varGamma )^*$ (equipped with the obvious $\delta$-operator), and

• (ii) the quadratic field extensions over K (equipped with the $\delta$-operator uniquely extending the one on K).

By Corollary 4.7 it remains to show that for $R=(K\varGamma )^*$ in (i), we have ${\mathfrak {g}}_0(R)^{\varGamma }\simeq {\mathfrak {g}}_0(K)$. Let $\mathrm{Map}(\varGamma , {\mathfrak {g}}_0(K))$ denote the $\varGamma$-set of all maps $\varGamma \rightarrow {\mathfrak {g}}_0(K)$, equipped with the action

$$\begin{aligned} \gamma f : \gamma ' \mapsto f(\gamma '\gamma ), \end{aligned}$$

where $\gamma , \gamma ' \in \varGamma$ and $f \in \mathrm{Map}(\varGamma , {\mathfrak {g}}_0(K))$. Regard this naturally as the direct product of $\#\varGamma$-copies of the $\delta \text {-}K$-Lie algebra ${\mathfrak {g}}_0(K)$. Then we see that associating to $x \otimes a \in {\mathfrak {g}}_0 \otimes _{{\mathbb {C}}} (K\varGamma )^*$, the map $\gamma \mapsto \gamma x \otimes a(\gamma )$ gives a $\varGamma$-equivariant isomorphism

$$\begin{aligned} {\mathfrak {g}}_0(R) \overset{\simeq }{\longrightarrow } \mathrm{Map}(\varGamma , {\mathfrak {g}}_0(K)) \end{aligned}$$

of $\delta \text {-}K$-Lie algebras, whose restriction to the $\varGamma$-invariants is the desired ${\mathfrak {g}}_0(R)^{\varGamma }\simeq {\mathfrak {g}}_0(K)$. This completes the proof.

4.5 Proof of Theorem 1.5, Part 3: Case $\varGamma =\mathfrak {S}_3$

In this case, let R be a right $(K\varGamma )^*$-Galois $\delta \text {-}K$-algebra. In view of Corollary 4.7 we wish to show that ${\mathfrak {g}}_0(R)^{\varGamma }$ is such as in Part 3 of the theorem. This is obvious when R is either trivial or a Galois field extension L/K with $\varGamma =\mathrm {Gal}(L/K)$; notice from the preceding case that ${\mathfrak {g}}_0(R)^{\varGamma }={\mathfrak {g}}_0(K)$ if R is trivial. We may thus exclude these two cases.

To describe R, note that R is artinian as a ring, and $\varGamma$-simple in the sense that it does not include any non-trivial $\varGamma$-stable ideal. Since the action by $\varGamma$ on R commutes with the $\delta$-operator, R is a module algebra over the ${\mathbb {C}}$-Hopf algebra ${\mathbb {C}}\varGamma \otimes _{{\mathbb {C}}} {\mathbb {C}}[\delta ]$, which is artinian simple or AS in the sense of [2, Definition 11.6]; see the original [1, Definition 2.6] as an alternate. This ${\mathbb {C}}$-Hopf algebra is the group algebra ${\mathbb {C}}\varGamma$ tensored with the polynomial algebra ${\mathbb {C}}[\delta ]$ in which $\delta$ is primitive. Choose arbitrary a maximal (or equally, minimal) ideal $\mathfrak {m}$ of R, and let $\varGamma '$ be the subgroup of $\varGamma$ consisting of all elements that stabilize $\mathfrak {m}$. By [2, Proposition 11.5] we have

$$\begin{aligned} \text {(a)}\ \varGamma ' \simeq {\mathbb {Z}}_2\quad \text {or} \quad \text {(b)}\ \varGamma ' \simeq {\mathbb {Z}}_3, \end{aligned}$$

with the extremal cases being excluded. Moreover, there exists a $\delta \text {-}K$-field L such that R is naturally isomorphic to the $(\delta ,\varGamma )$-K-algebra $\mathrm{Map}(\varGamma '\backslash \varGamma , L)$ consisting of all maps from the set of right cosets $\varGamma '\backslash \varGamma$ to L. This $\mathrm{Map}(\varGamma '\backslash \varGamma , L)$ is naturally isomorphic to the direct product of $[\varGamma :\varGamma ']$-copies of L, as $\delta \text {-}K$-algebra, and possesses the $\varGamma$-action presented below. Suppose that ${\mathbb {Z}}_2$ is an arbitrarily chosen subgroup of $\varGamma$ of order 2, and ${\mathbb {Z}}_3$ is the unique subgroup of $\varGamma$ of order 3, so that we have $\varGamma ={\mathbb {Z}}_3\rtimes {\mathbb {Z}}_2$.

