Abstract
We propose a general procedure to construct noncommutative deformations of an algebraic submanifold M of \(\mathbb {R}^{n}\), specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of \(\mathbb {R}^{n}\), arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883–1911, 2006), whereby the commutative pointwise product is replaced by the ⋆-product determined by a Drinfel’d twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds Mc that are level sets of the fa(x), where fa(x) = 0 are the polynomial equations solved by the points of M, employing twists based on the Lie algebra Ξt of vector fields that are tangent to all the Mc. The twisted Cartan calculus is automatically equivariant under twisted Ξt. If we endow \(\mathbb {R}^{n}\) with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted M is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and ⋆-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in \(\mathbb {R}^{3}\) except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean \(\mathbb {R}^{3}\) and twisted hyperboloids embedded in twisted Minkowski \(\mathbb {R}^{3}\) [the latter are twisted (anti-)de Sitter spaces dS2, AdS2].
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Acknowledgments
We are indebted with F. D’Andrea for stimulating discussions, critical advice and useful suggestions at all stages of the work. The third author would like to thank P. Aschieri for his kind hospitality at the University of Eastern Piedmont “Amedeo Avogadro”.
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Appendices
Appendix A: Real Nullstellensatz
First of all, we recall some basic notions and notation in algebraic geometry that we are using in this subsection. In what follows we fix a ground field \(\mathbb {K}\) of any characteristic (even though we work only over real and complex fields, all the notions and definitions we are going to review hold true in a much wider generality).
-
1.
(Algebraic Sets [38, p. 2]) A subset of \(\mathbb {K}^{n}\) is an algebraic set if it is defined as the set of common solutions of a system of polynomial equations. By Hilbert basis theorem [46, Theorem 3.3], algebraic sets can be also defined as
$$ Z(I):=\{x\in \mathbb{K}^{n} \mid P(x)=0, \forall P\in I \}, $$where I denotes an ideal of the polynomial ring \(\mathbb {K}[x^{1}, \dots , x^{n}]\).
-
2.
(Zariski topology [38, p. 2]) The affine space \(\mathbb {K}^{n}\) can be endowed with a topology, the so called Zariski topology, where closed sets coincide with algebraic sets. In this section we will equip algebraic sets with the induced topology.
-
3.
(Algebraic, or affine, varieties) An algebraic variety is an irreducible algebraic set, i.e. an algebraic set which is not the union of two proper (i.e. strictly contained) closed subsets. It turns out [38, Exercise I.1.6] that a non-empty open set of an affine variety is irreducible.
-
4.
(Decomposition of algebraic sets) An algebraic set M can be expressed uniquely as a union of varieties, no one containing another [38, Corollary 1.6]. Such varieties are called irreducible components of M.
-
5.
(Radicals) For any ideal \(I\leq \mathbb {K}[x^{1}, \dots , x^{n}]\) the radical of I [46, p. 3], [33, Section 1.3] is the ideal defined as
$$\text{Rad} (I):=\{P\in \mathbb{K}[x^{1}, \dots, x^{n}]: \exists k>0 \mid P^{k}\in I \}.$$A radical ideal is an ideal I s.t. I = Rad(I). By the very definition of prime ideal [46, p. 2], any such an ideal is radical.
-
6.
(Correspondence among varieties and prime ideals) An affine algebraic set is a variety if and only if its ideal is a prime ideal [38, Corollary 1.4].
-
7.
(Associated primes) Consider an algebraic set M = Z(I). An associated prime is a prime ideal of \(\mathbb {K}[x^{1}, \dots , x^{n}]\) which is the annihilator ann(x) of some element \(x\in \frac {{\mathbb {K}}[x^{1}, \dots , x^{n}]}{I}\). It turns out [46, Section 6] that there are two kinds of associated primes: minimal associated primes (they are in one to one correspondence with the irreducible components of M), and embedded associated primes (they have NOT a simple geometric interpretation).
-
8.
