Representation type and Auslander-Reiten theory of Frobenius-Lusztig kernels

Transcription

1 Representation type and Auslander-Reiten theory of Frobenius-Lusztig kernels Julian Külshammer University of Kiel, Germany

2 Notation Denote by: k an algebraically closed field (of characteristic p 0)

3 Notation Denote by: k an algebraically closed field (of characteristic p 0) Fix a Dynkin diagram A n,..., G 2

4 Notation Denote by: k an algebraically closed field (of characteristic p 0) Fix a Dynkin diagram A n,..., G 2 g corresponding Lie algebra

5 Notation Denote by: k an algebraically closed field (of characteristic p 0) Fix a Dynkin diagram A n,..., G 2 g corresponding Lie algebra G r r-th Frobenius kernel of algebraic group with Lie(G) = g

6 Notation Denote by: k an algebraically closed field (of characteristic p 0) Fix a Dynkin diagram A n,..., G 2 g corresponding Lie algebra G r r-th Frobenius kernel of algebraic group with Lie(G) = g l 3 an odd integer

7 Notation Denote by: k an algebraically closed field (of characteristic p 0) Fix a Dynkin diagram A n,..., G 2 g corresponding Lie algebra G r r-th Frobenius kernel of algebraic group with Lie(G) = g l 3 an odd integer ζ a primitive l-th root of unity

8 Definition of u ζ (g) Definition The Lusztig form of the quantised enveloping algebra is obtained in three steps: (i) Drinfeld-Jimbo quantum group U q (g) over Q(q) with generators E i, F i, K ±1 i, i = 1,..., n and quantum Serre relations

9 Definition of u ζ (g) Definition The Lusztig form of the quantised enveloping algebra is obtained in three steps: (i) Drinfeld-Jimbo quantum group U q (g) over Q(q) with generators E i, F i, K ±1 i, i = 1,..., n and quantum Serre relations (ii) a Z[q, q 1 ]-subalgebra U Z (g) generated by the divided powers E (n) i, F (n) i, K ±1 i

10 Definition of u ζ (g) Definition The Lusztig form of the quantised enveloping algebra is obtained in three steps: (i) Drinfeld-Jimbo quantum group U q (g) over Q(q) with generators E i, F i, K ±1 i, i = 1,..., n and quantum Serre relations (ii) a Z[q, q 1 ]-subalgebra U Z (g) generated by the divided powers E (n) i, F (n) i, K ±1 i (iii) U ζ (g) obtained by specialising to k via q ζ

11 Definition of u ζ (g) Definition The Lusztig form of the quantised enveloping algebra is obtained in three steps: (i) Drinfeld-Jimbo quantum group U q (g) over Q(q) with generators E i, F i, K ±1 i, i = 1,..., n and quantum Serre relations (ii) a Z[q, q 1 ]-subalgebra U Z (g) generated by the divided powers E (n) i, F (n) i, K ±1 i (iii) U ζ (g) obtained by specialising to k via q ζ The small quantum group U ζ (G 0 ) is now the k-subalgebra generated by E i, F i, K ±1 i

12 The sl 2 -case Example For g = sl 2 the small quantum group is the k-algebra with generators E, F, K ±1 and relations:

13 The sl 2 -case Example For g = sl 2 the small quantum group is the k-algebra with generators E, F, K ±1 and relations: (R2) KEK 1 = ζ 2 E (R3) KFK 1 = ζ 2 F

14 The sl 2 -case Example For g = sl 2 the small quantum group is the k-algebra with generators E, F, K ±1 and relations: (R2) KEK 1 = ζ 2 E (R3) KFK 1 = ζ 2 F (R4) EF FE = K K 1 ζ ζ 1

15 The sl 2 -case Example For g = sl 2 the small quantum group is the k-algebra with generators E, F, K ±1 and relations: (R2) KEK 1 = ζ 2 E (R3) KFK 1 = ζ 2 F K K 1 (R4) EF FE = ζ ζ 1 (s) E l = F l = K l 1 = 0

16 Why Frobenius-Lusztig kernel? In positive characteristic: k U ζ (G 0 ) U ζ (G r ) Fr ζ Dist(G r ) k

17 Why Frobenius-Lusztig kernel? In positive characteristic: k U ζ (G 0 ) U ζ (G r ) Fr ζ Dist(G r ) k Properties of U ζ (G r ):

