Growth of Differential Identities

Abstract

In this paper we study the growth of the differential identities of some algebras with derivations, i.e., associative algebras where a Lie algebra L (and its universal enveloping algebra U(L)) acts on them by derivations. In particular, we study in detail the differential identities and the cocharacter sequences of some algebras whose sequence of differential codimensions has polynomial growth. Moreover, we shall give a complete description of the differential identities of the algebra UT ₂ of 2 × 2 upper triangular matrices endowed with all possible action of a Lie algebra by derivations. Finally, we present the structure of the differential identities of the infinite dimensional Grassmann G with respect to the action of a finite dimensional Lie algebra L of inner derivations.

1 Introduction

Let A be an associative algebra over a field F of characteristic zero and assume that a Lie algebra L acts on it by derivations. Such an action can be naturally extended to the action of the universal enveloping algebra U(L) of L and in this case we say that A is an algebra with derivations or an L-algebra. In this context it is natural to define the differential identities of A, i.e., the polynomials in non-commutative variables x ^h = h(x), h ∈ U(L), vanishing in A.

An effective way of measuring the differential identities satisfied by a given L-algebra A is provided by its sequence of differential codimensions $c_{n}^{L}(A)$, n = 1, 2, …. The nth term of such sequence measures the dimension of the space of multilinear differential polynomials in n variables of the relatively free algebra with derivations of countable rank of A. Since in characteristic zero, by the multilinearization process, every differential identity is equivalent to a system of multilinear ones, the sequence of differential codimensions of A gives a quantitative measure of the differential identities satisfied by the given L-algebra. Maybe the most important feature of this sequence proved by Gordienko in [6] is that in case A is a finite dimensional L-algebra, $c_{n}^{L}(A)$ is exponentially bounded. Moreover, he determined the exponential rate of growth of the sequence of differential codimensions, i.e., he proved that for any finite dimensional L-algebra A, the limit $\lim _{n\to \infty }\sqrt [n]{c_{n}^{L}(A)}$ exists and is a non-negative integer. Such integer, denoted $\operatorname {\mathrm {exp}}^L(A)$, is called the differential PI-exponent of the algebra A and it provides a scale allowing us to measure the rate of growth of the identities of any finite dimensional L-algebra. As a consequence of this result it follows that the differential codimensions of a finite dimensional L-algebra A are either polynomially bounded or grow exponentially. Hence no intermediate growth is allowed.

When studying the polynomial identities of an L-algebra A, one is lead to consider $\operatorname {\mathrm {var}} ^{L}(A)$, the L-variety of algebras with derivations generated by A, that is the class of L-algebras satisfying all differential identities satisfied by A. Thus we define the growth of $\mathcal {V}= \operatorname {\mathrm {var}} ^{L}(A)$ to be the growth of the sequence $c_{n}^{L}(\mathcal {V})=c_{n}^{L}(A)$, n = 1, 2, … and we say that a variety $\mathcal {V}$ has almost polynomial growth if $\mathcal {V}$ has exponential growth but every proper subvariety has polynomial growth. Since the ordinary polynomial identities and corresponding codimensions are obtained by leting L act on A trivially (or L is the trivial Lie algebra), the algebra UT ₂ of 2 × 2 upper triangular matrices regarded as L-algebra where L acts trivially on it generates an L-variety of almost polynomial growth (see [4, 8]). Clearly another example of algebras generating an L-variety of almost polynomial growth is the infinite dimensional Grassmann algebra G where L acts trivially on it (see [8, 13]). Notice that in the ordinary case Kemer in [8] proved that UT ₂ and G are the only algebras generating varieties of almost polynomial growth.

Recently in [4] the authors introduced another algebra with derivations generating a L-variety of almost polynomial growth. They considered $UT_2^\varepsilon$ to be the algebra UT ₂ with the action of the 1-dimensional Lie algebra spanned by the inner derivation ε induced by 2⁻¹(e ₁₁ − e ₂₂), where the e _ij’s are the usual matrix units. Also they proved that when the Lie algebra $\operatorname {{\mathrm {Der}}}(UT_2)$ of all derivations acts on UT ₂, the variety with derivations generated by UT ₂ has no almost polynomial growth.

Notice that if δ is the inner derivation of UT ₂ induced by 2⁻¹ e ₁₂, then $\operatorname {{\mathrm {Der}}}(UT_2)$ is a 2-dimensional metabelian Lie algebra with basis {ε, δ}. Here we shall study the differential identities of $UT_2^\delta$, i.e., the algebra UT ₂ with the action of the 1-dimensional Lie algebra spanned by δ. In particular we shall prove that $UT_2^\delta$ does not generate an L-variety of almost polynomial growth. Moreover, in order to complete the description of the differential identities of UT ₂, we shall study the T _L-ideal of the differential identities of UT ₂ with the action of an arbitrary 1-dimensional Lie subalgebra of $\operatorname {{\mathrm {Der}}}(UT_2)$.

Furthermore, we shall study the differential identities of some particular L-algebras whose sequence of differential codimensions has polynomial growth. In particular we shall exhibit an example of a commutative algebra with derivations that generates a L-variety of linear growth.

Finally, we shall give an example of an infinite dimensional L-algebra of exponential growth. We shall present the structure of the differential identities of $\widetilde {G}$, i.e., the infinite dimensional Grassmann algebra with the action of a finite dimensional abelian Lie algebra and we shall show that, unlike the ordinary case, $\widetilde {G}$ does not generate an L-variety of almost polynomial growth.

2 L-Algebras and Differential Identities

Throughout this paper F will denote a field of characteristic zero. Let A be an associative algebra over F. Recall that a derivation of A is a linear map ∂ : A → A such that

$$\displaystyle \begin{aligned} \partial(ab)=\partial(a)b+a\partial(b), \qquad \mbox{ for all } a,b\in A. \end{aligned}$$

In particular an inner derivation induced by a ∈ A is the derivation $\operatorname {\mathrm {ad}} a:A\to A$ of A defined by $( \operatorname {\mathrm {ad}} a)(b)=[a,b]=ab-ba$, for all b ∈ A. The set of all derivations of A is a Lie algebra denoted by $\operatorname {{\mathrm {Der}}}(A)$, and the set $\operatorname {\mathrm {ad}} (A)$ of all inner derivations of A is a Lie subalgebra of $\operatorname {{\mathrm {Der}}}(A)$.

Let L be a Lie algebra over F acting on A by derivations. If U(L) is its universal enveloping algebra, the L-action on A can be naturally extended to an U(L)-action. In this case we say that A is an algebra with derivations or an L-algebra.

Let L be a Lie algebra. Given a basis $\mathcal {B}=\{h_{i}\ \vert \ i\in I \}$ of the universal enveloping algebra U(L) of L, we let F〈X|L〉 be the free associative algebra over F with free formal generators $x_{j}^{h_{i}}$, i ∈ I, $j\in \mathbb {N}$. We write $x_{i}=x_{i}^{1}$, 1 ∈ U(L), and then we set X = {x ₁, x ₂, … }. We let U(L) act on F〈X|L〉 by setting

$$\displaystyle \begin{aligned} \gamma(x_{j_{1}}^{h_{i_{1}}}x_{j_{2}}^{h_{i_{2}}}\dots x_{j_{n}}^{h_{i_{n}}})=x_{j_{1}}^{\gamma h_{i_{1}}}x_{j_{2}}^{h_{i_{2}}}\dots x_{j_{n}}^{h_{i_{n}}}+\dots+x_{j_{1}}^{h_{i_{1}}}x_{j_{2}}^{h_{i_{2}}}\dots x_{j_{n}}^{\gamma h_{i_{n}}}, \end{aligned}$$

where γ ∈ L and $x_{j_{1}}^{h_{i_{1}}}x_{j_{2}}^{h_{i_{2}}}\dots x_{j_{n}}^{h_{i_{n}}}\in F\langle X|L\rangle$. The algebra F〈X|L〉 is called the free associative algebra with derivations on the countable set X and its elements are called differential polynomials (see [4, 7, 9]).