(Hilbert’s Nullstellensatz [46, Section 5]) Assume now \(\mathbb {K}\) algebraically closed and define
$$ {\mathcal{I}}(S):=\{P\in \mathbb{K}[x^{1}, \dots, x^{n}]\mid P\mid_{S} \equiv 0 \}, $$for any subset \(S\subseteq \mathbb {K}^{n}\). Then we have
$$ {\mathcal{I}}(Z(I))= \text{Rad} (I), \forall I\leq \mathbb{K}[x^{1}, \dots, x^{n}]. $$A weak form of this result says that Z(I)≠∅, for any proper ideal \(I\leq \mathbb {K}[x^{1}, \dots , x^{n}]\) [33, Section 1.7].
-
9.
(Regular sequences) A set of polynomials \(P_{1}, {\dots } , P_{k}\) form a regular sequence in \(\frac {\mathbb {K}[x^{1}, \dots , x^{n}]}{I}\) [46, Section 16], if every Pi is not a zero divisor in \(\frac {\mathbb {K}[x^{1}, \dots , x^{n}]}{(I, P_{1}, {\dots } , P_{i-1})}\).
-
10.
(Cohen-Macaulay property [46, Section 17]) An affine variety M = Z(I), such that \(\dim M=m\), is said Cohen-Macaulay at x ∈ M if there is a regular sequence \(P_{1}, {\dots } , P_{m}\) in \(\frac {{\mathbb {K}}[x^{1}, \dots , x^{n}]}{I}\), such that Pi(x) = 0, ∀i. An affine variety M = Z(I) is said Cohen-Macaulay if it is Cohen-Macaulay at any point.
Now consider an algebraic submanifold, i.e. a smooth algebraic variety, \(M\subseteq \mathbb {R}^{n}\), defined by a system of polynomial (1), with \(f^{1}, {\dots } , f^{k}\in \mathbb {R}[x^{1}, \dots , x^{n}]\). Assume that \(\dim M=n-k\). Then, the hypersurfaces defined by each of the equations in (1) meet transversally et each point of M; in other words, the Jacobian matrix is of rank k at each point of M. Consider a further polynomial \(Q\in {\mathbb {R}}[x^{1}, \dots , x^{n}]\) and assume that
One may wonder whether the irreducibility in \(\mathbb {R}[x^{1}, \dots , x^{n}]\) of each polynomial \(f^{1}, {\dots } , f^{k}\) is a sufficient condition in order that Q lies in \((f^{1}, {\dots } , f^{k})\), the ideal generated by \(f^{1}, {\dots } , f^{k}\). The following example answers in the negative.
Example 11
Consider in \(\mathbb {R}^{3}\) the variety defined by the system
where the first equation represents a cubic cylinder C. Since the curve defined by
is smooth in \({\mathbb {P}_{\mathbb {C}}^{2}}\), the cylinder C is smooth and the polynomial 2x3 − y3 − 1 is irreducible in \(\mathbb {R} [x,y,z]\) (the same conclusion is obvious for y − 1). The real variety defined by (173) is the line
which is obviously smooth. Furthermore, the equation of the tangent plane to the cylinder C at the point (1, 1, t) ∈ l is 2(x − 1) − (y − 1) = 0, hence the intersection is transversal at each point of l. On the other hand, the plane π defined by x + y − 2 = 0 contains l but
since both 2x3 − y3 − 1 and y − 1 do vanish at the points \((\exp {\frac {2}{3}\pi i},1,t)\), \(\forall t\in \mathbb {R}\), and conversely x + y − 2 does not. In view of the previous example, it is interesting to ask for some sufficient condition in order that \(Q\in (f^{1}, {\dots } , f^{k})\). An answer is provided by Theorem 1, which we now prove.