18 Why Frobenius-Lusztig kernel? In positive characteristic: k U ζ (G 0 ) U ζ (G r ) Fr ζ Dist(G r ) k Properties of U ζ (G r ): finite dimensional

19 Why Frobenius-Lusztig kernel? In positive characteristic: k U ζ (G 0 ) U ζ (G r ) Fr ζ Dist(G r ) k Properties of U ζ (G r ): finite dimensional Hopf algebra, neither commutative nor cocommutative

20 Why Frobenius-Lusztig kernel? In positive characteristic: k U ζ (G 0 ) U ζ (G r ) Fr ζ Dist(G r ) k Properties of U ζ (G r ): finite dimensional Hopf algebra, neither commutative nor cocommutative symmetric

21 Normal Hopf subalgebras Theorem k B A C k. Assume simple B-modules lift to A.

22 Normal Hopf subalgebras Theorem k B A C k. Assume simple B-modules lift to A.Then: All simple A-modules are of the form M L, where M is a simple C-module and L is a simple B-module

23 Normal Hopf subalgebras Theorem k B A C k. Assume simple B-modules lift to A.Then: All simple A-modules are of the form M L, where M is a simple C-module and L is a simple B-module Assume simple B-modules have no self-extensions. Then: Ext 1 (M 1 L 1, M 2 L 2 ) = { Ext 1 (M 1, M 2 ) L 1 = L2 Hom(M 1, M 2 Ext 1 (L 1, L 2 )) L 1 L 2

24 Normal Hopf subalgebras Theorem k B A C k. Assume simple B-modules lift to A.Then: All simple A-modules are of the form M L, where M is a simple C-module and L is a simple B-module Assume simple B-modules have no self-extensions. Then: Ext 1 (M 1 L 1, M 2 L 2 ) = { Ext 1 (M 1, M 2 ) L 1 = L2 Hom(M 1, M 2 Ext 1 (L 1, L 2 )) L 1 L 2 Assume B has a simple projective module St. Then: There is a block embedding mod C mod A, M M St.

25 Support varieties If certain finiteness assumptions on cohomology hold, we can define a theory of support varieties for Hopf algebras, i.e. to each module M one can associate a variety V(M) reflecting certain properties of M. Conjecture (fg) The Hopf algebras U ζ (G r ) satisfy these finiteness assumptions.

26 Support varieties If certain finiteness assumptions on cohomology hold, we can define a theory of support varieties for Hopf algebras, i.e. to each module M one can associate a variety V(M) reflecting certain properties of M. Conjecture (fg) The Hopf algebras U ζ (G r ) satisfy these finiteness assumptions. Proven for r = 0 and r = 1 (in special cases). From now on assumed in general.

27 Representation type of Frobenius-Lusztig kernels Theorem Assume (fg). Then the blocks of U ζ (G r ) are: representation-finite only if simple, only St.

28 Representation type of Frobenius-Lusztig kernels Theorem Assume (fg). Then the blocks of U ζ (G r ) are: representation-finite only if simple, only St. tame for A 1 and r = 0: all but one block r = 1: all but one of the block in the image of St.

29 Representation type of Frobenius-Lusztig kernels Theorem Assume (fg). Then the blocks of U ζ (G r ) are: representation-finite only if simple, only St. tame for A 1 and r = 0: all but one block r = 1: all but one of the block in the image of St. wild in all other cases.

30 Only SL 2 with r = 0, 1 has tame blocks Proof. 1 If B is a block of an (fg)-hopf algebra A and there is M with dim V(M) 3, then B is wild. [Farnsteiner 2007, Feldvoss-Witherspoon 2009] This allows to restrict to A 1

31 Only SL 2 with r = 0, 1 has tame blocks Proof. 1 If B is a block of an (fg)-hopf algebra A and there is M with dim V(M) 3, then B is wild. [Farnsteiner 2007, Feldvoss-Witherspoon 2009] This allows to restrict to A 1 2 For A 1, one can compute the quiver of U ζ (G r ) by considering the quiver of U ζ (G 0 ) and the quiver of Dist(G r ), i.e. using the Ext-group result from before. Then one can identify a wild subquiver