Given an L-algebra A, a polynomial f(x ₁, …, x _n) ∈ F〈X|L〉 is a polynomial identity with derivation of A, or a differential identity of A, if f(a ₁, …, a _n) = 0 for all a _i ∈ A, and, in this case, we write f ≡ 0.

Let $\operatorname {\mathrm {Id}}^{L}(A)=\{f\in F\langle X|L\rangle \ \vert \ f\equiv 0 \mbox{ on } A\}$ be the set of all differential identities of A. It is readily seen that $\operatorname {\mathrm {Id}}^L (A)$ is a T _L-ideal of F〈X|L〉, i.e., an ideal invariant under the endomorphisms of F〈X|L〉. In characteristic zero every differential identity is equivalent to a system of multilinear differential identities. Hence $\operatorname {\mathrm {Id}}^L (A)$ is completely determined by its multilinear polynomial.

Let

$$\displaystyle \begin{aligned} P_{n}^{L}=\operatorname{\mathrm{span}}\{x_{\sigma(1)}^{h_{1}}\dots x_{\sigma(n)}^{h_{n}} \ \vert \ \sigma\in S_{n},h_{i}\in \mathcal{B} \} \end{aligned}$$

be the space of multilinear differential polynomials in the variables x ₁, …, x _n, n ≥ 1. We act on $P_n ^L$ via the symmetric group S _n as follows: for σ ∈ S _n, $\sigma (x_{i}^{h})=x_{\sigma (i)}^{h}$. For every L-algebra A, the vector space $P_{n}^{L}\cap \operatorname {\mathrm {Id}}^{L}(A)$ is invariant under this action. Hence the space $P_n ^L (A)= P_n ^L / ( P_{n}^{L}\cap \operatorname {\mathrm {Id}}^{L}(A))$ has a structure of left S _n-module. The non-negative integer $c_n ^L (A)= \dim P_n ^L (A)$ is called nth differential codimension of A and the character $\chi _n ^L (A)$ of $P_n^L (A)$ is called nth differential cocharacter of A. Since $\operatorname {\mathrm {char}} F=0$, we can write

$$\displaystyle \begin{aligned} \chi_n^{L}(A)=\sum_{\lambda\vdash n} m^L_{\lambda}\chi_{\lambda}, \end{aligned}$$

where λ is a partition of n, χ _λ is the irreducible S _n-character associated to λ and $m^L_{\lambda }\geq 0$ is the corresponding multiplicity.

Let L be a Lie algebra and H be a Lie subalgebra of L. If A is an L-algebra, then by restricting the action, A can be regarded as a H-algebra. In this case we say that A is an L-algebra where L acts on it as the Lie algebra H and we identify the T _L-ideal $\operatorname {\mathrm {Id}}^L(A)$ and the T _H-ideal $\operatorname {\mathrm {Id}}^H(A)$, i.e., in $\operatorname {\mathrm {Id}}^L(A)$ we omit the differential identities x ^γ ≡ 0, for all γ ∈ L∖H.

Notice that any algebra A can be regarded as L-algebra by letting L act on A trivially, i.e., L acts on A as the trivial Lie algebra. Hence the theory of differential identities generalizes the ordinary theory of polynomial identities.

We denote by P _n the space of multilinear ordinary polynomials in x ₁, …, x _n and by $\operatorname {\mathrm {Id}}(A)$ the T-ideal of the free algebra F〈X〉 of polynomial identities of A. We also write c _n(A) for the nth codimension of A and χ _n(A) for the nth cocharacter of A. Since the field F is of characteristic zero, we have χ _n(A) =∑_λ⊢n m _λ χ _λ, where m _λ ≥ 0 is the multiplicity of χ _λ in the given decomposition.

Since U(L) is an algebra with unit, we can identify in a natural way P _n with a subspace of $P_n^L$. Hence $P_n\subseteq P^L_n$ and $P_n\cap \operatorname {\mathrm {Id}}(A)=P_n\cap \operatorname {\mathrm {Id}}^L(A)$. As a consequence we have the following relations.

Remark 1

For all n ≥ 1,

• 1. $c_n(A)\leq c_n^L(A)$;

• 2. $m_\lambda \leq m_\lambda ^L$, for any λ ⊢ n.

Recall that if A is an L-algebra then the variety of algebras with derivations generated by A is denoted by $\operatorname {\mathrm {var}} ^{L}(A)$ and is called L-variety. The growth of $\mathcal {V}= \operatorname {\mathrm {var}} ^{L}(A)$ is the growth of the sequence $c_{n}^{L}(\mathcal {V})=c_{n}^{L}(A)$, n = 1, 2, ….

We say that the L-variety $\mathcal {V}$ has polynomial growth if $c_{n}^{L}(\mathcal {V})$ is polynomially bounded and $\mathcal {V}$ has almost polynomial growth if $c_{n}^{L}(\mathcal {V})$ is not polynomially bounded but every proper L-subvariety of $\mathcal {V}$ has polynomial growth.

3 On Algebras with Derivations of Polynomial Growth

In this section we study some algebras with derivations whose sequence of differential codimension has linear growth.

Let us first consider the algebra C = F(e ₁₁ + e ₂₂) ⊕ Fe ₁₂ where the e _ij’s are the usual matrix units. The Lie algebra $\operatorname {{\mathrm {Der}}} (C)$ of derivations of C is a 1-dimensional Lie algebra generated by ε where

$$\displaystyle \begin{aligned} \varepsilon (\alpha(e_{11}+e_{22})+\beta e_{12})=\beta e_{12}, \end{aligned}$$

for all α, β ∈ F.

Let C ^ε denote the L-algebra C where L acts on it as the Lie algebra $\operatorname {{\mathrm {Der}}} (C)$. Thus we have the following.

Theorem 1

• 1. $\operatorname {\mathrm {Id}}^L(C^\varepsilon )=\langle [x,y],\ x^\varepsilon y^\varepsilon ,\ x^{\varepsilon ^2}- x ^\varepsilon \rangle _{T_L}$.

• 2. $c_n ^L(C^\varepsilon )=n+1$.

• 3. $\chi _n^L(C^\varepsilon )=2\chi _{(n)}+\chi _{(n-1,1)}$.

Proof

Let $Q=\langle [x,y], x^\varepsilon y^\varepsilon ,\ x^{\varepsilon ^2}- x ^\varepsilon \rangle _{T_L}$. It is easily checked that $Q \subseteq \operatorname {\mathrm {Id}}^L(C^\varepsilon )$. Since x ^ε wy ^ε ∈ Q, where w is a (eventually trivial) monomial of F〈X|L〉, we may write any multilinear polynomial f, modulo Q, as a linear combination of the polynomials

$$\displaystyle \begin{aligned} x_1 \dots x_n, \; x_k ^\varepsilon x_{i_1}\dots x_{i_{n-1}}, \quad i_1<\dots < i_{n-1}. \end{aligned}$$

We next show that these polynomials are linearly independent modulo $\operatorname {\mathrm {Id}}^L(C^\varepsilon )$. Suppose that

By making the evaluation x _j = e ₁₁ + e ₂₂, for all j = 1, …, n, we get α = 0. Also for fixed k, the evaluation x _k = e ₁₂ and x _j = e ₁₁ + e ₂₂ for j ≠ k gives β _k = 0. Thus the above polynomials are linearly independent modulo $P_n ^L \cap \operatorname {\mathrm {Id}}^L(C^\varepsilon )$. Since $P_n ^L \cap Q \subseteq P_n ^L \cap \operatorname {\mathrm {Id}}^L(C^\varepsilon )$, this proves that $\operatorname {\mathrm {Id}}^L(C^\varepsilon )=Q$ and the above polynomials are a basis of $P_n ^L$ modulo $P_n ^L \cap \operatorname {\mathrm {Id}}^L(C^\varepsilon )$. Hence $c_n ^L(C^\varepsilon )=n+1$.

