Quasi-hereditary algebras

Dlab, Vlastimil; Ringel, Claus Michael

doi:10.1215/ijm/1255988725

Cited by 240 publications

(192 citation statements)

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“…Let T (x · λ) ∈ λ O be the indecomposable tilting module with M (x · λ) occurring as a submodule in any Verma flag. (For the classification we refer to [CI89], and for the general theory to [DR89], for example. )…”

mentioning

confidence: 99%

A categorification of finite-dimensional irreducible representations of quantum $${\mathfrak{sl}_2}$$ and their tensor products

Frenkel

Khovanov

Stroppel

2007

Sel. math., New ser.

103

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The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for sl 2 . The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra gl n . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed. We also give a categorical version of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.

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mentioning

confidence: 99%

A categorification of finite-dimensional irreducible representations of quantum $${\mathfrak{sl}_2}$$ and their tensor products

Frenkel

Khovanov

Stroppel

2007

Sel. math., New ser.

103

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“…Dlab and Ringel [13] have shown that every finite dimensional algebra B is of the form eAe for some quasi-hereditary algebra A. In a weak sense, the algebra A can be considered as a Schur algebra of B.…”

Section: Schur Algebras For Cellular Algebrasmentioning

confidence: 99%

Dominant Dimension and Almost Relatively True Versions of Schur’s Theorem

Koenig

2010

Milan J. Math.

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Perhaps the most fundamental problems of representation theory are to classify and to describe irreducible (=simple) representations and to determine cohomology. It is crucial to develop techniques that allow to transfer information from some (known) cases to other (unknown) cases. A classical result of this kind, due to Schur, recently has been extended widely, and put into a general context. These modern 'relative' versions of Schur's result will be presented. Moreover, the theoretical background behind these results, and the crucial invariant controlling the existence and strength of such equivalences, will be explained, and illustrated by an explicit example. Finally, some open problems will be stated and discussed.

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“…Similarly, a costandard module (λ) is defined as the greatest submodule of an injective enveloping module I(λ) for L(λ) such that all of its composition factors are indexed by weights µ ≤ λ. A fundamental notion of a quasihereditary algebra was introduced in [8,9]. Under the same initial data as above, A is said to be quasihereditary if kernels of natural epimorphisms P (λ) → (λ) have filtrations with factors (µ), µ > λ.…”

Section: Introductionmentioning

confidence: 99%

Borel Subalgebras of Schur Superalgebras

Zubkov

2005

Algebr Logic

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It is proved that any Schur superalgebra is representable as a product of two Borel subalgebras of that superalgebra, which are symmetric w.r.t. its natural anti-isomorphism (Bruhat-Tits decomposition). This readily implies that any simple module is uniquely defined by its highest weight, and all other weights are strictly less than is the highest under the dominant ordering. It is stated that the fundamental theorem of Kempf, which is valid for all classical Schur algebras, might be true for superalgebras only if they are semisimple. Nevertheless, a weaker theorem of Grothendieck holds true for superalgebras since Borel subalgebras are quasihereditary. Also we formulate an analog of the Donkin-Mathieu theorem for Schur superalgebras, and show that it is valid in the elementary non-classical case, that is, for the algebras S(1|1, r).

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Quasi-hereditary algebras

Cited by 240 publications

References 1 publication

A categorification of finite-dimensional irreducible representations of quantum $${\mathfrak{sl}_2}$$ and their tensor products

A categorification of finite-dimensional irreducible representations of quantum $${\mathfrak{sl}_2}$$ and their tensor products

Dominant Dimension and Almost Relatively True Versions of Schur’s Theorem

Borel Subalgebras of Schur Superalgebras

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