Noncommutative invariant theory and Auslander s Theorem

Transcription

1 Noncommutative invariant theory and Auslander s Theorem Miami University Algebra Seminar Robert Won Wake Forest University Joint with Jason Gaddis, Ellen Kirkman, and Frank Moore arxiv: November 14, / 34

2 Keep your friends close and your coauthors closer Images from and college.wfu.edu. November 14, / 34

3 Classical invariant theory k = k, char k = 0. All rings are k-algebras. Classically: G a finite subgroup of GL n (k) acting on A = k[x 1,..., x n ] as graded automorphisms. Study A G the subring of invariants. Example The symmetric group G = S n acts on A = k[x 1,..., x n ] by permuting the variables. A G is generated by the n elementary symmetric polynomials: e 0 = 1 e 1 = x i e 2 = i<j x i x j... e n = x 1 x 2 x n So A G = k[e 1,..., e n ] = A. November 14, 2017 Invariant theory 3 / 34

4 Classical invariant theory G a finite subgroup of GL n (k) acting on A = k[x 1,..., x n ] as automorphisms. A g G is a reflection if g fixes a codim 1 subspace of kx i. Theorem (Chevalley 1955; Shephard and Todd 1954) Let A = k[x 1,..., x n ] and G a finite subgroup of GL n (k). Then A G is a polynomial ring G is generated by reflections Noncommutative invariant theory? Replace A with a noncommutative ring. Replace G with a Hopf algebra H. November 14, 2017 Invariant theory 4 / 34

5 Noncommutative invariant theory What is a noncommutative polynomial ring? Definition A a f.g. connected graded k-algebra is Artin-Schelter regular if (1) gl. dim A = d < (2) GKdim A < and (3) Ext i A (k, A) = δ i,dk. Behaves homologically like a commutative polynomial ring. Commutative AS regular polynomial ring. Example The ( 1)-skew polynomial ring k 1 [x 1,..., x n ] = k x 1,..., x n x i x j = x j x i for i j. November 14, 2017 Invariant theory 5 / 34

6 Noncommutative invariant theory What is a reflection? Example Let S 2 = g. Then S 2 k 1 [x, y] by g(x) = y and g(y) = x. This gives an action since it preserves the relation xy = yx. This is different than S 2 k[x, y] as g(xy) = yx = xy. S 2 is a classical reflection group but A G is not AS regular. November 14, 2017 Invariant theory 6 / 34

7 Noncommutative invariant theory A a graded algebra, G a group of graded automorphisms. Each g G gives a linear map A j A j. Definition (Jing and Zhang 1997) The trace function of g is the formal power series Tr A (g, t) := tr(g Aj )t j. j=0 Definition (Kirkman, Kuzmanovich, and Zhang 2008) Let A be a graded algebra of GK dimension n. Then g G is a quasi-reflection if its trace function is of the form Tr A (g, t) := 1 (1 t) n 1, q(1) 0. q(t) November 14, 2017 Invariant theory 7 / 34

8 Noncommutative invariant theory Example S 2 k 1 [x, y] where g(x) = y, g(y) = x. Then g A1 = [ ] g A2 = g A3 = Tr A (g, t) = 1 t 2 + t 4 t = t 2. So g is not a quasi-reflection of A = k 1 [x, y]. November 14, 2017 Invariant theory 8 / 34

9 Noncommutative invariant theory So S 2 k 1 [x, y] contains no quasi-reflections. In fact: Example The symmetric group S n acting naturally on the ( 1)-skew polynomial ring k 1 [x 1,..., x n ] = k x 1,..., x n x i x j = x j x i for i j contains no quasi-reflections. Note: S n is a classical reflection group of k[x 1,..., x n ] (generated by reflections). November 14, 2017 Invariant theory 9 / 34

10 Noncommutative invariant theory Theorem (Chevalley-Shephard-Todd) Let A = k[x 1,..., x n ] and G a finite subgroup of GL n (k). Then A G is a polynomial ring G is generated by reflections Theorem (Kirkman, Kuzmanovich, and Zhang 2005) Let A be an AS regular algebra and G be a finite group of graded automorphisms of A. Then A G has finite global dimension G is generated by quasi-reflections of A. Conjecture (Kirkman) A G is AS regular G is generated by quasi-reflections of A. November 14, 2017 Invariant theory 10 / 34

