A generalization of Veldkamp's theorem for a class of Lie algebras
aa r X i v : . [ m a t h . QA ] J u l RIGIDITY AND DERIVED ISOMORPHISM PROBLEM FOR ENVELOPINGALGEBRAS
AKAKI TIKARADZE
Abstract.
We prove that there are no injective homomorphisms between enveloping algebras ofnon-isomorphic semi-simple Lie algebras of the same dimension. We also describe the center ofreduction modulo large prime p of the enveloping algebra of an algebraic Lie algebra g with theproperty that the center of its enveloping algebra is a polynomial algebra and Sym( g ) has no non-trivial semi-invariants with respect to the adjoint g action, in particular showing that it is generatedby the Harish-Chandra part and p -center, and is a complete intersection ring. As an application, wesolve the derived isomorphism problem of enveloping algebras for a class of Lie algebras. Introduction
The isomorphism problem for enveloping algebras of Lie algebras is a basic open question inring theory. Recall that it asks whether a C -algebra isomorphism between enveloping algebras ofLie algebras implies an isomorphism of the underlying Lie algebras. For the background and thedetailed discussion of this well-known problem we refer the reader to the survey article by Usefi [U].In analogy with the derived isomorphism problem for rings of differential operators for smoothaffine varieties, in [T] we considered the following natural generalization.
Conjecture 1.
Let g , g ′ be finite dimensional Lie algebras over C . If the derived categories ofbounded complexes of Ug -modules and Ug ′ -modules are equivalent, then g ∼ = g ′ . In our approach to the above conjecture we follow the well-known blueprint of ”dequantization”by reducing to the modulo large prime p technique, which allows a translation of questions aboutvarious quantizations to the ones about Poisson algebras. This is largely motivated by the proof ofequivalence between the Dixmier and the Jacobian conjectures given by Belov-Kanel and Kontsevich[BK]. Using this approach, we proved in [T] that certain class of Lie algebras that includes FrobeniusLie algebras (see Remark 1.2 below) are derived invariants of their enveloping algebras, also a finitesubgroup of automorphisms of Ug is isomorphic to a subgroup of Aut( g ) for a semi-simple g . Recall that given an associative flat Z -algebra R and a prime number p, then the center Z ( R/pR )of its reduction modulo p acquires the natural Poisson bracket, defined as follows. Given centralelements a, b ∈ Z ( R/pR ), let z, w ∈ R be their lifts respectively. Then the Poisson bracket { a, b } isdefined to be 1 p [ z, w ] mod p ∈ Z ( R/pR ) . In particular, given a a Lie algebra g over S ⊂ C –a finitely generated ring such that g is a finitefree S -module, then for a prime p > , the center Z ( Ug p ) = Z ( Ug /p Ug ) of the enveloping algebraof g p = g /p g is equipped with the natural S/pS -linear Poisson bracket defined as above. In thiscontext, the importance of understanding Z ( Ug p ) (besides its fundamental role in representationtheory of g p ) in regards with the derived isomorphism problem above lies in its derived invariance: After completing this manuscript, we learned that the full solution of the isomorphism problem of envelopingalgebras has been announced very recently in [CPNW]. if Ug and Ug ′ are derived equivalent, then Z ( Ug p ) ∼ = Z ( Ug ′ p ) as Poisson S/pS -algebras. This easilyfollows from the derived invariance of the Hochschild cohomology and the Gersenhaber bracket(see [[T] Lemma 4]). Now recall that if I ⊂ Z ( Ug p ) is a Poisson ideal, then the Poisson bracketinduces Lie bracket on I/I . In particular, if m is a Poisson ideal such that Z ( Ug p ) / m = S/pS, then m / m is a finite S/pS -Lie algebra. Therefore, the collection of isomorphisms classes of