We now determine the decomposition of the nth differential cocharacter of this algebra. Suppose that $\chi _{n}^{L}(C^\varepsilon )=\sum _{\lambda \vdash n} m_{\lambda }\chi _{\lambda }$. Let us consider the standard tableau

and the monomials

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_{(n)}=x^n, \qquad f_{(n)}^\varepsilon =x^\varepsilon x^{n-1} {} \end{array} \end{aligned} $$

(1)

obtained from the essential idempotents corresponding to the tableau T _(n) by identifying all the elements in the row. Clearly f _(n) and $f_{(n)}^\varepsilon$ are not identities of C ^ε. Moreover, they are linearly independent modulo $\operatorname {\mathrm {Id}}^L(C^\varepsilon )$. In fact, suppose that $\alpha f_{(n)} + \beta f_{(n)}^\varepsilon \equiv 0 (\bmod \; \operatorname {\mathrm {Id}}^L(C^\varepsilon ))$. By making the evaluation x = e ₁₁ + e ₂₂ we get α = 0. Moreover, if we evaluate x = e ₁₁ + e ₂₂ + e ₁₂, we obtain β = 0. Thus it follows that m _(n) ≥ 2.

Since deg χ _(n) = 1 and deg χ _(n−1,1) = n − 1, if we find a differential polynomial corresponding to the partition (n − 1, 1) which is not a differential identity of C ^ε, we may conclude that $\chi _n^L(C^\varepsilon )=2\chi _{(n)}+\chi _{(n-1,1)}$.

Let us consider the polynomial

$$\displaystyle \begin{aligned} f_{(n-1,1)}=(x^\varepsilon y - y^\varepsilon x)x^{n-2} \end{aligned}$$

obtained from the essential idempotent corresponding to the standard tableau

by identifying all the elements in each row of the tableau. Evaluating x = e ₁₁ + e ₂₂ and y = e ₁₂ we get f _(n−1,1) = −e ₁₂ ≠ 0 and f _(n−1,1) is not a differential identity of C ^ε. Thus the claim is proved. □

Let us now consider the algebra M ₁ = Fe ₂₂ ⊕ Fe ₁₂ and let ε and δ be derivations of M ₁ such that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varepsilon (\alpha e_{22}+ \beta e_{12})= \beta e_{12}, \qquad \delta (\alpha e_{22}+ \beta e_{12})= \alpha e_{12}, {} \end{array} \end{aligned} $$

(2)

for all α, β ∈ F.

Lemma 1

$\operatorname {{\mathrm {Der}}}(M_1)$ is a 2-dimensional metabelian Lie algebra spanned by ε and δ defined in (2).

Proof

Let us consider the Lie algebra D spanned by ε and δ. Since [ε, δ] = δ, D is a 2-dimensional metabelian Lie algebra and $D\subseteq \operatorname {{\mathrm {Der}}}(M_1)$.

Now consider $\gamma \in \operatorname {{\mathrm {Der}}}(M_1)$. Notice that γ(e ₂₂ e ₁₂) = γ(e ₂₂)e ₁₂ + e ₂₂ γ(e ₁₂) = e ₂₂ γ(e ₁₂). Since γ(e ₂₂ e ₁₂) = 0, it follows that

$$\displaystyle \begin{aligned} \gamma(e_{12})=\alpha e_{12}, \end{aligned}$$

for some α ∈ F. On the other hand, γ(e ₁₂) = γ(e ₁₂ e ₂₂) = αe ₁₂ + e ₁₂ γ(e ₂₂). Thus it follows that e ₁₂ γ(e ₂₂) = 0. Hence

$$\displaystyle \begin{aligned} \gamma(e_{22})=\beta e_{12},\end{aligned} $$

for some β ∈ F. Thus we have that γ = αε + βδ ∈ D and the claim is proved. □

Similarly, if we consider the algebra M ₂ = Fe ₁₁ ⊕ Fe ₁₂ and we assume that ε and δ are derivation of M ₂ such that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \varepsilon (\alpha e_{11}+ \beta e_{12})= \beta e_{12}, \qquad \delta (\alpha e_{11}+ \beta e_{12})= \alpha e_{12}, \end{array} \end{aligned} $$

(3)

for all α, β ∈ F, then we have the following.

Lemma 2

$\operatorname {{\mathrm {Der}}}(M_2)$ is a 2-dimensional metabelian Lie algebra spanned by ε and δ defined in (3).

Let L be any Lie algebra. We shall denote by M ₁ and M ₂ the L-algebras M ₁ and M ₂ where L acts trivially on them. Since x ^γ ≡ 0 for all γ ∈ L, in this case we are dealing with ordinary identities. Thus we have the following result.

Theorem 2 ([3, Lemma 3])

• 1. $Id^L(M_1)=\langle x[y,z]\rangle _{T_L}$ and $Id^L(M_2)=\langle [x,y] z \rangle _{T_L}$.

• 2. $c_n ^L(M_1)=c_n ^L(M_2)=n$.

• 3. $\chi _n^L(M_1)=\chi _n^L(M_2)=\chi _{(n)}+\chi _{(n-1,1)}$.

Denote by $M_1 ^\varepsilon$ and $M_2^\varepsilon$ the L-algebras M ₁ and M ₂ where L acts on them as the 1-dimensional Lie algebra spanned by the derivation ε defined in (2) and (3), respectively.

Theorem 3

• 1. $Id^L(M_1^\varepsilon )=\langle xy^{\varepsilon },\ x^\varepsilon y- y^\varepsilon x -[x,y],\ x^{\varepsilon ^{2}}-x^{\varepsilon }\rangle _{T_{L}}$ and $Id^L(M_2^\varepsilon )=\langle x^{\varepsilon }y,\ x y^\varepsilon - y x^\varepsilon - [x,y],\ x^{\varepsilon ^{2}}-x^{\varepsilon }\rangle _{T_{L}}$.

• 2. $c_n ^L(M_1 ^\varepsilon )=c_n ^L(M_2^\varepsilon )=n+1$.

• 3. $\chi _n^L(M_1^\varepsilon )=\chi _n^L(M_2^\varepsilon )=2\chi _{(n)}+\chi _{(n-1,1)}$.

Proof

If Q is the T _L-ideal generated by the polynomials $xy^{\varepsilon },\ x^\varepsilon y- y^\varepsilon x -[x,y],\ x^{\varepsilon ^{2}}-x^{\varepsilon }$, then it easy to check that $Q\subseteq \operatorname {\mathrm {Id}}^L (M_1 ^\varepsilon )$.

