On the defining relations for generalized q-Schur algebras (original) (raw)

Journal of Algebra, 1999

We obtain a presentation of certain affine q-Schur algebras in terms of generators and relations. The presentation is obtained by adding more relations to the usual presentation of the quantized enveloping algebra of type affine gl n. Our results extend and rely on the corresponding result for the q-Schur algebra of the symmetric group, which were proved by the first author and Giaquinto.

Presenting affine q-Schur algebras

Mathematische Zeitschrift, 2007

Cellular and quasihereditary structures of generalized quantized Schur algebras

2010

We show that cellular bases of generalized q-Schur algebras can be constructed by gluing arbitrary bases of the cell modules and their dual basis (with respect to the anti-involution giving the cell structure) along defining idempotents. For the rational form, over the field Q(v) of rational functions in an indeterminate v, our proof of this fact is self-contained and independent of the theory of quantum groups. In the general case, over a commutative ring k regarded as a Z[v, v −1 ]-algebra via specialization v → q for some chosen invertible q ∈ k, our argument depends on the existence of the canonical basis.

Constructing quantized enveloping algebras via inverse limits of finite dimensional algebras

2009

It is well known that a generalized q-Schur algebra may be constructed as a quotient of a quantized enveloping algebra U or its modified formU. On the other hand, we show here that both U andU may be constructed within an inverse limit of a certain inverse system of generalized q-Schur algebras. Working within the inverse limit U clarifies the relation betweenU and U. This inverse limit is a q-analogue of the linear dual R[G] * of the coordinate algebra of a corresponding linear algebraic group G.

A geometric construction of generalized 𝑞-Schur algebras

Contemporary Mathematics, 2014

We show that the algebras C (X × X) in [Li10] and L d in [Li12] are generalized q-Schur algebras as defined in [D03]. This provides a geometric construction of generalized q-Schur algebras in types A, D and E. We give a parameterization of Nakajima's Lagrangian quiver variety of type D associated to a certain highest weight.

Integral and graded quasi-hereditary algebras, II with applications to representations of generalized qqq-Schur algebras and algebraic groups

arXiv (Cornell University), 2009

Given a quasi-hereditary algebra B, we present conditions which guarantee that the algebra grB obtained by grading B by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that B might possess. The method involves working with a pair (A, a) consisting of a quasi-hereditary algebra A and a (positively) graded subalgebra a. The algebra B arises as a quotient B = A/J of A by a defining ideal J of A. Along the way, we also show that the standard (Weyl) modules for B have a structure as graded modules for a. These results are applied to obtain new information about the finite dimensional algebras (e. g., the q-Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated to semisimple algebraic groups in positive characteristic p. These results require, at least presently, considerable restrictions on the size of p.

Generators and relations for Schur algebras

2001

We obtain a presentation of Schur algebras (and q-Schur algebras) by generators and relations, one which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra gl n of n × n matrices. We also find several new bases of Schur algebras.

Cellular bases of generalized q-Schur algebras

We give a new proof that generalized quantized Schur algebras are cellular and their specializations (with respect to the Lusztig integral form) are quasihereditary over any field of characteristic zero. The proof is independent of the theory of quantum groups, and in particular does not depend on the existence of the canonical basis, in contrast with the earlier proof.

On representation types of q-Schur algebras

Journal of Pure and Applied Algebra, 1993

Xi, C., On representation types of q-Schur algebras, Journal of Pure and Applied Algebra 84 (1993) 73-84. A sufficient condition for a q-Schur algebra to have finitely many finitely generated nonisomorphic indecomposable modules is provided. Moreover, if the condition is satisfied, then the q-Schur algebra is quadratic and its structure can be determined. It is also proved that if a Schur algebra is representation infinite then it is wild.

Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl 2

2001

We obtain a presentation of the quantum Schur algebras Sv(2, d) by generators and relations. This presentation is compatible with the usual presentation of the quantized enveloping algebra U = Uv(gl2). In the process we find new bases for Sv(2, d). We also locate the Z[v, v −1 ]-form of the quantum Schur algebra within the presented algebra and show that it has a basis which is closely related to Lusztig’s basis of the Z[v, v −1 ]-form of U.

On the defining relations for generalized q-Schur algebras (original) (raw)

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