Proof of Theorem 1
Denote by
the ideal of \(\mathbb {C}[x^{1}, \dots , x^{n}]\) generated by \(f^{1}, {\dots } , f^{k}\). Since we are assuming that the zero locus of I is irreducible, there is only one minimal prime associated to the ideal I. The hypothesis that the hypersurfaces corresponding to the generators of I meet transversally at M imply that \(f^{1}, {\dots } , f^{k}\) form a regular sequence in \({\mathbb {C}}[x^{1}, \dots , x^{n}]\) [46, Section 16], hence the zero locus Z(I) of I is a complete intersection, i.e. an affine variety defined as the intersection of as many hypersurfaces as its codimension. This implies that Z(I) is Cohen-Macaulay, hence there is no embedded associated prime [46, Theorem 17.3] and the ideal I is primary, i.e. there is only one associated prime. Again, the hypothesis that the hypersurfaces defined by the equations in (1) meet transversally et each point of M imply that I is a prime ideal in \(\mathbb {C}[x^{1}, \dots , x^{n}]\).
On the other hand, by Hilbert’s Nullstellensatz [5, Exercise 7.14], [33, Section 1.7], [46, Theorem 5.4], the hypothesis Q∣M ≡ 0 amounts to
where Rad(I) denotes the radical of I [5, Exercise 1.12], [33, Section 1.3] and where the last equality follows because I is prime. This shows that \(Q\in I\cap {\mathbb {R}}[x^{1}, \dots , x^{n}]= (f^{1}, {\dots } , f^{k})\). Finally, for a complex-valued h = Q1 + iQ2 vanishing on M both Q1, Q2 do, and therefore h belongs to the complexification of \((f^{1}, {\dots } , f^{k})\).
As for the last statement, the projective closure \(X\subseteq {\mathbb {P}_{\mathbb {C}}^{n}}\) of the zero locus of (1) in \({\mathbb {P}_{\mathbb {C}}^{n}}\) has degree at least s. On the other hand, s is the maximum degree so X is a complete intersection in \({\mathbb {P}_{\mathbb {C}}^{n}}\). Then, there cannot be other components and the variety defined by (1) is irreducible in \({\mathbb {P}_{\mathbb {C}}^{n}}\). The statement follows at once, since a non-empty open set of an irreducible variety is irreducible [38, Exercise I.1.6]. □
Remark 12
Consider an algebraic smooth hypersurface M, defined by a single equation f(x) = 0, with \(d:= \deg f\). By the result above, in order that any polynomial h, such that h∣M ≡ 0, is a multiple of f it suffices that there exists a line meeting M in d points.
Appendix B: Proofs of Sections 2, 3, 4
1.1 B.1 Proof of Proposition 2
Using the definition \(\beta :={\mathcal {F}}_{1}\cdot S({\mathcal {F}}_{2})\), \(a^{\prime }\star a = ({\overline {\mathcal {R}}}_{1}\triangleright a)\star ({\overline {\mathcal {R}}}_{2}\triangleright a^{\prime })\) and the relation
valid for all triangular Hopf algebras, one can prove relations (90–91) as follows:
By (85) the action of either leg \({\mathcal {F}}_{1},{\mathcal {F}}_{2}\) of the twist, or \(\overline {\mathcal {F}}_{1},\overline {\mathcal {F}}_{2}\) of its inverse, as well as of any tensor factor in the (iterated) coproducts of \({\mathcal {F}}_{1},{\mathcal {F}}_{2},\overline {\mathcal {F}}_{1},\overline {\mathcal {F}}_{2}\), maps every homogeneous polynomial in ξi or ∂i into another one of the same degree, and every polynomial in xi into another one of the same degree: hence (92), (93) follow. Finally, the relations \(g\triangleright \partial _{i}^{\prime }=\tau ^{ij}[S_{{\mathcal F}}(g)]\partial _{j}^{\prime }\), (94) are straightforward consequences of (19), (29), (85):
