UNSW-2018

Transcript

1. Quantum groups via finite-dimensional algebras S.R. Doty Pure Maths Seminar

   UNSW 10 April 2018

2. Serre’s presentation of a semisimple Lie algebra Serre’s Theorem. A

   complex semisimple Lie algebra g with Cartan matrix (cα,β) is the free Lie algebra generated by {eα, fα, hα : α ∈ ∆} modulo the ideal generated by the relations: (1) [hα, hβ] = 0 (2) [eα, fβ] = δα,βhα (3) [hα, eβ] = cα,βeβ , [hα, fβ] = −cα,βfβ (4) (ad eα)1−cα,β eβ = 0 if α = β (5) (ad fα)1−cα,β fβ = 0 if α = β. Relations (4), (5) are often called the Serre relations. In U(g) they take the equivalent form: (4’) 1−cα,β k=0 1−cα,β k e1−cα,β−k α eβek α = 0 if α = β (5’) 1−cα,β k=0 1−cα,β k f 1−cα,β−k α fβf k α = 0 if α = β.

3. Quantum groups Let Uq = Uq(g) be a Drinfeld–Jimbo quantized

   enveloping algebra of a semisimple complex Lie algebra g. I.e., Uq = associative Q(q)-algebra generated by {Eα, Fα, Kα, K−1 α : α ∈ ∆} subject to defining relations KαK−1 α = 1 = K−1 α Kα, KαKβ = KβKα (1) EαFβ − FβEα = δα,β Kα − K−1 α qα − q−1 α (2) KαEβ = qcα,β α EβKα, KαFβ = q−cα,β α FβKα (3) 1−cα,β s=0 (−1)s 1 − cα,β s α E1−cα,β−s α EβEs α = 0 (α = β) (4) 1−cα,β s=0 (−1)s 1 − cα,β s α F1−cα,β−s α FβFs α = 0 (α = β). (5)

4. Lusztig’s modified form ˙ Uq Let X = set of

   integral weights for g. Lusztig constructed certain quotients 1λ ˙ Uq1µ of Uq (as vector spaces) and glued them together to form ˙ Uq = λ,µ∈X 1λ ˙ Uq1µ. This is an algebra without 1, with a family of pairwise orthogonal idempotents (1λ)λ∈X . The algebra ˙ Uq has two virtues over Uq: The representations of ˙ Uq are precisely the Type 1 integrable representations of Uq. The canonical basis extends from U− q to all of ˙ Uq, and this is a cellular basis of ˙ Uq.

5. Generalized q-Schur algebras Let W = Weyl group of g.

   Let π = a finite saturated set of dominant weights. Definition. The generalized q-Schur algebra Sq(π) is the quotient of ˙ Uq by the ideal generated by {1µ : µ ∈ X \ W π}. It makes sense to set q = 1 in the definition: S(π) := Sq(π)|q=1. This is an (unquantized) generalized Schur algebra in Donkin’s sense (over Q). It is the quotient of ˙ U(g) by the ideal generated by {1µ : µ ∈ X \ W π}. Main point is that (q)-Schur algebras localize representation theory: Sq(π)-mod = type 1 rep’s of Uq with weights in W π. S(π)-mod = rep’s of U(g) with weights in W π. By forming an inverse system (Sq(π))π of generalized q-Schur algebras, we can reconstruct Uq inside the inverse limit (D, 2009).

6. References Type A [1] D–Giaquinto: IMRN 2002 [2] Du–Parshall: Transactions

   AMS 2002 [3] Dipper-D: Rep Theory 2008 Types B, C D [4] D–Giaquinto–Sullivan: Adv. Math 2006 General type [5] D: Rep Theory 2003 [6] D–Giaquinto–Sullivan: Adv. Math 2009 [7] D: J Algebra 2009 [8] D-Giaquinto: Math Proc Camb Phil Soc 2017

7. Purpose of this talk In all of the listed papers

   [1]–[8], generators and relations are given for various generalized (q)-Schur algebras (with or without the q). References [1]–[7] are flawed, in that they impose (q)-Serre relations unnecessarily. This was pointed out recently by R. Rouquier. Reference [8] D-Giaquinto: Math Proc Camb Phil Soc 2017 sets the record straight, and gives derivations of the (q)-Serre relations as a consequence of the other defining relations. One purpose of this talk is to explain why the (q)-Serre relations are superfluous in presentations of generalized (q)-Schur algebras, and to publicize this fact.