Case (a). We may suppose $\varGamma '={\mathbb {Z}}_2$ (see [2, Proposition 11.5 (1)]) and $\varGamma '\backslash \varGamma ={\mathbb {Z}}_3$. If $\gamma \in \varGamma$, $\gamma '\in {\mathbb {Z}}_3$ and $f \in \mathrm{Map}({\mathbb {Z}}_3, L)$, then we have

$$\begin{aligned} \gamma f : \gamma ' \mapsto {\left\{ \begin{array}{ll} f(\gamma \gamma '), &{}\text {if}\ \gamma \in {\mathbb {Z}}_3\\ \gamma f(\gamma '^{-1}), &{}\text {if}\ 0\ne \gamma \in {\mathbb {Z}}_2. \end{array}\right. } \end{aligned}$$

Case (b). We have $\varGamma '={\mathbb {Z}}_3$, and we may suppose $\varGamma '\backslash \varGamma ={\mathbb {Z}}_2$. If $\gamma \in \varGamma$, $\gamma '\in {\mathbb {Z}}_2$ and $f \in \mathrm{Map}({\mathbb {Z}}_2, L)$, then we have

$$\begin{aligned} \gamma f : \gamma ' \mapsto {\left\{ \begin{array}{ll} f(\gamma \gamma '), &{}\text {if}\ \gamma \in {\mathbb {Z}}_2;\\ \gamma f(\gamma '), &{}\text {if}\ \gamma \in {\mathbb {Z}}_3, \ \gamma '=0~\text {in}~{\mathbb {Z}}_2;\\ \gamma ^{-1}f(\gamma '), &{}\text {if}\ \gamma \in {\mathbb {Z}}_3, \ \gamma '\ne 0~\text {in}~{\mathbb {Z}}_2. \end{array}\right. } \end{aligned}$$

In either case, since R is right $(K\varGamma )^*$-Galois, $\varGamma '$ must act non-trivially on L, so that L/K is a Galois field extension with $\varGamma '=\mathrm{Gal}(L/K)$. Conversely, if L/K is such, then R is seen to be a right $(K\varGamma )^*$-Galois $\delta \text {-}K$-algebra, being split by L. Moreover, ${\mathfrak {g}}_0(R)$ is naturally isomorphic to the $(\delta ,\varGamma )\text {-}R$-Lie algebra $\mathrm{Map}(\varGamma '\backslash \varGamma , {\mathfrak {g}}_0(L))$ equipped with the obviously induced structure. We see

$$\begin{aligned} {\mathfrak {g}}_0(R)^{\varGamma }&\simeq \mathrm{Map}(\varGamma '\backslash \varGamma , {\mathfrak {g}}_0(L))^{\varGamma } =\big ( \mathrm{Map}( \varGamma '\backslash \varGamma , {\mathfrak {g}}_0(L) )^{{\mathbb {Z}}_3} \big )^{{\mathbb {Z}}_2}\\&={\left\{ \begin{array}{ll} \{\text {all constant maps}~{\mathbb {Z}}_3\rightarrow {\mathfrak {g}}_0(L)\}^{{\mathbb {Z}}_2} &{} \text {in Case (a)}\\ \, \mathrm{Map}({\mathbb {Z}}_2, {\mathfrak {g}}_0(L)^{{\mathbb {Z}}_3})^{{\mathbb {Z}}_2} &{} \text {in Case (b)} \end{array}\right. }\\&={\mathfrak {g}}_0(L)^{\varGamma '}, \end{aligned}$$

which completes the proof.