11 Auslander s Theorem Another result in invariant theory involving reflections is Auslander s Theorem. G a finite subgroup of GL n (k) acting on A = k[x 1,..., x n ]. The skew group ring A#G is A kg as a k-vector space with multiplication (a g)(b h) = ag(b) gh. Define the Auslander map: γ A,G : A#G End A G(A) a#g (b ag(b)). In general, neither injective nor surjective. November 14, 2017 Invariant theory 11 / 34

12 Auslander s Theorem Theorem (Auslander 1962) Let G GL n (k) be a finite group acting on A = k[x 1,..., x n ]. If G is small (contains no reflections), then the Auslander map is an isomorphism. γ A,G : A#G End A G(A) a#g (b ag(b)). A key ingredient in the classical McKay correspondence. Let A = k[x 1, x 2 ] and let G be a small finite subgroup of GL 2 (k). Correspondences between irreducible representations of G, certain R G -modules, certain End R G(R)-modules, and the McKay quiver of G. When G SL 2 (k) these are of type ADE. November 14, 2017 Invariant theory 12 / 34

13 Auslander s Theorem A natural conjecture: Conjecture If A is an AS regular algebra and G GrAut(A) contains no quasi-reflections, then the Auslander map γ A,G is an isomorphism. When γ A,G is an isomorphism, there is a notion of the noncommutative McKay correspondence. (Chan, Kirkman, Walton, and Zhang, 2016) November 14, 2017 Invariant theory 13 / 34

14 Auslander s Theorem Some answers: Theorem (Chan, Kirkman, Walton, and Zhang 2016) If A a noetherian AS-regular algebra of global dimension 2 with trivial homological determinant, then the Auslander map is an isomorphism. Theorem (Mori and Ueyama 2016) A a noetherian AS regular algebra of dimension d 2 and G GrAut(A) with trivial homological determinant. TFAE: 1 G is ample for A. 2 A#G = End A G(A) and gl. dim tails A <. A#G 3 dim k < where f = 1 1#g. (f ) G g G November 14, 2017 Invariant theory 14 / 34

15 This seems pertinent Definition (Bao, He, and Zhang 2016) The pertinency p(a, G) = GKdim A GKdim A#G (f G ) where (f G ) is the two-sided ideal generated by f G = g G 1#g. (Actually defined for Hopf algebras) Under hypotheses of Mori and Ueyama, G is ample if and only if p(a, G) = n. November 14, 2017 Pertinency 15 / 34

16 This seems pertinent In fact: Theorem (Bao, He, and Zhang 2016) The Auslander map A#G End A G(A) is an isomorphism if and only if p(a, G) 2. November 14, 2017 Pertinency 16 / 34

17 This seems pertinent Recall: Theorem (Bao, He, and Zhang 2016) Let A = k 1 [x 1,..., x n ] and G = (1 2 n). Then p(a, G) 2 so the Auslander map is an isomorphism. Conjecture G contains no quasi-reflections, then the Auslander map γ A,G is an isomorphism. Example S n k 1 [x 1,..., x n ] contains no quasi-reflections. November 14, 2017 Pertinency 17 / 34

18 Question Let G = S n acting on A = k 1 [x 1,..., x n ]. Is the Auslander map γ A,G an isomorphism? November 14, 2017 Pertinency 18 / 34

19 This seems ideal Thanks to Bao, He, Zhang, we just need to understand the ideal (f G ) = 1#g. g G Theorem (Bao, He, and Zhang 2016) Let A be finitely generated over a central subalgebra T. Let A be the image of the map A A#G (A#G)/(f G ) and T A be the image of T. Then GKdim T = GKdim A = GKdim A#G (f G ). November 14, 2017 The ideal 19 / 34

20 This seems ideal So need only understand (f G ) A or even (f G ) T. A minus sign and a quotient: p(a, G) = GKdim A GKdim A#G (f G ) = GKdim A GKdim A (f G ) A (f G ) A A p(a, G). (f G ) A Idea: Construct enough elements in (f G ) so p(a, G) 2. November 14, 2017 The ideal 20 / 34