1.2 B.2 Proof of Proposition 3
All statements up to (106) and the statement that the ⋆-polynomials \(\hat {f_{c}^{J}}(a\star )\) have the same degrees in xi, ξi, eα as the polynomials \({f_{c}^{J}}(a)\) are straightforward consequences of (85) and of what precedes the proposition. Under \(U{\mathfrak {g}}\) the fi transform as the ∂i; in fact, since g⊳f = ε(g)f, we find g ⊳ fi = g ⊳ (∂if − f∂i) = (g ⊳ ∂i)f − f(g ⊳ ∂i) = τih(Sg)(∂hf − f∂h) = τih(Sg)fh. Hence
this can be computed more explicitly using the relation (see e.g. Equation (126) in [29])
1.3 B.3 (a) Family of Parabolic Cylinders
1.3.1 Proof of Proposition 4
Since L13 and L23 are commuting anti-Hermitian vector fields it follows that \(\mathcal {F}\) is a unitary abelian twist on \(U{\mathfrak {g}}\). We find \(S(\beta )=\exp (-i\nu L_{12}L_{23})\), and
for all n > 0, since [L13, L12] = L23, [L13, L23] = 0. This implies
where in the last equation we use \(\mathcal {F}({\textbf {1}}\otimes L_{12}) =({\textbf {1}}\otimes L_{12})\mathcal {F}\) since the second leg of the twist is central. Moreover \(\mathcal {F}(L_{13}\otimes {\textbf {1}}) ={\sum }_{n=0}^{\infty }\frac {(i\nu )^{n}}{n!}L_{13}^{n+1}\otimes L_{23}^{n} =(L_{13}\otimes {\textbf {1}})\mathcal {F}\) and \(\mathcal {F}(L_{23}\otimes {\textbf {1}}) =(L_{23}\otimes {\textbf {1}})\mathcal {F}\) show that \({\Delta }_{{\mathcal F}}(L_{13})={\Delta }(L_{13})\) and \({\Delta }_{{\mathcal F}}(L_{23})={\Delta }(L_{23})\). We have thus proved the claimed coproducts \({\Delta }_{{\mathcal F}}(L_{ij})\). Next, the latter and the antipode property \( \mu [(S_{{\mathcal F}}\otimes \text {id}){\Delta }_{{\mathcal F}}(g)] =\epsilon (L_{ij}){\textbf {1}}=0\) easily determine the twisted antipode \(S_{{\mathcal F}}(L_{12})\) as in (110), and other ones \(S_{{\mathcal F}}(L_{ij})=S(L_{ij})=-L_{ij}\). Furthermore, since the L23 contained in the second leg of the twist commutes with Lkℓ we conclude that Lij ⋆ Lkℓ = LijLkℓ and [Lij, Lkℓ]⋆ = [Lij, Lkℓ] for all 1 ≤ i < j ≤ 3 and 1 ≤ k < ℓ ≤ 3. For the same reason one gets for all 1 ≤ i, j, k ≤ 3
On the other hand by (109) we obtain
for all 1 ≤ i, j, k ≤ 3. The commutation relations respectively follow. Furthermore this means we can express the generators of the Lie algebra in terms of the twisted module action, namely L12 = x1∂2 = x1 ⋆ ∂2, L13 = x1∂3 + b∂1 = x1 ⋆ ∂3 + b∂1 and L23 = b∂2, while \(f_{c}(x)=\frac {1}{2}(x^{1})^{2}-bx^{3}-c =\frac {1}{2}x^{1}\star x^{1}-bx^{3}-c\), dfc = x1ξ1 − bξ3 = x1 ⋆ ξ1 − bξ3 and
hold. Again by (109) L12L23 ⊳ xi = 0, L12L23 ⊳ ξi = 0, L12L23 ⊳ ∂i = 0 for all i = 1, 2, 3, which implies that the ∗-structure on \({{\mathcal Q}}^{\bullet }_{\star }\) remains the same as on \({{\mathcal Q}_{M_{c}}^{\bullet }}\): ∗⋆ = ∗.