4.6 Explicit Non-trivial Forms

Let us describe explicitly (up to quasi-isomorphism) the non-trivial twisted forms of ${\mathfrak {g}}_0(K)$ listed in (ii) of Part 2 and (ii)–(iv) of Part 3 of the theorem, separately for four types. For all those, quadratic field extensions are needed. Such an extension L/K is of the form

$$\begin{aligned} L=K(\sqrt{\alpha })=\{ a+b\sqrt{\alpha }\mid a, b \in K\}, \end{aligned}$$

where $\alpha \in K^{\times }{\setminus } (K^{\times })^2$. The generator of $\mathrm {Gal}(L/K)\, (={\mathbb {Z}}_2)$ sends each element $x=a+b\sqrt{\alpha }$ to

$$\begin{aligned} \overline{x}:= a-b\sqrt{\alpha }. \end{aligned}$$

(4.7)

We will use this symbol $\overline{x}$, regardless of $\alpha$.

4.6.1 Type $\mathrm {A}_{\ell }\ (\ell \ge 2)$

We have ${\mathfrak {g}}_0=\mathfrak {sl}_n({\mathbb {C}})$, where $n=\ell +1\ge 3$. The order 2 outer-automorphism is conjugate to $X \mapsto -{}^tX$. For a quadratic extension field $L=K(\sqrt{\alpha })$ over K as above, the generator of $\varGamma \, (={\mathbb {Z}}_2)$ may supposed to act on ${\mathfrak {g}}_0(L)$ by $X=\big (x_{ij}\big )_{i,j} \mapsto -{}^t\overline{X}=\big (-\overline{x}_{ji}\big )_{i,j}$; see [9, Chapter IX, Theorem 5]. We see

$$\begin{aligned} {\mathfrak {g}}_0(L)^{\varGamma }=\mathfrak {o}_n(K)~\oplus \sqrt{\alpha } \big (\mathrm{Sym}_n(K)\cap \mathfrak {sl}_n(K)\big ), \end{aligned}$$

where $\mathrm{Sym}_n(K)$ (resp., $\mathfrak {o}_n(K)$) denotes the K-subspace of $\mathfrak {gl}_n(L)$ consisting of all matrices X with entries in K that are symmetric (resp., skew-symmetric, ${}^tX=-X$).

4.6.2 Type $\mathrm {D}_{\ell }\ (\ell \ge 5)$

Let $m=2\ell$. We have ${\mathfrak {g}}_0=\mathfrak {o}_{m}({\mathbb {C}}),$ which consists of all skew-symmetric $m\times m$ complex matrices. The order 2 outer-automorphism is conjugate to $X\mapsto DXD$, where $D=\mathrm{diag}(-1,1, \ldots ,1)$; see [9, Chapter IX, Theorem 6]. For a quadratic extension field $L=K(\sqrt{\alpha })$ over K as above, we see

$$\begin{aligned} {\mathfrak {g}}_0(L)^{\varGamma }=\left\{ \, \begin{pmatrix} 0 &{} -\sqrt{\alpha } \, {}^tX \\ \sqrt{\alpha } \, X &{}Y \end{pmatrix} \, \biggm | X\in K^{m-1},\ Y \in \mathfrak {o}_{m-1}(K) \, \right\} , \end{aligned}$$

(4.8)

where by writing $X \in K^{m-1}$, we mean that X is an $(m-1)$-columned vector with entries in K.

4.6.3 Type $\mathrm {E}_6$

Here we follow Jacobson [10, Section 7] for the construction. Let $\mathfrak {J}$ be the exceptional central simple Jordan algebra over ${\mathbb {C}}$, and let $\mathfrak {J}^+$ denote the subspace of $\mathfrak {J}$ which consists of the elements a with trace zero, $T(a)=0$. We have the general linear complex Lie algebra $\mathfrak {gl}(\mathfrak {J})$ on the ${\mathbb {C}}$-vector space $\mathfrak {J}$. Given an element $a \in \mathfrak {J}^+$, we have an element $R_a \in \mathfrak {gl}(\mathfrak {J})$ given by $R_a(x)= xa\, (=ax)$, $x \in \mathfrak {J}$. Let $R_{\mathfrak {J}^+}$ be the subspace of $\mathfrak {gl}(\mathfrak {J})$ which consists of all $R_a$, $a\in \mathfrak {J}^+$. The complex simple Lie algebra ${\mathfrak {g}}_0$ of type $\mathrm {E}_6$ is the Lie subalgebra of $\mathfrak {gl}(\mathfrak {J})$ generated by $R_{\mathfrak {J}^+}$. We have