S/pS -Lie algebras m / m , as m ranges over Poisson ideals of Z ( Ug p ) such that Z ( Ug p ) / m = S/pS, is aderived invariant of Ug . The significance of this derived invariant of g is highlighted by the factthat given Poisson ideal m ⊂ Z ( Ug p ) as above, as the key computation by Kac and Radul shows(see Lemma 1.1 below), there is a canonical Lie algebra homomorphism g p → m/m (when g is analgebraic Lie algebra). There is one distinguished such maximal Poisson ideal–the augmentationideal m ( g p ) = Z ( Ug p ) ∩ g p Ug p . Now we recall a key computation of the Poisson bracket for restricted Lie algebras due to Kacand Radul [KR]. Let R be a commutative reduced ring of characteristic p > . Let g be a restrictedLie algebra over R with the restricted structure map x → x [ p ] , x ∈ g . As usual, Z p ( g ) denotes the p -center of Ug : the central R -subalgebra of the enveloping algebra Ug generated by elements ofthe form x p − x [ p ] , x ∈ g . It is well-known that the map x → x p − x [ p ] induces homomorphism of R -algebras i : Sym( g ) → Z p ( g ) , where Z p ( g ) is viewed as an R -algebra via the Frobenius map F : R → R. The homomorphism i is an isomorphism when R is perfect. Recall also that the Lie algebra bracket on g defines theKirillov-Kostant Poisson bracket on the symmetric algebra Sym( g ) . The following is the above mentioned key result from [KR].
Lemma 1.1.
Let S be a finitely generated integral domain over Z . Let g be an algebraic Lie algebraover S. Then Z p ( g p ) is a Poisson subalgebra of Z ( Ug p ) , moreover the induced Poisson bracketcoincides with the negative of the Kirillov-Kostant bracket: { a p − a [ p ] , b p − b [ p ] } = − ([ a, b ] p − [ a, b ] [ p ] ) , a ∈ g p , b ∈ g p . Given an algebraic Lie algebra g over S ⊂ C –a finitely generated ring, by Z HC ( g p ) (the Harish-Chandra part of the center) we denote the image of Z ( Ug ) in Z ( Ug p ) . Towards the goal of understand Z ( Ug p ) , it is a natural and important problem to establish whether Z ( Ug p ) is generated over its p -center Z p ( g p ) by Z HC ( g p ) (see the discussion in [K].) Although this is not always the case (seeRemark 1.1), we show that it does hold for a class of Lie algebras satisfying the following assumption. Assumption 1.
Let S ⊂ C be a finitely generated ring, and g be an algebraic Lie algebra over S .Let Sym( g ) g = O be generated by f , · · · , f m over S. We have the quotient map π : g ∗ = Spec Sym( g ) → Spec O = Y. Denote by Y sm the smooth locus of Y. Let U = { x ∈ g ∗ , π ( x ) ∈ Y sm , dπ x is onto } . Assume that
Sym g has no nontrivial g -semi-invariants, and g ∗ \ π − ( Y sm ) has codimension ≥ in g ∗ . Then we have the following.
Theorem 1.1.
Let a Lie algebra g be as in Assumption 1. Let g i ∈ Z ( Ug ) be the symmetrizationof f i . Then for all p ≫ , we have Sym( g p ) g p = Sym( g p ) p [ f , · · · , f m ] , Z ( Ug p ) = Z p ( g k )[ g , · · · , g m ] . Given a base change S → k to an algebraically closed field of characteristic p and a Poisson maximalideal m in Z ( Ug k ) (under the Poisson bracket induced from the Poisson bracket on Z ( Ug p ) , ) the Lie IGIDITY AND DERIVED ISOMORPHISM PROBLEM FOR ENVELOPING ALGEBRAS 3 algebra m / m is isomorphic to a quotient of g k /Z ( g k ) ⊕ k m , where k m is viewed as an abelian Liealgebra. In view of Theorem 1.1 it is temping to make the following
Conjecture 2.
Let g be an algebraic Lie algebra over a finitely generated ring S ⊂ C , such that Sym g has no nontrivial g -semi-invariants Then for all large enough primes p ≫ , Z ( Ug p ) isgenerated over Z p by the image of Z ( Ug ) . The following is a significant strengthening of Theorem 1.1 for Lie algebras g with additionalassumption that (Sym g ) g is a polynomial algebra. Theorem 1.2.