1.4 B.4 (b) Family of Elliptic Paraboloids
1.4.1 Proof of Proposition 5
Since L13, L23 are commuting anti-Hermitian vector fields \(\mathcal {F}\) is a unitary abelian twist on \(U{\mathfrak {g}}\). By a direct calculation one finds \( \beta =S(\beta ) =\mathcal {F}_{2}S(\mathcal {F}_{1}) =\exp (-i\nu L_{13}L_{23}). \) The commutation relations (114) also imply \(\mathcal {F}{\Delta }(L_{13})={\Delta }(L_{13})\mathcal {F}\), \(\mathcal {F}{\Delta }(L_{23})={\Delta }(L_{23})\mathcal {F}\), resulting in \({\Delta }_{{\mathcal F}}(L_{13})={\Delta }(L_{13})\) and \({\Delta }_{{\mathcal F}}(L_{13})={\Delta }(L_{13})\), respectively. Moreover,
for n > 0, which follow by iteratively applying (114). Then
imply (117)1. The twisted antipodes (117)2 follow using the properties \(\mu \circ (S_{{\mathcal F}}\otimes \text {id})\circ {\Delta }_{{\mathcal F}} =\eta \circ \epsilon =\mu \circ (\text {id}\otimes S_{{\mathcal F}})\circ {\Delta }_{{\mathcal F}}\). The twisted tensor and star products coincide with the untwisted ones as soon as one of the factors is L13 or L23. This is because the latter commute with both legs of the twist. Among all star products of generators of \({\mathfrak {g}}\) only the one
is different. By a similar direct calculation one can prove (118). The latter imply (119-120) and that the submanifold constraints coincide with their twisted analogues, namely (121) holds. The twisted star involutions coincide with the untwisted ones, since \(L_{13}L_{23}\rhd x^{i}=L_{13}L_{23}\rhd \xi ^{i}=L_{13}L_{23}\rhd \partial _{i}=0\). This concludes the proof of the proposition.
1.5 B.5 (c) Family of Elliptic Cylinders
1.5.1 Proof of Proposition 16 in [32], see Section 4.3
Since [∂3, L12] = 0 and ∂3, L12 are anti-Hermitian, \(\mathcal {F}=\exp (i\nu \partial _{3}\otimes L_{12})\) is a unitary abelian twist on \(U{\mathfrak {g}}\). As ∂3, L12 commute with both legs of the twist, \({\Delta }_{\mathcal {F}}(\partial _{3})={\Delta }(\partial _{3})\), \({\Delta }_{{\mathcal F}}(L_{12})={\Delta }(L_{12})\), \(S_{{\mathcal F}}(\partial _{3})=S(\partial _{3})\), \(S_{{\mathcal F}}(L_{12})=S(L_{12})\) and all twisted tensor and star products as well as Lie brackets where one of the factors is ∂3, L12 coincide with the untwisted ones. Furthermore the star products of xi, ξj, ∂k, L12 coincide with the classical ones unless the first leg of the twist acts on x3. Consequently, (115)b=0 implies (124) and the equations (125) coincide with their classical analogues. The twisted star involutions are trivial since \(\partial _{3}L_{12}\rhd x^{i}=\partial _{3}L_{12}\rhd \xi ^{i} =\partial _{3}L_{12}\rhd \partial _{i}=0\). This concludes the proof.