$$\begin{aligned} {\mathfrak {g}}_0= R_{\mathfrak {J}^+} \oplus ~\mathfrak {f}_0, \end{aligned}$$

where we set $\mathfrak {f}_0:=[R_{\mathfrak {J}^+}, R_{\mathfrak {J}^+}]$; this is a Lie subalgebra of ${\mathfrak {g}}_0$ such that $[R_{\mathfrak {J}^+},\mathfrak {f}_0]=R_{\mathfrak {J}^+}$, and is in fact the complex simple Lie algebra of type $\mathrm {E}_6$. The order 2 outer-automorphism of ${\mathfrak {g}}_0$ is conjugate to $X\mapsto -X^*$, where $X^*$ denotes the operator adjoint to X with respect to the trace form $(a, b)\mapsto T(ab)$. More explicitly this is given by

$$\begin{aligned} X \mapsto {\left\{ \begin{array}{ll} -X &{} \text {if}\ X\in R_{\mathfrak {J}^+};\\ \ X &{} \text {if}\ X \in \mathfrak {f}_0. \end{array}\right. } \end{aligned}$$

Therefore, we have

$$\begin{aligned} {\mathfrak {g}}_0(L)^{\varGamma }= (R_{\mathfrak {J}^+}\, \otimes _{{\mathbb {C}}} K\sqrt{\alpha }) \oplus ~\mathfrak {f}_0(K). \end{aligned}$$

4.6.4 Type $\mathrm {D}_4$

The complex Lie algebra ${\mathfrak {g}}_0$ is the Lie algebra $\mathfrak {o}_{8}({\mathbb {C}})$ of skew-symmetric $8 \times 8$ complex matrices. We follow É. Cartan [8] for the explicit description of outer-automorphisms. We discuss for each group action, separately as in Part 3 of the theorem.

(ii) Action by ${\mathbb {Z}}_2$. The argument above for $\mathrm {D}_{\ell }\ (\ell \ge 5)$ works for $\ell =4$, as well, so that ${\mathfrak {g}}_0(L)^{{\mathbb {Z}}_2}$ is given by the right-hand side of (4.8) with $m=8$.

(iii) Action by ${\mathbb {Z}}_3$. Choose a generator $\sigma$ of the group. The relevant Galois extension is a cubic one, and it is of the form $L=K(\root 3 \, \of {\beta })$, where $\beta \in K^{\times }{\setminus } (K^{\times })^3$. The generator $\sigma$ acts on L so that $\root 3 \, \of {\beta }\mapsto \omega \root 3 \, \of {\beta }$, where $\omega$ is a primitive 3rd root of 1.

We suppose that the rows and the columns of matrices in ${\mathfrak {g}}_0\, (=\mathfrak {o}_8({\mathbb {C}}))$ are indexed by the eight integers $0,1,\ldots , 7$. Given a matrix $X=\big ( x_{ij} \big )_{0\le i,j\le 7}$ in ${\mathfrak {g}}_0$, we define seven vectors in ${\mathbb {C}}^4$ by

$$\begin{aligned} X_i= {}^t\big (x_{0,i},\ x_{i+1,i+5},\ x_{i+4,i+6}, \ x_{i+2,i+3}\big ), \quad i=1,2,\ldots ,7, \end{aligned}$$

(4.9)

where the index $i+p$ greater than 7 is understood to be $i+p-7$; e.g., the third entry $x_{i+4,i+6}$ in $i=3$ in understood to be $x_{7,2}\, (=-x_{2,7})$. Then every matrix X as above is uniquely determined by these seven vectors. This holds when ${\mathfrak {g}}_0$ is replaced by its base extension. We will use in (4.13)–(4.15) the notation $X_i$ for the seven vectors which are associated as above with a matrix X in such a base extension.