Since x ^ε y ^ε, x[y, z] ∈ Q, the polynomials

$$\displaystyle \begin{aligned} x_j x_{i_1}\dots x_{i_n-1}, \; x_1 ^\varepsilon x_2 \dots x_n, \quad i_1<\dots < i_{n-1},\end{aligned} $$

span $P_n ^L$ modulo $P_n ^L \cap Q$ and we claim that they are linearly independent modulo $\operatorname {\mathrm {Id}}^L (M_1 ^\varepsilon )$. In fact, let $f\in P_n ^L \cap \operatorname {\mathrm {Id}}^L (M_1 ^\varepsilon )$ be a linear combination of these polynomials, i.e.,

For fixed j ≠ 1, from the substitutions x _j = e ₁₂ and x _k = e ₂₂ for k ≠ j we get α _j = 0, j ≠ 1. By making the evaluation x _k = e ₂₂ for all k = 1, …, n, we obtain α ₁ = 0. Finally by evaluating x ₁ = e ₁₂ and x _k = e ₂₂ for k ≠ 1, we get β = 0. Thus the above polynomials are linearly independent modulo $P_{n}^{L} \cap \operatorname {\mathrm {Id}}^{L} (M_{1}^{\varepsilon })$. Since $P_n ^L \cap Q \subseteq P_n ^L \cap \operatorname {\mathrm {Id}}^L(M_1^\varepsilon )$, this proves that $\operatorname {\mathrm {Id}}^L(M_1^\varepsilon )=Q$ and the above polynomials are a basis of $P_n ^L$ modulo $P_n ^L \cap \operatorname {\mathrm {Id}}^L(M_1^\varepsilon )$. Clearly $c_n ^L(M_1 ^\varepsilon )=n+1$.

We now determine the decomposition of the nth differential cocharacter $\chi _n^L(M_1^\varepsilon )$ of this algebra. Suppose that $\chi _{n}^{L}(M_1^\varepsilon )=\sum _{\lambda \vdash n} m_{\lambda }\chi _{\lambda }$. We consider the tableau T _(n) defined in Theorem 1 and let f _(n) and $f_{(n)}^\varepsilon$ be the corresponding polynomials defined in (1). It is clear that f _(n) and $f_{(n)}^\varepsilon$ are not identities of $M_1^\varepsilon$. Moreover, they are linearly independent modulo $\operatorname {\mathrm {Id}}^L(M_1^\varepsilon )$. In fact, suppose that $\alpha f_{(n)} + \beta f_{(n)}^\varepsilon \equiv 0 (\bmod \; \operatorname {\mathrm {Id}}^L(M_1^\varepsilon ))$. By making the evaluation x = e ₂₂ we get α = 0. Moreover, if we evaluate x = e ₂₂ + e ₁₂, we obtain β = 0. Thus it follows that m _(n) ≥ 2. By Remark 1 and Theorem 2 we have m _(n−1,1) ≥ 1. Thus, since deg χ _(n) = 1 and deg χ _(n−1,1) = n − 1, it follows that $\chi _n^L(M_1^\varepsilon )=2\chi _{(n)}+\chi _{(n-1,1)}$.

A similar proof holds for the algebra $M_2 ^\varepsilon$. □

Let $M_1 ^\delta$ and $M_2^\delta$ be the L-algebras M ₁ and M ₂ where L acts on them as the 1-dimensional Lie algebra spanned by the derivation δ defined in (2) and (3), respectively. The proof on the next theorem is similar to the above proof and is omitted.

Theorem 4

• 1. $Id^L(M_1^\delta )=\langle x[y,z],\ xy^{\delta },\ x^\delta y - y^\delta x,\ x^{\delta ^{2}}\rangle _{T_{L}}$ and $Id^L(M_2^\delta )=\langle [x,y]z,\ x^{\delta }y,\ x y^\delta - y x^\delta ,\ x^{\delta ^{2}}\rangle _{T_{L}}$.

• 2. $c_n ^L(M_1 ^\delta )=c_n ^L(M_2^\delta )=n+1$.

• 3. $\chi _n^L(M_1^\delta )=\chi _n^L(M_2^\delta )=2\chi _{(n)}+\chi _{(n-1,1)}$.

Let now L be a 2-dimensional metabelian Lie algebra. Let denote by $M_1^D$ the L-algebra M ₁ where L acts on it as the Lie algebra $\operatorname {{\mathrm {Der}}}(M_1)$ and $M_2^D$ the L-algebra M ₂ where L acts on it as the Lie algebra $\operatorname {{\mathrm {Der}}}(M_2)$.

Remark 2

• 1. $x^\delta y - y^\delta x \in \langle xy^{\varepsilon },\ x^\varepsilon y- y^\varepsilon x -[x,y], \ x^{ \varepsilon \delta }- x^{\delta }\rangle _{T_{L}}$.

• 2. $x y^\delta - y x^\delta \in \langle x^{\varepsilon }y,\ x y^\varepsilon - y x^\varepsilon - [x,y],\ x^{ \varepsilon \delta }- x^{\delta } \rangle _{T_{L}}$.

Proof

First notice that $[x,y]^\delta \in \langle xy^{\varepsilon },\ [x,y]^\varepsilon -[x,y]\rangle _{T_{L}}$. Thus, since $[x,y]^\varepsilon \equiv [x,y] (\bmod \; \langle xy^{\varepsilon },\ x^\varepsilon y- y^\varepsilon x -[x,y]\rangle _{T_{L}})$, it follows that

$$\displaystyle \begin{aligned}{}[x,y]^\delta \in \langle xy^{\varepsilon},\ x^\varepsilon y- y^\varepsilon x -[x,y],\ x^{ \varepsilon\delta}- x^{\delta}\rangle_{T_{L}}.\end{aligned} $$

Moreover, since $xy^\delta \in \langle xy^{\varepsilon },\ x^{ \varepsilon \delta }- x^{\delta }\rangle _{T_{L}}$, we get

$$\displaystyle \begin{aligned} x^\delta y - y^\delta x \in \langle xy^{\varepsilon},\ x^\varepsilon y- y^\varepsilon x -[x,y],\ x^{ \varepsilon\delta}- x^{\delta}\rangle_{T_{L}}.\end{aligned} $$

A similar proof holds for the other statement. □

We do not present the proof of next theorem since it can be easily deduced by using the strategy of proof given in Theorem 3.

Theorem 5

• 1. $Id^L(M_1^D)=\langle xy^{\varepsilon },\ x^\varepsilon y- y^\varepsilon x -[x,y],\ x^{\varepsilon ^{2}}-x^{\varepsilon },\ x^{\delta \varepsilon },\ x^{ \varepsilon \delta }- x^{\delta }\rangle _{T_{L}}$ and $Id^L(M_2^D)=\langle x^{\varepsilon }y,\ x y^\varepsilon - y x^\varepsilon - [x,y],\ x^{\varepsilon ^{2}}-x^{\varepsilon },\ x^{\delta \varepsilon },\ x^{ \varepsilon \delta }- x^{\delta } \rangle _{T_{L}}$.

• 2. $c_n ^L(M_1 ^D)=c_n ^L(M_2^D)=n+2$.

• 3. $\chi _n^L(M_1^D)=\chi _n^L(M_2^D)=3\chi _{(n)}+\chi _{(n-1,1)}$.

4 The Algebra of 2 × 2 Upper Triangular Matrices and Its Differential Identities

In this section we study the growth of differential identities of the algebra UT ₂ of 2 × 2 upper triangular matrices over F.

Let L be any Lie algebra over F and denote by UT ₂ the L-algebra UT ₂ where L acts trivially on it. Since x ^γ ≡ 0, for all γ ∈ L, is a differential identity of UT ₂, we are dealing with ordinary identities. Thus by Malcev [11], Kemer [8] and by the proof of Lemma 3.5 in [1], we have the following results.

Theorem 6

• 1. $\operatorname {\mathrm {Id}}^L(UT_{2})=\langle [x_{1},x_{2}][x_{3},x_{4}]\rangle _{T_L}$.