Let g be a Lie algebra of a connected algebraic group G over a finitely generatedring S ⊂ C , such that Sym g has no nontrivial g -semi-invariants, and Sym( g ) g = S [ f , · · · , f n ] is apolynomial algebra. Let g i ∈ Z ( Ug ) be the symmetrization of f i . Then for all primes p ≫ , Z ( Ug p ) is a free Z p ( g p ) -module with a basis { g α = g α · · · g α n n , ≤ α i < p } . Moreover given a base change S → k to an algebraically closed field k of characteristic p, then Z ( Ug k ) = Z ( Ug p ) ⊗ S/pS k is acomplete intersection and Z ( Ug k ) ∼ = Z p ( g k ) ⊗ Z p ( g k ) ∩ Z HC ( g k ) Z HC ( g k ) , Z HC ( g k ) = ( Ug ) G k . Given a Poisson maximal ideal m in Z ( Ug k ) (under the Poisson bracket induced from the Poissonbracket on Z ( Ug p ) , ) the Lie algebra m / m is isomorphic to g k /Z ( g k ) ⊕ V, where V is a trivial Liealgebra spanned by images of g i − a i , a i ∈ k , where g i − a i ∈ m . The proof of Theorems 1.1, 1.2 crucially relies on the first Kac-Weisfeler conjecture, which wasestablished in [MSTT] for p ≫ . Remark 1.1.
It is easy to see that assumption about non-existence of nontrivial G -semi-invariantsof Sym( g ) is essential. Let g be the following 3-dimensional Lie algebra g = C x ⊕ C y ⊕ C z with thebracket [ z, x ] = x, [ z, y ] = y, [ x, y ] = 0 . Then it is easy to see that Z ( Ug k ) is not generated by Z p and Z HC , as Z HC = k , while xy p − ∈ Z ( Ug k ) \ Z p . It was proved in [AP] that given a finite subgroup W of automorphisms of the enveloping algebraof a semi-simple Lie algebra g , such that ( Ug ) W is isomorphic to an enveloping algebra of a Liealgebra g ′ , then W must be trivial and g ′ = g . On the other hand, Caldero [C] showed that givensemi-simple Lie algebras g , g ′ and finite subgroups of automorphisms of corresponding envelopingalgebras W ⊂ Aut( Ug ) , W ′ ⊂ Aut( Ug ′ ) such that ( Ug ) W ∼ = ( Ug ′ ) W ′ , then g ∼ = g ′ . Recall that the index of a Lie algebra g is defined as the smallest dim g χ over χ ∈ g ∗ . We have the following rigidity result about homomorphisms between enveloping algebras.
Theorem 1.3.
Let g be a semi-simple Lie algebra over C , g ′ be a Lie algebra satisfying Assumption1 and dim g = dim g ′ . Let f : Ug → Ug ′ be a C -algebra homomorphism such that, either f isinjective, or f | Z ( Ug ) is injective and index( g ) ≤ index( g ′ ) . Then g ∼ = g ′ and f ( Z ( Ug )) ⊂ Z ( Ug ′ ) . As an application of Theorem 1.2 to the derived equivalence problem we have the following.
Theorem 1.4.
Let g , g ′ be algebraic Lie algebras satisfying assumptions in Theorem 1.2. If Ug isderived equivalent to Ug ′ , then g /Z ( g ) ∼ = g ′ /Z ( g ′ ) . We recall the following simple result that illustrates usefulness of ”dequantizing” Ug to Sym g k in regards with the isomorphism problem for enveloping algebras. AKAKI TIKARADZE
Lemma 1.2.
Let m be a Poisson maximal ideal of Sym g k . Then m/m ∼ = g k as Lie algebras. Inparticular, if g ′ is another Lie algebra over k such that Sym( g ) ∼ = Sym( g ′ ) as Poisson k -algebras,then g ∼ = g ′ . Proof.