The action by ${\mathbb {Z}}_3$ on ${\mathfrak {g}}_0$ is (up to conjugation) such that $\sigma$ acts on the vectors above as the ${\mathbb {C}}$-linear automorphisms given by the matrix

$$\begin{aligned} S=\frac{1}{2} \begin{pmatrix} -1&{}-1&{}-1&{}-1\\ \ 1&{}\ 1&{}-1&{}-1\\ \ 1&{}-1&{}\ 1&{}-1\\ \ 1&{}-1&{}-1&{}\ 1 \end{pmatrix}, \end{aligned}$$

(4.10)

which is seen to have $1,1,\omega$ and $\omega ^2$ as eigen-values; see [8, Section 4]. Set $\sqrt{-3}:=1+2\omega$, a square root of $-3$. Then we have the eigen-vectors

$$\begin{aligned} \varvec{v}^{(1)}_1=\begin{pmatrix} 0\\ 1\\ -1\\ 0 \end{pmatrix},\ \varvec{v}^{(2)}_1=\begin{pmatrix} 0\\ 1 \\ 0 \\ -1 \end{pmatrix},\ \varvec{v}_{\omega }=\begin{pmatrix} \sqrt{-3} \\ 1\\ 1\\ 1 \end{pmatrix},\ \varvec{v}_{\omega ^2}=\begin{pmatrix} -\sqrt{-3}\\ 1\\ 1\\ 1 \end{pmatrix} \end{aligned}$$

(4.11)

of S which are associated with 1, 1, $\omega$ and $\omega ^2$, respectively; these form a basis of ${\mathbb {C}}^4$. Let $L^4$ denote the L-vector space of all 4-columned vectors with entries in L. Define a 4-dimensional K-subspace of $L^4$ by

$$\begin{aligned} \varXi _{L/K}= K \varvec{v}^{(1)}_1+K\varvec{v}^{(2)}_1 +K\root 3 \, \of {\beta }\, \varvec{v}_{\omega ^2} +K(\root 3 \, \of {\beta })^2 \varvec{v}_{\omega }. \end{aligned}$$

(4.12)

We see now easily

$$\begin{aligned} {\mathfrak {g}}_0(L)^{\varGamma } = \{ \, X \in {\mathfrak {g}}_0(L)\, \mid \, X_i \in \varXi _{L/K},\ 1\le i \le 7 \, \}. \end{aligned}$$

(4.13)

(iv) Action by $\varGamma \, (=\mathfrak {S}_3)$. The relevant Galois extension is described by the following.

Lemma 4.8

A Galois extension field over K with Galois group $\varGamma \, (=\mathfrak {S}_3)$ is the same as a field L of the form $L=K(\sqrt{\alpha }, \root 3 \, \of {\beta })$, where

• (a) $\alpha \in K^{\times }{\setminus } (K^{\times })^2$, so that $K(\sqrt{\alpha })/K$ is a quadratic field extension,

• (b) $\beta \in M^{\times }{\setminus } (M^{\times })^3$, where $M=K(\sqrt{\alpha })$, and

• (c) $\beta \overline{\beta }\in (K^{\times })^3$, where $\overline{\beta }$ is such as given by (4.7).

For such an L, we have

• (x) an order 3 element $\sigma$ and an order 2 element $\tau$ of $\varGamma$, which necessarily generate $\varGamma$, satisfying $\sigma \tau =\tau \sigma ^2$,

• (y) a primitive 3rd root $\omega$ of 1, and

• (z) an element $\gamma$ of $K^{\times }$ such that $\gamma ^3=\beta \overline{\beta }$ (see (c) above),

with which the action by $\varGamma$ is presented as

$$\begin{aligned} \sigma : \sqrt{\alpha }\mapsto \sqrt{\alpha },\ \root 3 \, \of {\beta }\mapsto \omega \root 3 \, \of {\beta };\quad \tau : \sqrt{\alpha }\mapsto -\sqrt{\alpha }\, (=\overline{\sqrt{\alpha }}),\ \root 3 \, \of {\beta }\mapsto \frac{\gamma }{\root 3 \, \of {\beta }}. \end{aligned}$$