21 Constructing elements We borrow an idea from Brown and Lorenz. Assume A is commutative. For g G, define I(g) generated by {a g.a a A}. Lemma (Brown and Lorenz 1994) I(g) (f G ) A. g G,g e Issue 1: Proof requires commutativity. Issue 2: Produces elements of degree G 1 = n! 1 (often much higher than lowest degree element in (f G )). November 14, 2017 The ideal 21 / 34

22 Example: k 1 [x 1, x 2, x 3 ] and S 3 Let T = k[x 2 1, x2 2, x2 3 ] C(V 3) and f = σ S 3 1#σ. Define f 1 = x 2 1 f fx2 2 = (x 2 1 x2 2 )#(1) + (x2 1 x2 2 )#(13) + (x 2 1 x2 3 )#(23) + (x2 1 x2 3 )#(123) f 2 = x 2 1 f 1 f 1 x 2 3 = (x 2 1 x2 2 )(x2 1 x2 3 )#(1) + (x2 1 x2 3 )(x2 1 x2 2 )#(23) f 3 = x 2 2 f 2 f 2 x 2 3 = (x 2 1 x2 2 )(x2 1 x2 3 )(x2 2 x2 3 )#(1) (f ) C(A). This provides only one of the elements we need. We must use noncommutativity to obtain the second element. November 14, 2017 An example is worth a thousand theorems 22 / 34

23 Example: k 1 [x 1, x 2, x 3 ] and S 3 Recall Now f 2 = (x 2 1 x2 2 )(x2 1 x2 3 )#(1) + (x2 1 x2 3 )(x2 1 x2 2 )#(23). g 23 = (x 2 f 2 f 2 x 3 )(x 2 x 3 ) = (x 2 1 x2 2 )(x2 1 x2 3 )(x 2 x 3 ) 2 #(1) = (x 2 1 x2 2 )(x2 1 x2 3 )(x2 2 + x2 3 )#(1) (f ) C(A). We can similarly construct g 12 and g 13. Set g = g 12 + g 13 + g 23. November 14, 2017 An example is worth a thousand theorems 23 / 34

24 Example: k 1 [x 1, x 2, x 3 ] and S 3 The elements f 3 = (x 2 1 x2 2 )(x2 1 x2 3 )(x2 2 x2 3 ) g = (x 2 1 x2 2 )(x2 1 x2 3 )(x2 2 + x2 3 ) + (x 2 1 x2 2 )(x2 1 + x2 3 )(x2 2 x2 3 ) + (x x2 2 )(x2 1 x2 3 )(x2 2 x2 3 ) are relatively prime in T = k[x 2 1, x2 2, x2 3 ] and GKdim T/(f 3, g) 1. Theorem (Gaddis, Kirkman, Moore, and W) Let G be any subgroup of S n acting on A = k 1 [x 1,... x n ] as permutations. Then p(a, G) 2 so the Auslander map is an isomorphism. November 14, 2017 An example is worth a thousand theorems 24 / 34

25 Example: S(a, b, c) and (1 2 3) Let S(a, b, c) be the three-dimensional Sklyanin algebra S(a, b, c) = k x 1, x 2, x 3 ax 1 x 2 + bx 2 x 1 + cx 2 3 ax 2 x 3 + bx 3 x 2 + cx 2 1 ax 3 x 1 + bx 1 x 3 + cx 2 2. acted on by (1 2 3). Let f = 1#e + 1#(1 2 3) + 1#(1 3 2). Then f 1 = x 1 f fx 3 = (x 1 x 3 )#e + (x 1 x 2 )#(1 3 2). (x 1 x 2 )f 1 + f 1 (x 2 x 3 ) = (x 1 x 2 )(x 1 x 3 ) + (x 1 x 3 )(x 2 x 3 )#e = (x 2 1 x2 3 x 2x 1 x 1 x 2 )#e (f ). November 14, 2017 An example is worth a thousand theorems 25 / 34