Let m be a maximal Poisson ideal. Then m = ( g − χ ( g ) , g ∈ g ) for some χ ∈ g ∗ . It followsthat χ must be a character of g , hence the homomorphism g → g − χ ( g ) defines a Lie algebraisomorphism g → m/m as desired. (cid:3) Remark 1.2.
Theorems 1.2 and 1.4 were proved in [T] under much more restrictive assumptions on g : there we assumed that ( f , · · · , f n ) is a regular sequence, Sym( g ) / ( f , · · · , f n ) is a normal domainand the coadjoint action of G on Spec(Sym( g ) / ( f , · · · , f n )) has an open orbit. In particular, nonilpotent Lie algebras can satisfy these conditions. On the other hand, Theorems 1.2 and 1.4 canbe applied to many nilpotent algebras, as the class of Lie algebras with the property that Sym( g ) g is a polynomial algebra is large (see[O]).2. The first Kac-Weisfeiler conjecture
In this section we recall some results associated with the first Kac-Weisfeiler conjecture that areused in proof of our main results. Recall that the first Kac-Weisfeiler conjecture asserts that fora p -restricted Lie algebra g over k , the maximal possible dimension of an irreducible g -module is p (dim( g ) − index( g )) . Equivalently the rank of Ug over its center equals p dim( g ) − index( g ) . Let g be a Lie algebra of an algebraic group G defined over a finitely generated ring S ⊂ C . Thenfor all p ≫ S → k to an algebraically closed field of characteristic p, the firstKac-Weisfeiler conjecture was established for g k in [MSTT]. Namely we have proved the following.As usual, D ( g ) denotes the skew field of fractions of Ug . Similarly, by C ( g ) we will denote the fieldof fractions of Sym( g ) . Theorem 2.1 ([MSTT], Theorems 3.8 and 3.9) . Let g be an algebraic Lie algebra over a finitelygenerated ring S ⊂ C . Then for all p ≫ and a base change S → k to an algebraically closedfield of characteristic p , the fraction field of Z ( U ( g k )) is generated by the image of Z ( D ( g )) and thefraction field of Frac( Z p ( g k )) Also, the following equality of degrees of field extensions holds: [Frac( Z ( Ug k )) : Frac( Z p ( g k ))] = p index ( g ) = [ C ( g k ) g k : C ( g k ) p ] Remark 2.1.
The equality [ C ( g k ) g k : C ( g k ) p ] = p index( g ) follows from the fact that on the one hand (as proved in [[MSTT] Theorem 3.8, 3.9])[Frac(gr Z ( Ug k )) : C ( g k ) p ] ≥ p index( g ) , and on the other hand it was proved in [PS] that[ C ( g k ) g k : C ( g k ) p ] ≥ p dim( g ) − index( g ) . We also need to recall the following simple result from commutative algebra (for a proof see[[MSTT] Lemma 3.11]. )
Lemma 2.1.
Let S ⊂ C be a finitely generated ring. Let A be a finitely generated commutativealgebra over S such that A C is a domain. Let B ⊂ A be a finitely generated S -subalgebra. Then forall p ≫ and a base change S → k to an algebraically closed field k of characteristic p the rank of A k over B k A p k is p dim( A ) − dim( B ) . IGIDITY AND DERIVED ISOMORPHISM PROBLEM FOR ENVELOPING ALGEBRAS 5 The proofs
We crucially rely on the following result of Panyushev-Yakimova [[PY], Remark 1.3] (see also[[PY], Proposition 1.2], [[JS] Proposition 5.2]).
Proposition 3.1.
Let g be a an Lie algebra over C such that Sym( g ) has no proper g -semi-invariants. Let O = Sym( g ) g be finitely generated. Put Y = Spec O . We have the quotient map π : g ∗ = Spec Sym( g ) → Spec O = Y. Denote by Y sm the smooth locus of Y. Let U = { x ∈ g ∗ , π ( x ) ∈ Y sm , dπ x is onto } . Then g ∗ \ U has codimension ≥ in g ∗ . Proof of Theorems 1.1, 1.2.