Proof

Given $\beta$ such as in (b), we have a cubic extension $M(\root 3 \, \of {\beta })/M$. One sees that $\beta \overline{\beta }\in (M^{\times })^3$ if and only if $M(\root 3 \, \of {\beta })=M(\root 3 \, \of {\overline{\beta }})$. If $\gamma \in M^{\times }$ and $\gamma ^3=\beta \overline{\beta }$, then $\gamma /\root 3 \, \of {\beta }$ is a 3rd root of $\overline{\beta }$. A point is only to see that $\root 3 \, \of {\beta }\mapsto \gamma /\root 3 \, \of {\beta }$ gives an involution which extends $M \rightarrow M, x \mapsto \overline{x}$ if and only if $\gamma \in K^{\times }$. $\square$

Example 4.9

Recall $K={\mathbb {C}}(t)$. One can prove that

$$\begin{aligned} \alpha =1-t^3\quad \text {and}\quad \beta =1+\sqrt{1-t^3} \end{aligned}$$

satisfy the conditions above. Indeed, a point is to prove $\beta \notin (M^{\times })^3$, reducing it to show directly that there is no triple of polynomials a, b, c in ${\mathbb {C}}[t]$ with c monic, such that

$$\begin{aligned} a^3 + 3ab^2\alpha =c^3,\quad 3a^2b+b^3\alpha =c^3. \end{aligned}$$

The result shows that there exists a Galois extension L/K with $\mathrm{Gal}(L/K)=\mathfrak {S}_3$.

Let $L=K(\sqrt{\alpha },\ \root 3 \, \of {\beta }),\ M=K(\sqrt{\alpha })$, $\sigma$, $\tau$, $\omega$ and $\gamma$ be as in Lemma 4.8. Recall from the proof of the lemma that $\gamma /\root 3 \, \of {\beta }$ is a 3rd root of $\overline{\beta }$, and denote it by $\root 3 \, \of {\overline{\beta }}$, so that one has

$$\begin{aligned} \tau (\root 3 \, \of {\beta })=\root 3 \, \of {\overline{\beta }},\quad \tau \left( \root 3 \, \of {\overline{\beta }}\right) =\root 3 \, \of {\beta } . \end{aligned}$$

The action by $\varGamma$ on ${\mathfrak {g}}_0\, (=\mathfrak {o}_8({\mathbb {C}}))$ is (up to conjugation) such that the generators $\sigma$ and $\tau$ act on the seven vectors in (4.9) as the K-linear automorphisms given by the matrix S in (4.10) and the diagonal matrix $D=\mathrm{diag}(-1,1,1,1)$, respectively. The latter action by $\tau$ on ${\mathfrak {g}}_0$ coincides with the above mentioned outer-automorphism $X\mapsto DXD$ for type $\mathrm {D}_{\ell }$, when $\ell =4$.

Note $L=M(\root 3 \, \of {\beta })$, and apply the previous result for the action by ${\mathbb {Z}}_3$ to the action by $\langle \sigma \rangle$ (on L/M). Then, by using the M-subspace $\varXi _{L/M}$ of $L^4$ defined by (4.12) (modified into the present situation), we have

$$\begin{aligned} {\mathfrak {g}}_0(L)^{\langle \sigma \rangle } = \{ \, X \in {\mathfrak {g}}_0(L)\, \mid \, X_i \in \varXi _{L/M},\ 1\le i \le 7 \,\}. \end{aligned}$$

(4.14)

By using the vectors given in (4.11) we define a 4-dimensional K-subspace of $L^4$ by

$$\begin{aligned} \varTheta _{L/K}&= K\varvec{v}^{(1)}_1+K\varvec{v}^{(2)}_1 +K\left( \root 3 \, \of {\beta }\, \varvec{v}_{\omega ^2} +\root 3 \, \of {\overline{\beta }}\, \varvec{v}_{\omega }\right) \\&\quad +K\sqrt{\alpha }\left( \root 3 \, \of {\beta }\, \varvec{v}_{\omega ^2} -\root 3 \, \of {\overline{\beta }}\, \varvec{v}_{\omega }\right) . \end{aligned}$$

We see now easily

$$\begin{aligned} {\mathfrak {g}}_0(L)^{\varGamma } = ({\mathfrak {g}}_0(L)^{\langle \sigma \rangle })^{\langle \tau \rangle } =\{ \, X \in {\mathfrak {g}}_0(L)\, \mid \, X_i \in \varTheta _{L/K},\ 1\le i \le 7 \, \}. \end{aligned}$$

(4.15)