1.6 B.6 (d) Family of Hyperbolic Paraboloids
1.6.1 Proof of Proposition 7
The anti-Hermitian vector fields H and E satisfy (133), which implies that \(\mathcal {F}=\exp (H/2\otimes \log ({\textbf {1}}+i\nu E))\) is a unitary Jordanian twist on \(U\mathfrak {g}\). We note that
for all m > 0, which follows by iteratively applying (133). In particular this implies
where we have made use of the expansions
Both are well-defined in the ν-adic topology. Applying this result iteratively we obtain
for all n > 0, whence
This and \({\mathcal {F}}(H{\otimes } {\textbf {1}})=(H{\otimes } {\textbf {1}}){\mathcal {F}}\) determine \({\Delta }_{{\mathcal F}}(H)\) as in (137). For the twisted coproduct of E we first remark that \(\left (\frac {H}{2}\right )^{n}E =E\left (\frac {H}{2}+{\textbf {1}}\right )^{n}\) for all n ≥ 0, which is proven by induction. Then
This and \({\mathcal {F}}({\textbf {1}}{\otimes } E)=({\textbf {1}}{\otimes } E){\mathcal {F}}\) determine \({\Delta }_{{\mathcal F}}(E)\) as in (137). Similarly one proves
and \({\mathcal {F}}({\textbf {1}}{\otimes } E^{\prime })=({\textbf {1}}{\otimes } E^{\prime }){\mathcal {F}}\), which determine \({\Delta }_{{\mathcal F}}(E^{\prime })\) as in (137). Next, it is straightforward to check that the coproducts \({\Delta }_{{\mathcal F}}(g)\), with \(g=H,E,E^{\prime }\), and the antipode property \( \mu [(S_{{\mathcal F}}\otimes \text {id}){\Delta }_{{\mathcal F}}(g)] =\epsilon (g){\textbf {1}}=0\) determine the twisted antipodes \(S_{{\mathcal F}}(g)\) as in (137). To compute the twisted tensor and star products we first make only the first leg \(\overline {\mathcal {F}}_{1}\) of \(\overline {\mathcal {F}}\) to act on its eigenvectors HCode \(H,E,E^{\prime },u^{i},\partial _{i}\) (generators of \(U{\mathfrak {g}}\) and of \({{\mathcal Q}}\)), and find
for all ui ∈{yi, ηi}, 1 ≤ i ≤ 3; note that the exponents ± λi/2 take the values ± 1, 0. This simplifies the computation of the action of the second leg \(\overline {\mathcal {F}}_{2}\) on the second factor; using (133) and (136) and noting that only the terms of degree lower than two in the power expansion of 1/(1 + iνE) contribute to its action on the \(H,E,E^{\prime },u^{i},\partial _{i}\), by a direct computation one thus finds the star products (138-140). In particular the twisted tensor and star products are trivial if H, u3 or \(\tilde \partial _{3}\) appears in the first factor. The twisted commutation relations (141)-(142) and the twisted submanifold constraints (143) follow. For the twisted star involution note that
since \(E\rhd y^{i}={\delta ^{i}_{3}}y^{1}+2b{\delta ^{i}_{2}}\) and λ3 = 0, λ2 = − 1, while \(E\rhd \tilde {\partial }_{i}=-\delta _{i1}\tilde {\partial }_{3}\), and λ1 = 1.
1.7 B.7 (d-e-f) Elliptic Cone and Hyperboloids
1.7.1 B.7.1 Proof of Proposition 17 in [32], see Section 4.6
From the anti-Hermiticity of the vector fields H, E and from [H, E] = 2E it follows that \(\mathcal {F}=\exp (H/2\otimes \log ({\textbf {1}}+i\nu E))\) is a unitary Jordanian twist on \(U\mathfrak {g}\) and that the coproducts and antipodes of H, E are exactly as in case (d). Similarly, (176) holds, because it is only based on the relation \([H,E^{\prime }]=-2E^{\prime }\).
To compute \({\Delta }_{{\mathcal F}}(E^{\prime })\) we first determine \({\mathcal {F}}({\textbf {1}}{\otimes } E^{\prime })\). We use [H, E] = 2E, \(EE^{\prime }=E^{\prime }E-H\), and find by induction first that EnH = (H − 2n)En, then
for all n ≥ 0, by the “little Gauss” \({\sum }_{h=1}^{n-1}h=\frac {n^{2}-n}{2}\). Consequently, using the series expansions
we obtain
and in turn
Hence
On the other hand, using \(\left (\frac {H}{2}\right )^{n}E^{\prime }=E^{\prime }\left (\frac {H}{2}-{\textbf {1}}\right )^{n}\) we obtain
Summing the last two equations we find that \({\Delta }_{\mathcal F}(E^{\prime })\) is as in (151). The antipode \(S_{{\mathcal F}}(E^{\prime })\) follows from the antipode property \( \mu [(S_{{\mathcal F}}\otimes \text {id}){\Delta }_{{\mathcal F}}(E^{\prime })]=0\).