We start by proving Theorem 1.1. Since g has no nontrivial semi-invariants in Sym( g ), it follows easily that(Frac(Sym g )) g = Frac((Sym g ) g ) . Put O = k [ f , · · · , f m ] and n = index( g ) . In particular n = dim( O ) . Also put A = Sym( g k ) , B = Sym( g k ) p O , B ′ = (Sym g k ) g k . Clearly B ⊂ B ′ . Let K = Frac( A ) and K = Frac( B ) . We first claim that [ K : K p ] = p n . Indeed, this follows at once since the degree of K over K is p dim( g ) − n by Lemma 2.1, and[ K : K p ] = p dim g . Next we argue that B a normal domain. Indeed, it follows from Theorem 3.1 that for all p ≫ S → k , the compliment of U k = { x ∈ g ∗ k , π ( x ) ∈ Y sm , dπ x is onto } . in g ∗ k has codimension ≥ . We will argue that for a prime ideal I in U k , the ring B I is regular ring,while for a prime in π − ( Y sm ) , B I is a Cohen-Macaulay ring. Let I be a prime ideal in A such that I ∈ π − ( Y sm ) . Put I ′ = I ∩ A p , I ′′ = I ∩ O . Also put C = O pI ′′ ⊂ A pI ′ ∩ O I ′′ . We claim that the multiplication map φ induces an isomorphism φ : ( A p ) I ′ ⊗ C O I ′′ ∼ = B I . Indeed φ is a surjective map from a free A p -module of rank p n (as O I ′′ is a free C -module of rank p n since I ′′ belongs to the smooth locus of Y ) onto a A p -module of rank p n . Thus φ must be anisomorphism. In particular, it follows that B I is a Cohen-Macaulay. Now if in addition I ∈ U k , then it follows easily that A pI ′ ⊗ C O I ′′ is a regular ring. Hence B is regular in codimension 1 andCohen-Macaulay in codimension 2. Therefore it is a normal domain by Serre’s criterion of normality.It follows that B ′ and B have the same field of fractions as they are both extensions of degree p n of Frac( Z p ) by Theorem 2.1 and B ⊂ B ′ . Thus normality of B yields that B = B ′ . Hence(Sym( g k )) g k = (Sym( g k )) p [ f , · · · , f m ] , Z ( Ug k ) = Z p [ g , · · · , g m ] . Now we proceed to prove Theorem 1.2. We first show that B is a free A p -module with basis { f α = Q ni =1 f α i i , α i < p } . Indeed, since [ K : K p ] = p n , it follows that { f α , α i < p } are linearlyindependent over K p . In particular they form a basis of B over A p . Hence, Z ( Ug k ) is a free moduleover Z p with a basis { g α , α i < p } as desired. Thus we obtain thatgr Z ( Ug k ) = B = (Sym( g k )) g k . AKAKI TIKARADZE
We have the natural surjective homomorphism (that restricts to the Frobenius homomorphismon A and sends x i to f pi ) A [ x , · · · , x n ] / ( x p − f , · · · , x pn − f n ) → B, which must be an isomorphism since both rings are free A p -modules of rank p n . Since ( x p − f , · · · , x pn − f n ) is a regular sequence in A [ x , · · · , x n ] , it follows that B is a complete intersec-tion. Therefore Z ( Ug k ) is also a complete intersection.Our next goal is