32 Webb s Theorem Theorem (Kerner-Zacharia 2009) Let A be an (fg)-hopf algebra. Then the components of the stable Auslander-Reiten quiver Γ s (A) are of the form: periodic: finite or Z[A ]/τ m (tubes)

33 Webb s Theorem Theorem (Kerner-Zacharia 2009) Let A be an (fg)-hopf algebra. Then the components of the stable Auslander-Reiten quiver Γ s (A) are of the form: periodic: finite or Z[A ]/τ m (tubes) non-periodic: Z[ ], where is Euclidean or infinite Dynkin

34 Webb s Theorem Theorem (Kerner-Zacharia 2009) Let A be an (fg)-hopf algebra. Then the components of the stable Auslander-Reiten quiver Γ s (A) are of the form: periodic: finite or Z[A ]/τ m (tubes) non-periodic: Z[ ], where is Euclidean or infinite Dynkin Theorem The following hold for Γ s (U ζ (G r )) not associated to A 1 : There are no Euclidean components.

35 Webb s Theorem Theorem (Kerner-Zacharia 2009) Let A be an (fg)-hopf algebra. Then the components of the stable Auslander-Reiten quiver Γ s (A) are of the form: periodic: finite or Z[A ]/τ m (tubes) non-periodic: Z[ ], where is Euclidean or infinite Dynkin Theorem The following hold for Γ s (U ζ (G r )) not associated to A 1 : There are no Euclidean components. For r = 0 the periodic components are of the form Z[A ]/τ.

36 Webb s Theorem Theorem (Kerner-Zacharia 2009) Let A be an (fg)-hopf algebra. Then the components of the stable Auslander-Reiten quiver Γ s (A) are of the form: periodic: finite or Z[A ]/τ m (tubes) non-periodic: Z[ ], where is Euclidean or infinite Dynkin Theorem The following hold for Γ s (U ζ (G r )) not associated to A 1 : There are no Euclidean components. For r = 0 the periodic components are of the form Z[A ]/τ. For r = 0 the components containing U ζ (g)-modules, e.g. simples, are of the form Z[A ].

37 Application Corollary If S is a simple module for U ζ (G r ) in a Z[A ]-component, then ht P(S) = rad P(S)/ soc P(S) is indecomposable.

38 Indecomposable Heart Proof. 0 rad P ht P P P/ soc P 0 is the standard almost split sequence, i.e. if P/ soc P has only one predecessor in Γ s (U ζ (G r )), then ht P is indecomposable.

39 Indecomposable Heart Proof. 0 rad P ht P P P/ soc P 0 is the standard almost split sequence, i.e. if P/ soc P has only one predecessor in Γ s (U ζ (G r )), then ht P is indecomposable. Applying Ω to this sequence this is equivalent to Ω(P/ soc P) = S has only one predecessor.

40 Indecomposable Heart Proof. 0 rad P ht P P P/ soc P 0 is the standard almost split sequence, i.e. if P/ soc P has only one predecessor in Γ s (U ζ (G r )), then ht P is indecomposable. Applying Ω to this sequence this is equivalent to Ω(P/ soc P) = S has only one predecessor. [Kawata] Symmetric algebras with a non-quasi-simple simple module S in a Z[A ]-component have a very special shape of projective modules.

41 Indecomposable Heart Proof. 0 rad P ht P P P/ soc P 0 is the standard almost split sequence, i.e. if P/ soc P has only one predecessor in Γ s (U ζ (G r )), then ht P is indecomposable. Applying Ω to this sequence this is equivalent to Ω(P/ soc P) = S has only one predecessor. [Kawata] Symmetric algebras with a non-quasi-simple simple module S in a Z[A ]-component have a very special shape of projective modules. This contradicts a certain Ext-symmetry for U ζ (G r ).

42 The tame small quantum group Theorem (...) The tame blocks are as follows: x x y y 1 2 ; xy = yx = 0, x 2 = y 2

43 The tame small quantum group Theorem (...) The tame blocks are as follows: special biserial x x y y 1 2 ; xy = yx = 0, x 2 = y 2

44 The tame small quantum group Theorem (...) The tame blocks are as follows: special biserial x x y y 1 2 ; xy = yx = 0, x 2 = y 2 stable Auslander-Reiten quiver: 2 components of type Z[Ã 12 ] and two P 1 -families of homogeneous tubes.

45 Stable Auslander-Reiten quiver of u ζ (sl 2 )