To compute the twisted tensor and star products we first make only the first leg \(\overline {\mathcal {F}}_{1}\) of \(\overline {\mathcal {F}}\) to act on its eigenvectors HCode \(H,E,E^{\prime },u^{i}\) (generators of \(U{\mathfrak {g}}\) and of \({{\mathcal Q}}\)), and find
where \(u^{i}\in \{y^{i},\eta ^{i},\tilde {\partial }^{i}\}\), 1 ≤ i ≤ 3; note that the exponents ± λi/2 take the values ± 1, 0. By the first relation the twisted tensor or star products are trivial if H or some u2 is the first factor. The following two imply (153). Moreover, for all \(u^{i},v^{i}\in \{y^{i},\eta ^{i},\tilde {\partial }^{i}\}\), we find
By explicit calculations these imply relations (154-159), as well as (160), once one notes that
To determine \(u^{i}{}^{*_{\star }}= S(\beta )\rhd u^{i}{}^{*}\) recall that \(\beta =\mathcal {F}_{1}S(\mathcal {F}_{2})\). Then
1.7.2 B.7.2 Metric and Principal Curvatures on the Circular Hyperboloids and Cone
One can easily check the statements of the first paragraph of Section 4.6.1 using e.g. the basis S := {v1, v2, v3} of Ξ, where
we have abbreviated ρ2 ≡ (x1)2 + (x2)2. S is orthogonal with respect to g, while St := {v1, v2} is an orthogonal basis of Ξt with respect to gt, since, by an easy computation,
where E(x) ≡ fi(x)fi(x) = xixi. Since E = 2c on Mc, Mc is indeed Riemannian if c < 0, Lorentzian if c > 0, whereas the metric induced on the cone M0 is degenerate. One can easily check (161), (162) on such a St by explicit computations. The dual basis consists of
The principal curvatures on Mc are indeed HCode \(-\text {sign}(c)/\sqrt {|2c|}\), \(1/\sqrt {|2c|}\), because in the ‘orthonormal’ basis S := {e1, e2, e3}, with \(e_{1}:=\frac 1{\rho } v_{1}\), \(e_{2}:=\frac 1{\rho \sqrt {|{{\mathsf E}}|}} v_{2}\), \(e_{3}:=N_{\scriptscriptstyle \perp }=\frac 1{\sqrt {|{{\mathsf E}}|}} V_{\perp }\), one finds
On \(H,E,E^{\prime }\) the metric gives
and the same results in the last line if we flip the arguments. To prove (165) we use (67), (56), (177), the \({{\mathcal X} }\)-linearity of g. To prove (163) we use (67), (177). The undeformed version of (164) follows from (162) by \({{\mathcal X}}\)-linearity. We prove (164) using (67), (18), the definition of \({\overline {\mathcal R}}\):
To prove (166) we note that classically ∇ is the Levi-Civita covariant derivative \(\nabla _{X^{i}\partial _{i}}(Y^{i}\partial _{i}) =X^{i}\partial _{i}(Y^{j})\partial _{j}\) and \(\overline {\mathcal {F}}(\triangleright {\otimes }\triangleright )(H{\otimes } X)=H{\otimes } X\), \(\overline {\mathcal {F}}(\triangleright {\otimes }\triangleright )(X{\otimes } E)=X{\otimes } E\) for all X ∈Ξ, while by (177), (146) \(\overline {\mathcal {F}}(\triangleright {\otimes }\triangleright )(E{\otimes } H) =E{\otimes } H+2i\nu E{\otimes } E\), \(\overline {\mathcal {F}}(\triangleright {\otimes }\triangleright )(E{\otimes } E^{\prime }) =E{\otimes } E^{\prime }+i\nu E{\otimes } H-2\nu ^{2}E{\otimes } E\), \(\overline {\mathcal {F}}(\triangleright {\otimes }\triangleright )(E^{\prime }{\otimes } H) =E^{\prime }{\otimes } H-2i\nu E^{\prime }{\otimes } E\) and \(\overline {\mathcal {F}}(\triangleright {\otimes }\triangleright )(E^{\prime }{\otimes } E^{\prime }) =E^{\prime }{\otimes } E^{\prime }-i\nu E^{\prime }{\otimes } H\).