• 2. $c^L _{n}(UT_{2})=2^{n-1}(n-2)+2.$

• 3. If $\chi _{n}^{L}(UT_{2})=\sum _{\lambda \vdash n} m_{\lambda } \chi _{\lambda }$ is the nth differential cocharacter of UT ₂ , then

  $$\displaystyle \begin{aligned} m_{\lambda}=\begin{cases} 1, & \mathit{\mbox{ if }} \lambda=(n) \\ q+1, & \mathit{\mbox{ if }} \lambda=(p+q,p) \mathit{\mbox{ or }} \lambda=(p+q,p,1) \\ 0 & \mathit{\mbox{ in all other cases}} \end{cases}. \end{aligned}$$

Theorem 7

$\operatorname {\mathrm {var}}^L(UT_2)$ has almost polynomial growth.

Let now ε be the inner derivation of UT ₂ induced by 2⁻¹(e ₁₁ − e ₂₂), i.e.,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varepsilon(a)=2^{-1}[e_{11}-e_{22},a], \qquad \mbox{for all } a\in UT_2, {} \end{array} \end{aligned} $$

(4)

where the e _ij’s are the usual matrix units. We shall denote by $UT_2^\varepsilon$ the L-algebra UT ₂ where L acts on it as the 1-dimensional Lie algebra spanned by ε. In [4] the authors proved the following.

Theorem 8 ([4, Theorems 5 and 12])

• 1. $\operatorname {\mathrm {Id}}^{L}(UT_2^{\varepsilon })=\langle [x,y]^{\varepsilon }-[x,y],\ x^{\varepsilon }y^{\varepsilon },\ x^{\varepsilon ^{2}}-x^{\varepsilon }\rangle _{T_{L}}$.

• 2. $c_n^{L}(UT_2^{\varepsilon })=2^{n-1}n+1$.

• 3. If $\chi _{n}^{L}(UT_{2}^\varepsilon )=\sum _{\lambda \vdash n} m_{\lambda }^\varepsilon \chi _{\lambda }$ is the nth differential cocharacter of $UT_{2}^{\varepsilon }$ , then

  $$\displaystyle \begin{aligned} m_{\lambda}^{\varepsilon}=\begin{cases} n+1, & \mathit{\mbox{ if }} \lambda=(n) \\ 2(q+1), & \mathit{\mbox{ if }} \lambda=(p+q,p) \\ q+1, & \mathit{\mbox{ if }} \lambda=(p+q,p,1) \\ 0 & \mathit{\mbox{ in all other cases}} \end{cases}. \end{aligned}$$

Theorem 9 ([4, Theorem 15])

$\operatorname {\mathrm {var}}^L(UT_2^\varepsilon )$ has almost polynomial growth.

Let now δ be the inner derivation of UT ₂ induced by 2⁻¹ e ₁₂, i.e.,

$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta(a)=2^{-1}[e_{12},a], \qquad \mbox{for all } a\in UT_2. {} \end{array} \end{aligned} $$

(5)

Denote by $UT_2^\delta$ the L-algebra UT ₂ where L acts as the 1-dimensional Lie algebra spanned by δ. The following remarks are easily verified.

Remark 3

[x, y][z, w] ≡ 0, [x, y]^δ ≡ 0, x ^δ y ^δ ≡ 0, x ^δ[y, z] ≡ 0 and $x^{\delta ^2} \equiv 0$ are differential identities of $UT_{2}^{\delta }$.

Remark 4

$x^{\delta }y[z,w],\ [x,y]zw^{\delta },\ x^{\delta }yz^{\delta }\in \langle x^\delta y^\delta ,\ x^\delta [y,z],\ [x,y]^{\delta } \rangle _{T_{L}}$.

Remark 5

For any permutations σ ∈ S _t, we have

Proof

Let u ₁, u ₂, u ₃ be monomials. We consider w = u ₁ x _i x _j u ₂ y ^δ u ₃. Since x _i x _j = x _j x _i + [x _i, x _j], it follows that . In the same way we can show that . Hence in every monomial

$$\displaystyle \begin{aligned} x_{i_{1}}\dots x_{i_{t}}y^{\delta}z_{j_{1}}\dots z_{j_{p}} \end{aligned}$$

we can reorder the variables to the left and to the right of y ^δ. Since [x, y]^δ = [x ^δ, y] − [y ^δ, x], we can reorder all the variables in any commutator $[x^{\delta }_{i_{1}},x_{i_{2}},\dots ,x_{i_{t}}]$ as claimed. □

Lemma 3

The T _L -ideal of identities of $UT_2^\delta$ is generated by the following polynomials

$$\displaystyle \begin{aligned}{}[x,y][z,w],\; [x,y]^{\delta},\; x^\delta [y,z],\; x^{\delta}y^{\delta},\; x^{\delta^{2}}. \end{aligned}$$

Proof

Let $Q=\langle [x,y][z,w],\ [x,y]^{\delta },\ x^\delta [y,z],\ x^{\delta }y^{\delta },\ x^{\delta ^{2}}\rangle _{T_{L}}$. By Remark 3, $Q\subseteq \operatorname {\mathrm {Id}}^{L}(UT_{2}^\delta )$.

By the Poincaré-Birkhoff-Witt Theorem (see [12]) every differential multilinear polynomial in x ₁, …, x _n can be written as a linear combination of products of the type

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} x_{i_{1}}^{\alpha_{1}}\dots x_{i_{k}}^{\alpha_{k}} w_{1}\dots w_{m} \end{array} \end{aligned} $$

(6)

where α ₁, …, α _k ∈ U(L), w ₁…, w _m are left normed commutators in the $x_{i}^{\alpha _{j}}$s, α _j ∈ U(L), and i ₁ < ⋯ < i _k. Since $[x_{1}^{\alpha _{1}},x_{2}^{\alpha _{2}}][x_{3}^{\alpha _{3}},x_{4}^{\alpha _{4}}]\in Q$, with α ₁, α ₂, α ₃, α ₄ ∈{1, δ}, then, modulo $\langle [x_{1}^{\alpha _{1}},\ x_{2}^{\alpha _{2}}][x_{3}^{\alpha _{3}},x_{4}^{\alpha _{4}}],\ x^{\delta ^{2}}\rangle _{T_{L}}$, in (6) we have α _j ∈{1, δ} and m ≤ 1, so, only at most one commutator can appear in (6). Thus by Remark 4 every multilinear monomial in $P_{n}^{L}$ can be written, modulo Q, as linear combination of the elements of the type

$$\displaystyle \begin{aligned} x_{1}\dots x _{n},\quad x_{h_{1}}\dots x_{h_{n-1}}x_{j}^{\delta}, \quad x_{i_{1}}\dots x_{i_{k}}[x_{j_{1}}^{\gamma},x_{j_{2}},\dots,x_{j_{m}}],\end{aligned} $$

where h ₁ < ⋯ < h _n−1, i ₁ < ⋯ < i _k, m + k = n, m ≥ 2, γ ∈{1, δ}.

Let us now consider the left normed commutators $[x_{j_{1}}^{\gamma },x_{j_{2}},\dots ,x_{j_{m}}]$ and suppose first that γ = 1. Since [x ₁, x ₂][x ₃, x ₄] ∈ Q, then it is already known that (see for example [5, Theorem 4.1.5])

where k > h ₁ < ⋯ < h _m−1.