26 Example: S(a, b, c) and (1 2 3) So x 2 1 x2 3 x 2x 1 x 1 x 2 (f ) S(a, b, c) x 2 2 x2 3 x 1x 2 x 2 x 1 (f ) S(a, b, c). Now a Gröbner basis argument implies dim k S(a, b, c) (f ) S(a, b, c) <. Theorem (Gaddis, Kirkman, Moore, and W) Let G = (1 2 3) acting on A = S(a, b, c) for generic (a : b : c) P 2. Then p(a, G) = 3 2 so the Auslander map is an isomorphism. November 14, 2017 An example is worth a thousand theorems 26 / 34

27 All the Auslanders! Theorem (Gaddis, Kirkman, Moore, and W) The Auslander map is an isomorphism for the following: 1 subgroups of S n acting on k 1 [x 1,..., x n ], 2 subgroups of S n acting on the ( 1)-quantum Weyl algebra, 3 subgroups of S 3 acting on the three-dimensional Sklyanin algebra S(1, 1, 1), 4 the cyclic group (1 2 3) acting on a generic three-dimensional Sklyanin algebra S(a, b, c), 5 subgroups of weighted permutations acting on the down-up algebra A(2, 1), 6 I n, (1 3)(2 4) acting on k 1 [x 1, x 2, x 3, x 4 ]. November 14, 2017 Results 27 / 34

28 Graded isolated singularities Definition (Ueyama 2013) A G is a graded isolated singularity if gl. dim tails A G <. Theorem (Ueyama 2016) If A G is a graded isolated singularity, then A G is an AS Gorenstein algebra of dimension d 2, A CM gr (A G ) is a (d 1)-cluster tilting module, and Ext 1 A G (A, M) and Ext 1 A G (M, A) are f.d. for M CM gr (A G ). Theorem (Mori and Ueyama 2016) If GKdim A 2, A G is a graded isolated singularity if and only dim k A#G/(f G ) < if and only if p(a, G) = n. November 14, 2017 Results 28 / 34

29 Graded isolated singularities Theorem (Bao, He, and Zhang 2016) Let A = k 1 [x 1,..., x 2 n] and G = (1 2 2 n ). Then p(a, G) = 2 n so A G is a graded isolated singularity. Theorem (Gaddis, Kirkman, Moore, and W) For the following, A G is a graded isolated singularity: 1 (1 2)(3 4), (1 3)(2 4) acting on k 1 [x 1, x 2, x 3, x 4 ], 2 (1 2)(3 4) (2n 1 2n) acting on k 1 [x 1,..., x 2n ], 3 (1 2 3) acting on a generic Sklyanin algebra S(a, b, c), 4 I n, (1 3)(2 4) acting on k 1 [x 1, x 2, x 3, x 4 ]. November 14, 2017 Results 29 / 34

30 Whither the upper bounds? Constructing elements of (f G ) gives lower bounds for p(a, G). Upper bounds? Theorem (Gaddis, Kirkman, Moore, and W) If G G then p(a, G) p(a, G ). (This resolves a conjecture of Bao-He-Zhang for the group case.) Corollary (Gaddis, Kirkman, Moore, and W) Let A be a noetherian connected graded algebra and suppose G contains a reflection g. If A and A g have finite global dimension, then the Auslander map γ A,G is not an isomorphism. November 14, 2017 Results 30 / 34

31 Computing pertinency exactly Lower bounds: constructing elements of (f G ) Upper bounds: subgroup theorem Subgroups of S 3 acting on k 1 [x 1, x 2, x 3 ]: conjugacy class p(a, G) (12) 2 (123) 2 or 3 (12), (23) 2 November 14, 2017 Results 31 / 34

32 Computing pertinency exactly Subgroups of S 4 acting on k 1 [x 1, x 2, x 3, x 4 ]: conjugacy class p(a, G) (12) 2 (12)(34) 4 (123) 2 or 3 (1234) 4 (12), (34) 2 (12)(34), (13)(24) 4 (1234), (24) 2 (123), (124) 2 or 3 (123), (12) 2 (1234), (12) 2 November 14, 2017 Results 32 / 34

33 Questions Other actions on other AS regular algebras? Permutation actions on graded skew Clifford algebras? (The squares are central). Pertinency bounds for Hopf actions? November 14, 2017 Results 33 / 34

34 Thanks! November 14, 2017 Results 34 / 34