1.7.3 Proof of Proposition 10
\([D,{\mathfrak {g}}]=0\) implies \({\mathcal {F}}(g{\otimes } {\textbf {1}})=(g{\otimes } {\textbf {1}}){\mathcal {F}}\) for all \(g\in {\mathfrak {g}}\). Moreover, since \({\mathcal {F}}({\textbf {1}}{\otimes } H)=({\textbf {1}}{\otimes } H){\mathcal {F}}\), \({\mathcal {F}}({\textbf {1}}{\otimes } D)=({\textbf {1}}{\otimes } D){\mathcal {F}}\), it follows that \({\Delta }_{{\mathcal F}}(H)={\Delta }(H)\), \(S_{{\mathcal F}}(H)=S(H)=-H\), \({\Delta }_{{\mathcal F}}(D)={\Delta }(D)\), \(S_{{\mathcal F}}(D)=S(D)=-D\). On the other hand, HE = E(H + 2), \(HE^{\prime }=E^{\prime }(H-2)\) imply
which together with \({\mathcal {F}}(g{\otimes } {\textbf {1}})=(g{\otimes } {\textbf {1}}){\mathcal {F}}\) and \( \mu [(S_{{\mathcal F}}\otimes \text {id}){\Delta }_{{\mathcal F}}(g)]=0\) for \(g=E,E^{\prime }\) imply (168).
\([D,{\mathfrak {g}}]=0\) also implies \(\overline {\mathcal {F}}_{1}\rhd g\otimes \overline {\mathcal {F}}_{2}=g\otimes {\textbf {1}}\) for all \(g\in {\mathfrak {g}}\), whence g ⋆ α = gα for all \(\alpha \in U{\mathfrak {g}},{{\mathcal Q}}\), in particular for the α appearing in the formulas of the proposition. \(D\rhd \tilde {\partial }_{i}=-\tilde {\partial }_{i}\), \(D\rhd u^{i}=u^{i}\) for ui = yi, ηi imply \(\overline {\mathcal {F}}_{1}\rhd \tilde {\partial }_{i}\otimes \overline {\mathcal {F}}_{2}=\tilde {\partial }_{i}\otimes e^{i\nu H/2}\), \(\overline {\mathcal {F}}_{1}\rhd u^{i}{\otimes } \overline {\mathcal {F}}_{2}=u^{i}{\otimes } e^{-i\nu H/2}\), whence HCode \(\tilde {\partial }_{i}\star \alpha =\tilde {\partial }_{i} (e^{i\nu H/2}\rhd \alpha )\), \(u^{i}\star \alpha =u^{i} (e^{-i\nu H/2}\rhd \alpha )\). Since \(D,H,E,E^{\prime }\) and \(y^{i},\eta ^{i},\tilde {\partial }_{i}\) (generators of \({{\mathcal Q}}\)) are all eigenvectors of \(H\rhd \), choosing α as each of them, we immediately find the remaining formulae in (169-170). One finds the involution ∗⋆ using the following results:
The commutation relations (171), the realization of \(D,H,E,E^{\prime }\) as combinations of \(y^{i}\star \tilde {\partial }_{i}\), and the relations (172) characterizing \({{\mathcal Q}_{{M_{c}}\star }}\) follow from (169-170) by direct computations.
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Fiore, G., Franco, D. & Weber, T. Twisted Quadrics and Algebraic Submanifolds in \(\mathbb {R}^{n}\). Math Phys Anal Geom 23, 38 (2020). https://doi.org/10.1007/s11040-020-09361-3
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DOI: https://doi.org/10.1007/s11040-020-09361-3