Suppose now γ = δ, then by Remark 5 we get

It follows that $P_{n}^{L}$ is spanned, modulo $P_{n}^{L} \cap Q$, by the polynomials

$$\displaystyle \begin{aligned} \begin{array}{rcl} & x_{1}\dots x_{n}, \qquad x_{i_{1}}\dots x_{i_{m}}[x_{k},x_{j_{1}},\dots,x_{j_{n-m-1}}], \\ & x_{h_{1}}\dots x_{h_{n-1}}x_{r}^{\delta},\qquad x_{i_{1}}\dots x_{i_{m}}[x_{l_{1}}^{\delta},x_{l_{2}},\dots,x_{l_{n-m}}],{} \end{array} \end{aligned} $$

(7)

where i ₁ < ⋯ < i _m, k > j ₁ < ⋯ < j _n−m−1, h ₁ < ⋯ < h _n−1, l ₁ < ⋯ < l _n−m, m ≠ n − 1, n.

Next we show that these polynomials are linearly independent modulo $\operatorname {\mathrm {Id}}^{L}(UT_{2} ^\delta )$. Let I = {i ₁, …, i _m} be a subset of {1, …, n} and k ∈{1, …, n}∖ I such that $k>\min ( \{1,\dots ,n\}\setminus I)$, then set $X_{I,k}=x_{i_{1}}\dots x_{i_{m}}[x_{k},x_{j_{1}},\dots ,x_{j_{n-m-1}}]$. Also for I ^′ = {i ₁, …, i _m}⊆{1, …, n}, 0 ≤|I ^′| < n − 1, set $X_{I^{\prime }}^{\delta }= x_{i_{1}}\dots x_{i_{m}}[x_{l_{1}}^{\delta },x_{l_{2}},\dots ,x_{l_{n-m}}]$ and suppose that

In order to show that all coefficients α _I,k, $\alpha _{I^{\prime }}^{\delta }$, $\alpha _{r}^{\delta }$, β are zero we will make some evaluations. If we evaluate x ₁ = ⋯ = x _n = e ₁₁ + e ₂₂ we get β = 0. For a fixed r, by setting $x_{h_{1}}=\dots =x_{h_{n-1}}=e_{11}+e_{22}$ and x _r = e ₂₂ we get $\alpha _{r}^{\delta }=0$. Also, for a fixed I ^′ = {i ₁, …, i _m}, by making the evaluations $x_{i_{1}}=\dots =x_{i_{m}}=e_{11}+e_{22}$, $x_{l_{1}}=\dots =x_{l_{n-m}}=e_{22}$ we obtain $\alpha _{I^{\prime }}^{\delta }=0$. Finally, for fixed I = {i ₁, …, i _m} and J = {j ₁, …, j _n−m−1}, from the substitutions $x_{i_{1}}=\dots =x_{i_{m}}=e_{11}+e_{22}$, x _k = e ₁₂, $x_{j_{1}}=\dots =x_{j_{n-m-1}}=e_{22}$, it follows that α _I,k = 0.

We have proved that $\operatorname {\mathrm {Id}}^{L}(UT_{2}^\delta )=Q$ and the elements in (7) are a basis of $P_{n}^{L}$ modulo $P_{n}^{L}\cap \operatorname {\mathrm {Id}}^{L}(UT_{2}^\delta )$. □

We now compute the nth differential cocharacter of $UT_2^\delta$. Write

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \chi_{n}^{L}(UT_{2}^\delta)=\sum_{\lambda\vdash n} m_{\lambda}^\delta \chi_{\lambda}. \end{array} \end{aligned} $$

(8)

In the following lemmas we compute the non-zero multiplicities of such cocharacter.

Lemma 4

In (8) $m_{(n)}^\delta \geq n+1$.

Proof

We consider the following tableau:

We associate to T _(n) the monomials

$$\displaystyle \begin{aligned} a(x)=x^{n}, \end{aligned} $$

(9)

$$\displaystyle \begin{aligned} a_{k}^{(\delta)}(x)=x^{k-1}x^{\delta}x^{n-k},\end{aligned} $$

(10)

for all k = 1, …, n. These monomials are obtained from the essential idempotents corresponding to the tableau T _(n) by identifying all the elements in the row. It is easily checked that a(x), $a_{k}^{(\delta )}(x)$, k = 1, …, n, do not vanish in $UT_{2}^{\delta }$.

Next we shall prove that the n + 1 monomials a(x), $a_{k}^{(\delta )}(x)$, k = 1, …, n, are linearly independent modulo $\operatorname {\mathrm {Id}}^{L}(UT_{2}^\delta )$. In fact, suppose that

By setting x = e ₁₁ + e ₂₂ it follows that α = 0. Moreover, if we substitute x = βe ₁₁ + e ₂₂ where β ∈ F, β ≠ 0, we get $\sum _{k=1}^{n} (1-\beta )\beta ^{k-1}\alpha ^{\delta }_{k}=0$. Since |F| = ∞, we can choose β ₁, …, β _n ∈ F, where β _i ≠ 0 and β _i ≠ β _j, for all 1 ≤ i ≠ j ≤ n. Then we get the following homogeneous linear system of n equations in the n variables $\alpha ^{\delta }_{k}$, k = 1, …, n,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \sum_{k=1}^{n}\beta_{i}^{k-1}\alpha^{\delta}_{k}=0,\quad i=1,\dots,n. \end{array} \end{aligned} $$

(11)

Since the matrix associated to the system (11) is a Vandermonde matrix, it follows that $\alpha ^{\delta }_{k}=0$, for all k = 1, …, n. Thus the monomials a(x), $a_{k}^{(\delta )}(x)$, k = 1, …, n, are linearly independent modulo $\operatorname {\mathrm {Id}}^{L}(UT_{2}^\delta )$. This says that $m_{(n)}^\delta \geq n+1$. □

Lemma 5

Let p ≥ 1 and q ≥ 0. If λ = (p + q, p) then in (8) we have $m_{\lambda }^\delta \geq 2(q+1)$.

Proof

For every i = 0, …, q we define $T_{\lambda }^{(i)}$ to be the tableau

We associate to $T_{\lambda }^{(i)}$ the polynomials

$$\displaystyle \begin{aligned} b^{(p,q)}_{i}(x,y)=x^{i}\underbrace{\overline{x}\dots\widetilde{x}}_{p-1}[x,y]\underbrace{\overline{y}\dots\widetilde{y}}_{p-1}x^{q-i}, \end{aligned} $$

(12)

$$\displaystyle \begin{aligned} b^{(p,q,\delta)}_{i}(x,y)=x^{i}\underbrace{\overline{x}\dots\widetilde{x}}_{p-1}(x^{\delta}y-y^{\delta}x)\underbrace{\overline{y}\dots\widetilde{y}}_{p-1}x^{q-i}, \end{aligned} $$

(13)

where the symbols − or $\thicksim$ means alternation on the corresponding variables. The polynomials $b^{(p,q)}_{i}$, $b^{(p,q,\delta )}_{i}$ are obtained from the essential idempotents corresponding to the tableau $T_{\lambda }^{(i)}$ by identifying all the elements in each row of the tableau. It is clear that $b^{(p,q)}_{i}$, $b^{(p,q,\delta )}_{i}$, i = 0, …, q, are not differential identities of $UT_{2}^{\delta }$. We shall prove that the above 2(q + 1) polynomials are linearly independent modulo $\operatorname {\mathrm {Id}}^{L}(UT_{2}^\delta )$. Suppose that

If we set x = βe ₁₁ + e ₂₂, with β ∈ F, β ≠ 0, and y = e ₁₁, we obtain

$$\displaystyle \begin{aligned} \sum_{i=0}^{q}(-1)^{p-1}\beta^{i} \alpha^{\delta}_{i}=0. \end{aligned}$$