8. Serre’s presentation of a semisimple Lie algebra (again) Serre’s Theorem.

   A complex semisimple Lie algebra g with Cartan matrix (cα,β) is the free Lie algebra generated by {eα, fα, hα : α ∈ ∆} modulo the ideal generated by the relations: (1) [hα, hβ] = 0 , [eα, fβ] = δα,βhα (2) [hα, eβ] = cα,βeβ , [hα, fβ] = −cα,βfβ (3) (ad eα)1−cα,β eβ = 0 if α = β (4) (ad fα)1−cα,β fβ = 0 if α = β. Relations (3), (4) are often called the Serre relations. In U(g) they take the equivalent form: (3’) 1−cα,β k=0 1−cα,β k e1−cα,β−k α eβek α = 0 if α = β (4’) 1−cα,β k=0 1−cα,β k f 1−cα,β−k α fβf k α = 0 if α = β.

9. Corollary In their 1997 book, Representation Theory and Complex Geometry,

   Chriss–Ginzburg give the following explicit corollary to Serre’s theorem. Corollary. If A is any finite-dimensional associative algebra containing elements {eα, fα, hα : α ∈ ∆} satisfying relations (1), (2) in Serre’s presentation, then the Serre relations (3), (4) must hold in A. Why? This follows from the same argument that proves the Serre relations for the Lie algebra. One considers the adjoint representation ad : U(g) → End C (g), (ad x)y := [x, y]. Then, for fixed α, the subalgebra generated by {eα, fα, hα} forms a copy of U(sl2) and g (or A) is a finite-dimensional representation of that sl2. Now apply well known facts about sl2 representations. [Kashiwara, Miwa, Peterson, and Yung: Selecta Math 1996] prove a q-analogue of the above corollary.

10. Application to generalized (q)-Schur algebras Since generalized Schur algebras are

    finite-dimensional quotients of U(g), it comes for free from the Corollary that they must satisfy the Serre relations. Similarly, generalized q-Schur algebras are finite-dimensional quotients of Uq(g), so again it comes for free that they must satisfy the q-Serre relations.

11. Generalized q-Schur algebras (again) (Improved) Theorem (D, 2003) Sq(π) is

    the associative algebra over Q(q) on generators Eα, Fα (α ∈ ∆) and 1λ (λ ∈ W π) subject to the relations: (1) 1λ1µ = δλ,µ1λ, λ∈W π 1λ = 1 (2) EαFβ − FβEα = δα,β λ∈W π [α∨(λ)]α1λ (3) Eα1λ = 1λ+αEα, 1λEα = 1λ−αEα, (4) Fα1λ = 1λ−αFα, 1λFα = Fα1λ+α (5) q-Serre relations for any λ, µ ∈ W π, and α, β ∈ ∆. In the right hand side of relations (3), (4) we agree that 1λ±αi = 0 whenever λ ± αi / ∈ W π. By setting q = 1, the unquantized S(π) is defined by the same set of generators, subject to the same relations (1)–(4), except in (2) the quantum integer [α∨(λ)]α becomes the ordinary integer α∨(λ).

12. The Dipper–James q-Schur algebra (type A) Theorem (D–Giaquinto, 2002; Du–Parshall,

    2003) The Dipper–James q-Schur algebra Sq(n, d) is isomorphic to the quotient of Uq(gln ) by the ideal generated by (∗) K1 · · · Kn − qd , Ki (Ki − q)(Ki − q2) · · · (Ki − qd ). (Corrected) Corollary. The Dipper–James q-Schur algebra Sq(n, d) is the associative algebra on generators Ei , Fi (i = 1, . . . , n − 1), Ki (i = 1, . . . , n) subject to relations: (1) Usual “easy” defining relations for Uq(gln ). (2) The above K-relations (∗). (3) q-Serre relations We don’t need to impose the Serre relations to present the algebra.

13. History: generalized Schur algebras Schur algebras in type A were

    popularized in the influential 1980 monograph Polynomial Representations of GLn, by J.A. Green (SLN 830). They have important applications to the modular representations of symmetric groups, coming from Schur–Weyl duality. During the mid-1980s, Donkin generalized them to other types. Donkin’s construction: G = algebraic group over a field k A = k[G] = coordinate algebra (a Hopf algebra) π = finite saturated set of dominant weights A(π) = finite-dimensional sub-coalgebra of A (truncate) S(π) := A(π)∗ (linear dual). S(π) is a finite dimensional algebra, called a generalized Schur algebra.

14. More history: generalized Schur algebras Donkin proved that S(π) is

    always quasi-hereditary over a field. (Equivalently, S(π)-mod is a highest weight category.) Taking G = GLn and π to be the set of dominant weights of V ⊗d , V the vector representation, one recovers the type A Schur algebra S(n, d). Donkin also showed that: Theorem For a field k, the algebra S(π) is a quotient of the ‘hyperalgebra’ Uk(g). The hyperalgebra (AKA ‘algebra of distributions’ in Jantzen’s book) is defined as Uk(g) := k ⊗ Z U Z (g) where U Z (g) = Kostant’s integral form of U(g), generated by divided powers of the Chevalley generators eα, fα. Here g = complex Lie algebra of the same type as G.