show that Sym( g k ) G k = k [ f , · · · , f n ] , which implies Ug G k = Z HC ( g k ) . Let x ∈ Sym( g k ) G k ⊂ B. As B is a free Sym( g k ) p module with basis { f α = Q ni =1 f α i i , α i < p } , we maywrite x = X α x α f α , x α ∈ (Sym g k ) p , α i < p. Then x α ∈ (Sym( g k ) G k ) p . Replacing x by x p α and continuing in this manner, we obtain that x ∈ \ n (Sym( g k )) p n O = O . We also get that Z p ( g k ) ∩ Z HC ( g k ) = Z p ( g k ) G k . So gr( Z p ( g k ) ∩ Z HC ( g k )) = k [ f p , · · · , f pn ] . Hence Z HC ( g k ) is a free Z p ( g k ) ∩ Z HC (( g k )-module of rank p n . Therefore the natural ring homomor-phism Z p ( g k ) ⊗ Z p ( g k ) ∩ Z HC ( g k ) Z HC ( g k ) → Z ( Ug k )is a surjective homomorphism of free Z p ( g k )-modules of rank p n . Thus it is a Poisson algebraisomorphism.Let m ⊂ Z ( Ug k ) be a Poisson maximal ideal. Put m ′ = m ∩ Z p ( g k ) , m ′′ = m ∩ Z HC ( g k ) , m = m ′ ∩ m ′′ . Then the Poisson algebra isomorphism above implies the following Lie algebra isomorphism m / m ∼ = m ′ / m ′ × m / m m ′′ / m ′′ . On the other hand, since the Poisson bracket vanishes on Z HC , we conclude that m / m is a trivialcentral extension of the image of m ′ / m ′ ∼ = g k and the kernel of the natural homomorphism m ′ / m ′ → m / m is central. On the other hand, if g ∈ Z ( m ′ / m ′ ), then g = f p with f ∈ m , hence the imageof g in m / m is 0. Thus m / m is a trivial central extension of g k /Z ( g k ) . (cid:3) Proof of Theorem 1.4.
We may assume that Lie algebras g , g ′ and the corresponding derived iso-morphism are defined over a finitely generated ring S ⊂ C . Let Z ( Ug ) = S [ f , · · · , f n ] and Z ( Ug ′ ) = S [ f ′ , · · · , f ′ n ] . Thus we have an S -algebra isomorphism S [ f , · · · , f n ] ∼ = S [ f ′ , · · · , f ′ n ] anda Poisson algebra isomorphism Z ( Ug k ) ∼ = Z ( Ug ′ k ) . Moreover these isomorphisms are compatiblewith reduction modulo p maps so that the following diagram commutes: IGIDITY AND DERIVED ISOMORPHISM PROBLEM FOR ENVELOPING ALGEBRAS 7 S [ f , · · · , f n ] S [ f ′ , · · · , f ′ n ] Z ( Ug k ) Z ( Ug ′ k )Let m be a maximal Poisson ideals in Z ( Ug k ), and let m ′ ⊂ Z ( Ug ′ k ) be the correspondingmaximal Poisson ideal under the above isomorphisms. Hence we get an isomorphism of Lie algebras m / m ∼ = m ′ / m ′ . On the other hand, by Theorem 1.2, m / m (respectively m ′ / m ′ ) is isomorphic to a atrivial central extension of g k /Z ( g k ) (resp. of g ′ k /Z ( g ′ k )) Since we have the compatible isomorphism k [ g , · · · , g n ] ∼ = k [ g ′ , · · · , g ′ n ] we get that g k /Z ( g k ) ⊕ V ∼ = g k /Z ( g ′ k ) ⊕ V ′ with V ∼ = V ′ being abelian k -Lie algebra. Thus g k /Z ( g k ) ∼ = g ′ k /Z ( g ′ k ) . Hence g /Z ( g ) ∼ = g ′ /Z ( g ′ ) . (cid:3) Proof of Theorem 1.3.
