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.
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.
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).
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
[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.
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.
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.
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.
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).
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.
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.
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.