Since |F| = ∞, we can take β ₁, …, β _q+1 ∈ F, where β _j ≠ 0, β _j ≠ β _k, for all 1 ≤ j ≠ k ≤ q + 1. Then we obtain the following homogeneous linear system of q + 1 equations in the q + 1 variables $\alpha ^{\delta }_{i}$, i = 0, …, q,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \sum_{i=0}^{q}\beta_{j}^{i}\alpha^{\delta}_{i}=0,\quad j=1,\dots,q+1. \end{array} \end{aligned} $$

(14)

Since the matrix of this system is a Vandermonde matrix, it follows that $\alpha ^{\delta }_{i}=0$, for all i = 0, …, q. Hence we may assume that the following identity holds

If we evaluate x = βe ₁₁ + e ₁₂ + e ₂₂, where β ∈ F, β ≠ 0, and y = e ₁₁, then we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \sum_{i=0}^{q}(-1)^{p-1}\beta^{i}\alpha_{i}=0. \end{array} \end{aligned} $$

(15)

Since |F| = ∞, we choose β ₁, …, β _q+1 ∈ F, where β _j ≠ 0, β _j ≠ β _k, for all 1 ≤ j ≠ k ≤ q + 1. Then from (15) we obtain a homogeneous linear system of q + 1 equations in the q + 1 variables α _i, i = 0, …, q, equivalent to the linear system (14). Therefore α _i = 0, for all i = 0, …, q. Hence the polynomials $b^{(p,q)}_{i}$, $b^{(p,q,\delta )}_{i}$, i = 0, …, q, are linearly independent modulo $\operatorname {\mathrm {Id}}^{L}(UT_{2}^\delta )$ and, so, $m_{\lambda }^\delta \geq 2(q+1)$. □

As an immediate consequence of Remark 1 and Theorem 6 we have the following.

Lemma 6

Let p ≥ 1 and q ≥ 0. If λ = (p + q, p, 1), then in (8) we have $m_{\lambda }^\delta \geq q+1$.

We are now in a position to prove the following theorem about the L-algebra $UT_2^\delta$.

Theorem 10

• 1. $\operatorname {\mathrm {Id}}^{L}(UT_2^\delta )=\langle [x,y][z,w],\ [x,y]^{\delta },\ x^\delta [y,z],\ x^{\delta }y^{\delta },\ x^{\delta ^{2}}\rangle _{T_{L}}$.

• 2. $c_n^{L}(UT_2^\delta )=2^{n-1}n+1$.

• 3. If $\chi _{n}^{L}(UT_{2}^\delta )=\sum _{\lambda \vdash n} m_{\lambda }^\delta \chi _{\lambda }$ is the nth differential cocharacter of $UT_{2}^{\delta }$ , then

  $$\displaystyle \begin{aligned} m_{\lambda}^\delta =\begin{cases} n+1, & \mathit{\mbox{ if }} \lambda=(n) \\ 2(q+1), & \mathit{\mbox{ if }} \lambda=(p+q,p) \\ q+1, & \mathit{\mbox{ if }} \lambda=(p+q,p,1) \\ 0 & \mathit{\mbox{ in all other cases}} \end{cases}. \end{aligned} $$

  (16)

Proof

By Lemma 3 the T _L-ideal of differential identities of $UT_2^\delta$ is generated by the polynomials $[x,y][z,w],[x,y]^{\delta },x^\delta [y,z], x^{\delta }y^{\delta }, x^{\delta ^{2}}$ and the elements in (7) are a basis of $P_{n}^{L}$ modulo $P_{n}^{L}\cap \operatorname {\mathrm {Id}}^{L}(UT_{2}^\delta )$. Thus by counting these elements we get that $c_n^{L}(UT_2^\delta )=2^{n-1}n+1$.

Finally, as a consequence of Lemmas 4, 5, 6 and by following verbatim the proof of [4, Theorem 12] we get the decomposition into irreducible characters of $\chi _{n}^{L}(UT_{2}^\delta )$. □

Notice that $\operatorname {\mathrm {var}}^L(UT_2^\delta )$ has exponential growth, nevertheless it has no almost polynomial growth. In fact, the algebra UT ₂ (ordinary case) is an algebra with Fδ-action where δ acts trivially on UT ₂, i.e., x ^δ ≡ 0 is differential identity of UT ₂. Then it follows that $UT_2\in \operatorname {\mathrm {var}}^L(UT_2^\delta )$, but $\operatorname {\mathrm {var}}^L(UT_2)$ growths exponentially. Thus we have the following result.

Theorem 11

$\operatorname {\mathrm {var}}^L(UT_2^\delta )$ has no almost polynomial growth.

Now denote by $UT_2 ^\eta$ the L-algebra UT ₂ where L acts on it as the 1-dimensional Lie algebra spanned by a non-trivial derivation η of UT ₂. Notice that since any derivation of UT ₂ is inner (see [2]), it can be easily checked that the algebra $\operatorname {{\mathrm {Der}}}(UT_2)$ of all derivations of UT ₂ is the 2-dimensional metabelian Lie algebra with basis {ε, δ} defined in (4) and (5), respectively. Thus

$$\displaystyle \begin{aligned} \eta =\alpha \ \varepsilon + \beta \ \delta, \quad \mbox{for some } \alpha, \beta \in F \mbox{ not both zero} . \end{aligned}$$

Remark 6

$[x,y]^{\eta }-\alpha [x,y]\equiv 0 ,\ x^{\eta }y^{\eta } \equiv 0,\ x^{\eta ^{2}}-\alpha x^{\eta }\equiv 0, \ [x,y][z,w]\equiv 0, \ x^\eta [y,z]\equiv 0$ are differential identities of $UT_2 ^\eta$. Moreover, if α ≠ 0, then $[x,y][z,w], \ x^\eta [y,z] \in \langle [x,y]^{\eta }-\alpha [x,y],\ x^{\eta }y^{\eta }\rangle _{T_{L}}$.

We do not present the proof of next theorem since it can be deduced by using the strategy of proofs given in [4, Theorems 5 and 12] and Theorem 10.

Theorem 12

• 1. If α ≠ 0, then $\operatorname {\mathrm {Id}}^{L}(UT_2^{\eta })=\langle [x,y]^{\eta }-\alpha [x,y],\ x^{\eta }y^{\eta },\ x^{\eta ^{2}}-\alpha x^{\eta }\rangle _{T_{L}}$ . Otherwise, $\operatorname {\mathrm {Id}}^{L}(UT_2^{\eta })=\langle [x,y][z,w], \ x^\eta [y,z], \ [x,y]^{\eta },\ x^{\eta }y^{\eta },\ x^{\eta ^{2}}\rangle _{T_{L}}$.

• 2. $c_n^{L}(UT_2^{\eta })=2^{n-1}n+1$.

• 3. If $\chi _{n}^{L}(UT_{2}^\eta )=\sum _{\lambda \vdash n} m_{\lambda }^\eta \chi _{\lambda }$ is the nth differential cocharacter of $UT_{2}^{\eta }$ , then

  $$\displaystyle \begin{aligned} m_{\lambda}^{\eta}=\begin{cases} n+1, & \mathit{\mbox{ if }} \lambda=(n) \\ 2(q+1), & \mathit{\mbox{ if }} \lambda=(p+q,p) \\ q+1, & \mathit{\mbox{ if }} \lambda=(p+q,p,1) \\ 0 & \mathit{\mbox{ in all other cases}} \end{cases}. \end{aligned}$$

Notice that if α = 0, $\operatorname {\mathrm {var}}^L( UT_2 ^\eta )= \operatorname {\mathrm {var}}^L (UT_2 ^\delta )$. Thus by Theorem 11 and by following closely the proof of [4, Theorem 15], taking into account the due changes, we get the following.

Theorem 13

If α ≠ 0, then $\operatorname {\mathrm {var}}^L(UT_2^\eta )$ has almost polynomial growth. Otherwise it has no almost polynomial growth.