Put Z ( Ug ′ ) = S [ g ′ , · · · , g ′ n ] . Let φ : Ug → Ug ′ be a C -algebra embedding with g semi-simple, g ′ as above and dim g =dim g ′ . It is well-known that the maximal Krull dimension of a commutative subalgebra of Ug ′ is atmost dim( g ′ + index( g ′ )) , on the other hand Ug contains a commutative subalgebra of dimension dim( g + index( g )) as proved by Rybnikov [R]. So we may conclude that rank( g ) ≤ index( g ′ ) . Thusfor the remainder of the proof we may (and will) assume that we a have a C -algebra homomorphism φ : Ug → Ug ′ , such that rank( g ) ≤ index( g ′ )) and φ | Z ( Ug ) is injective.We may also assume that φ, g , g ′ are defined over S -a large enough finitely generated subring of C . Denote by φ k : Ug k → Ug ′ k the base change of φ to an algebraically closed field of characteristic p ≫ . The crucial step of the proof of Theorem 1.3 will be showing that φ k ( Z ( Ug k )) ⊂ Z ( Ug ′ k ) . This part of the proof will mimic the corresponding part of the proof equivalence between theJacobian and the Dixmier conjectures [[BK] Proposition 2], except we have to work little harder as Ug k , Ug ′ k are not Azumaya algebras.First we argue that for all p ≫ , there exists δ p ∈ Z HC ( g k ) that vanishes on the non-Azumayalocus of Z ( Ug k ) and φ k ( δ p ) = 0 . For this purpose we recall some well-knwon facts about the singularlocus of Spec Z ( Ug k ) following [Ta].Let Ψ : ( Ug k ) G k → Sym( h ) W be the usual isomorphism, W is the Weyl group and h is a Cartansubalgebra, Σ is the set of roots and h α ∈ h , α ∈ Σ are the corresponding elements. Put δ a = Ψ − (Π α ∈ Σ ( h α − a )) , a ∈ F p ; δ p = Y a ∈ F p δ a . Now recall that δ p vanishes on the singular locus of Spec Z ( Ug k ) [[Ta] Theorem 1], and since thesmooth and Azumaya loci of Z ( Ug k ) coincide, δ p vanishes on the compliment of the Azumaya locus.Denote by ∆ , · · · , ∆ l ∈ ( Ug ) g images under Ψ − of all elementary symmetric functions on h α , α ∈ Σ(viewed as elements of Sym( h ) W .) Then each δ a belongs to the F p -span of images of ∆ , · · · , ∆ l in Ug k . Since φ k (∆ ) , · · · , φ k (∆ l ) are linearly independent for p ≫ . Thus φ k ( δ a ) = 0 for all a ∈ F p and p ≫ . Hence φ k ( δ p ) = 0 . Recall that by the first Kac-Weisfeiler conjecture, the PI-degree of Ug k , (respectively Ug ′ k ) is (dim( g ) − index( g )) (resp. (dim( g ) − index( g ))). Since index( g ) ≤ index( g ′ ), It follows from thatthe PI-degree of Ug ′ k is at most the PI-degree of Ug ′ k . Let z ∈ Z ( Ug k ) be such that φ k ( z ) / ∈ Z ( Ug ′ k ) . AKAKI TIKARADZE
Put S = ( Ug k ) δ p and denote by S a localization of Ug ′ k so that φ k ( δ p ) is invertible in S , and S is an Azumaya algebra.So we have a homomorphism of Azumaya algebras φ k : S → S and z ∈ Z ( S ) such that φ k ( z ) / ∈ Z ( S ) and PI-degree( S ) ≤ PI-degree( S ) . Let V be a simple S -module on which φ k ( z )does not act like a scalar. To construct such a module suffices to take a simple module afforded bya character χ : Z ( S ) → k such that φ k ( z ) has a nonzero image in( S /Z ( S )) χ = S /Z ( S ) ⊗ Z ( S ) Z ( S ) / Ker( χ ) . Then V viewed as an S -module must be simple (forcing PI − deg( S ) = PI − deg( S )) on which z acts as a non-scalar, a contradiction.Thus we have a k -Poisson algebra homomorphism ˜ φ k : Z p ( g k ) → Z ( Ug ′ k ) . Recall that we havedistinguished Poisson maximal ideals–the augmentation ideals m = g k Ug k ∩ Z p ( g k ) , m ′ = g ′ k Ug ′ k ∩ Z ( Ug ′ k ) . Thus g k ∼ = m / m with the isomorphism given by x → x p − x [ p ] , x ∈ g k . Denote by g (1) k ⊂ m theimage of g k under the map x → x p − x [ p ] , x ∈ g k . Since g k is perfect, It follows that the φ k ( m ) ⊂{ Z ( Ug ′ k ) , Z ( Ug ′ k ) } . On the other hand, { Z ( Ug ′ k ) , Z ( Ug ′ k ) } is contained in the ideal generated by[ g ′ k , g ′ k ] , g ′ , · · · , g ′ n . The latter is clearly contained in m ′ . Thus we have a homomorphism of Liealgebras ¯ φ k : m / m → m ′ / m ′ . Let ˜ I denote the kernel of ¯ φ k . Thus [ ˜ I, ˜ I ] = ˜ I since g k is semi-simple. Let I ⊂ g (1) k be the lift of ˜ I. Thus φ k ( I ) ⊂ m ′ . Let φ k ( I ) ⊂ m ′ n for n > . Since φ k is a Poisson homomorphism, we get that φ k ( I ) = φ k ([ I, I ]) ⊂ { m ′ n , m ′ n } ⊂ m ′ n − . Hence φ k ( I ) = 0 . So, we have an ideal I ′ ⊂ g k , so that φ k ( g p − g [ p ] ) = 0 for all g ∈ I ′ . Hence φ k ( I ′ ) = 0 , so I = 0 . As by Theorem 1.2 the Lie algebra m ′ / m ′ is a quotient of g ′ k ⊕ k m anddim g = dim g ′ , we may conclude that g k ∼ = g ′ k . Hence g ∼ = g ′ . (cid:3) Acknowledgements.
I am very grateful to Lewis Topley for making numerous useful suggestions.
References [AP] J. Alev, P. Polo,
A rigidity theorem for finite group actions on enveloping algebras of semi-simple Lie algebras ,Advances in Math. 111(1995) no.2 208–226.[BK] A. Belov-Kanel, M Kontsevich,
The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture , MoscowMathematical Journal, 7 (2007), no.2, 209–218.[C] P. Caldero,
Isomorphisms of finite invariants for enveloping algebras, semi-simple case , Advances in Mathematics,Vol 134, No 2, (1998), 294-307.[CPNW] R. Campos, D. Peterson, D. Robert-Nicoud, F. Wierstra,
Lie, associative and commutative quasi-isomorphism , arXiv:1904.03585.[JS] A. Joseph, D. Shafrir,
Polynomiality of invariants, unimodularity and adapted pairs , Transform. Groups, 15, no.4 (2010), 851–882.[KR] V. Kac, A. Radul,
Poisson structures for restricted Lie algebras , The Gelfand Mathematical Seminars, 1996–1999.[K] V. Kac, Featured Math. review of Irreducible representations of reductive Lie algebras of reductive groups andthe Kac-Weisfeiler conjecture by A. Premet., Invent. Math. 121 (1995), no. 1, 79–117.[MSTT] B. Martin, D. Stewart, L. Topley, A. Tikaradze,
A proof of the first Kac-Weisfeiler conjecture in largecharacteristics , (2018) arXiv:1810.12632.
IGIDITY AND DERIVED ISOMORPHISM PROBLEM FOR ENVELOPING ALGEBRAS 9 [O] A. Ooms,
The polynomiality of the Poisson center and semi-center of a Lie algebra and Dixmier’s fourth problem ,J. Algebra 477 (2017), 95–46.[PY] D. Panyushev, O. Yakimova,
Takiff algebras with polynomial rings of symmetric invariants , TransformationGroups (2019).[PS] A. Premet, S. Skryabin,
Representations of restricted Lie algebras and families of associative L -algebras , J.Reine Angew. Math. 507 (1999), 189–218.[R] L. Rybnikov, Argument shift method and Gaudin model , Functional Analysis and Its Applications (2006), 188–199.[Ta] R. Tange,
The Zassenhaus variety of a reductive Lie algebra in positive characteristic , Adv. in Math. 224 (2010),no. 1, 340–354.[T] A. Tikaradze,
On automorphisms of enveloping algebras , IMRN (2019), arXiv:1705.08035.[U] H. Usefi,
Isomorphism invariants of enveloping algebras , In Noncommutative rings and their applications, Vol.634 of Contemp. Math., pages 253–265. AMS, 2015
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