Finally let us assume that L is a 2-dimensional metabelian Lie algebra and denote by $UT_2^D$ the L-algebra UT ₂ where L acts on it as the Lie algebra $\operatorname {{\mathrm {Der}}}(UT_2)$. Giambruno and Rizzo in [4] proved the following result.

Theorem 14 ([4, Theorems 19 and 25])

• 1. $\operatorname {\mathrm {Id}}^{L}(UT_{2}^D)=\langle [x,y]^{\varepsilon }-[x,y],\ x^{\varepsilon }y^{\varepsilon },\ x^{\varepsilon ^{2}}-x^{\varepsilon },\ x^{\delta \varepsilon },\ x^{\varepsilon \delta }- x^{\delta }\rangle _{T_{L}}$.

• 2. $c_{n}^{L}(UT_{2}^D)=2^{n-1}(n+2).$

• 3. If $\chi _{n}^{L}(UT_{2}^D)=\sum _{\lambda \vdash n} m_{\lambda }^D \chi _{\lambda }$ is the nth differential cocharacter of $UT_{2}^{D}$ , then

  $$\displaystyle \begin{aligned} m_{\lambda}^{D}=\begin{cases} 2n+1, & \mathit{\mbox{ if }} \lambda=(n) \\ 3(q+1), & \mathit{\mbox{ if }} \lambda=(p+q,p) \\ q+1, & \mathit{\mbox{ if }} \lambda=(p+q,p,1) \\ 0 & \mathit{\mbox{ in all other cases}} \end{cases}. \end{aligned}$$

Since x ^δ ≡ 0 is a differential identity of $UT_2 ^\varepsilon$, $\operatorname {\mathrm {var}}^L(UT_2 ^{\varepsilon }) \subseteq \operatorname {\mathrm {var}}^L (UT_2^D)$. Then by Theorem 8, we have the following.

Theorem 15 ([4, Theorem 26])

$\operatorname {\mathrm {var}}^L(UT_2^D)$ has no almost polynomial growth.

5 On Differential Identities of the Grassmann Algebra

In this section we present an example of infinite dimensional algebra with derivations of exponential growth.

Let L be a finite dimensional abelian Lie algebra and G the infinite dimensional Grassmann algebra over F. Recall that G is the algebra generated by 1 and a countable set of elements e ₁, e ₂, … subjected to the condition e _i e _j = −e _j e _i, for all i, j ≥ 1.

Notice that G can be decomposed in a natural way as the direct sum of the subspaces

$$\displaystyle \begin{aligned} G_{0}=\operatorname{\mathrm{span}}_{F} \{e_{i_{1}}\dots e_{i_{2k}} \ \vert \ i_{1}<\dots < i_{2k},k\geq 0\} \end{aligned}$$

and

$$\displaystyle \begin{aligned} G_{1}=\operatorname{\mathrm{span}}_{F} \{e_{i_{1}}\dots e_{i_{2k+1}} \ \vert \ i_{1}< \dots < i_{2k+1},k\geq 0\}, \end{aligned}$$

i.e., G = G ₀ ⊕ G ₁.

Now consider the algebra G where L acts trivially on it. Since x ^γ ≡ 0, for all γ ∈ L, is a differential identity of G, we are dealing with ordinary identities. Thus by Krakowski and Regev [10] we have the following results.

Theorem 16

• 1. $\operatorname {\mathrm {Id}}^L(G)=\langle [x,y,z]\rangle _{T}$.

• 2. $c^L _{n}(G)=2^{n-1}.$

• 3. $\chi _{n}^L(G)=\sum _{j=1}^{n} \chi _{(j, 1^{n-j})}$.

Theorem 17

$\operatorname {\mathrm {var}}^L(G)$ has almost polynomial growth.

Recall that if $g=e_{i_{1}}\dots e_{i_{n}}\in G$, the set $\operatorname {{\mathrm {Supp}}}\{g\}=\{e_{i_{1}},\dots ,e_{i_{n}}\}$ is called the support of g. Let now g ₁, …, g _t ∈ G ₁ be such that $\operatorname {{\mathrm {Supp}}}\{g_{i}\}\cap \operatorname {{\mathrm {Supp}}}\{g_{j}\}=\emptyset$, for all i, j ∈{1, …, t}. We set

$$\displaystyle \begin{aligned} \delta_{i}=2^{-1}\operatorname{\mathrm{ad}} g_{i},\quad i=1,\dots,t. \end{aligned}$$

Then for all g ∈ G we have

$$\displaystyle \begin{aligned} \delta_{i}(g)=\begin{cases} 0, &\mbox{ if } g\in G_{0} \\ g_{i}g , &\mbox{ if } g\in G_{1} \end{cases}, \quad i=1,\dots,t. \end{aligned}$$

Since for all g ∈ G, [δ _i, δ _j](g) = 0, i, j ∈{1, …, t}, $L= \operatorname {\mathrm {span}}_{F}\{\delta _{1},\dots ,\delta _{t}\}$ is a t-dimensional abelian Lie algebra of inner derivations of G. We shall denote by $\widetilde {G}$ the algebra G with this L-action.

Recall that for a real number x we denote by ⌊x⌋ its integer part.

Theorem 18 ([13, Theorems 3 and 9])

• 1. $\operatorname {\mathrm {Id}}^{L}(\widetilde {G})=\langle [x,y,z],\ [x^{\delta _i}, y],\ x^{\delta _i \delta _j} \rangle _{T_{L}}$ , i, j = 1, …, t.

• 2. $c_{n}^{L}(\widetilde {G})=2^{t}2^{n-1}-\sum _{j=1}^{\lfloor t/2\rfloor }\sum _{i=2j}^{t}\binom {t}{i}\binom {n}{i-2j}.$

• 3. If $\chi _{n}^{L}(\widetilde {G})=\sum _{\lambda \vdash n} m_{\lambda }^{L}\chi _{\lambda }$ is the nth differential cocharacter of $\widetilde {G}$ , then

  $$\displaystyle \begin{aligned} m_{\lambda}^L=\begin{cases} \sum_{i=0}^{r}\binom{t}{i}, & \mathit{\mbox{ if }} \lambda=(n-r+1,1^{r-1}) \mathit{\mbox{ and }} r<t \\ 2^{t}, & \mathit{\mbox{ if }} \lambda=(n-r+1,1^{r-1}) \mathit{\mbox{ and }} r\geq t\\ 0 & \mathit{\mbox{ in all other cases}} \end{cases}. \end{aligned}$$

Recall that two functions φ ₁(n) and φ ₂(n) are asymptotically equal and we write φ ₁(n) ≈ φ ₂(n) if lim_n→∞ φ ₁(n)∕φ ₂(n) = 1. Then the following corollary is an obvious consequence of the previous theorem.

Corollary 1

$c_{n}^{L}(\widetilde {G})\approx 2^{t}2^{n-1}$.

Notice that by Corollary 1 $\operatorname {\mathrm {var}}^L(\widetilde {G})$ has exponential growth, nevertheless it has no almost polynomial growth. In fact, the Grassmann algebra G (ordinary case) is an algebra with L-action where δ _i, i = 1, …, t, acts trivially on G, i.e., $x^{\delta _{i}}\equiv 0$, i = 1, …, t, are differential identities of G. Then it follows that $G\in \operatorname {\mathrm {var}}^L(\widetilde {G})$, but by Theorem 16 $c_n^L(G)=2^{n-1}$. Thus we have the following result.

Theorem 19 ([13, Theorem 6])

$\operatorname {\mathrm {var}}^{L}(\widetilde {G})$ has no almost polynomial growth.