Categorification of (induced) cell modules and the rough structure of generalised Verma modules

Mazorchuk, Volodymyr; Stroppel, Catharina

doi:10.1016/j.aim.2008.06.019

Cited by 68 publications

(136 citation statements)

References 48 publications

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“…For the definition and basic properties of this dimension we refer to [Ja]. The idea of constructing such filtrations is not new and was worked out earlier in much more generality for modules over symmetric groups and Hecke algebras ( [GGOR,6.1.3], [MS2,7.2]) and over Lie algebras [CR,Proposition 4.10], [Rou,Theorem 5.8]).…”

Section: Fattened Vermas and The Cohomology Rings Of Grassmanniansmentioning

confidence: 99%

“…In terms of categorification, we have already seen the weight space decomposition as a decomposition of categories into blocks. Here, we give a categorical analogue of the decomposition (9) based on Gelfand-Kirillov dimension (which is directly connected with Lusztig's afunction, see [MS2,Remark 42]). For the definition and basic properties of this dimension we refer to [Ja].…”

Section: Fattened Vermas and The Cohomology Rings Of Grassmanniansmentioning

confidence: 99%

“…The above filtration should be compared with the more general, but coarser filtrations [CR,Proposition 5.10] and [MS2,7.2]. Although the above filtrations carry over without problems to the quantum/graded version, the notion of minimal categorification is not yet completely developed in this context, but see [Rou].…”

Section: Graphical Calculus Gk-dimension and Lusztig's A-functionmentioning

confidence: 99%

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Categorifying fractional Euler characteristics, Jones–Wenzl projectors and $3j$-symbols

Frenkel¹,

Stroppel²,

Sussan³

2012

Quantum Topol.

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Abstract. We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as graded Euler characteristics. The 3j-symbols appear as new generalizations of Kazhdan-Lusztig polynomials.A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones-Wenzl projector, Θ-networks and tetrahedron networks. Networks and their evaluations play an important role in the Turaev-Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of 3-manifolds will be studied in detail in subsequent papers.

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Section: Fattened Vermas and The Cohomology Rings Of Grassmanniansmentioning

confidence: 99%

Section: Fattened Vermas and The Cohomology Rings Of Grassmanniansmentioning

confidence: 99%

Section: Graphical Calculus Gk-dimension and Lusztig's A-functionmentioning

confidence: 99%

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Categorifying fractional Euler characteristics, Jones–Wenzl projectors and $3j$-symbols

Frenkel¹,

Stroppel²,

Sussan³

2012

Quantum Topol.

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“…In [19], the answer to Kostant's problem was given for all simple modules in O 0 for sl n , n ≤ 5, and partial results were obtained for sl 6 . In type A the answer to Kostant's problem is a left cell invariant by [27,Theorem 60]. Furthermore, since in type A there is a unique involution in each left cell, it suffices to solve Kostant's problem for involutions.…”

mentioning

confidence: 99%

“…For a review of this theory, see [27], in particular Section 3. Hence, given a right cell R of W , all composition factors in the form L(x), x ∈ R of (e) occur in degree at least a(x), and there is precisely one such factor which occur in degree a(x), namely the one corresponding to the Duflo involution in R.…”

mentioning

confidence: 99%

Kostant's Problem and Parabolic Subgroups

Kåhrström

2009

Glasgow Math. J.

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Abstract. Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let g I be the corresponding semisimple subalgebra of g. Denote by W I the Weyl group of g I and let w • and w I• be the longest elements of W and W I , respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight g I -module L I (x) of highest weight x · 0, x ∈ W I , as the answer for the simple highest weight g-module L(xwWe also give a new description of the unique quasi-simple quotient of the Verma module (e) with the same annihilator as L(y), y ∈ W .

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Parabolic category O for classical Lie superalgebras

Mazorchuk

2014

Advances in Lie Superalgebras

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Categorification of (induced) cell modules and the rough structure of generalised Verma modules

Cited by 68 publications

References 48 publications

Categorifying fractional Euler characteristics, Jones–Wenzl projectors and $3j$-symbols

Categorifying fractional Euler characteristics, Jones–Wenzl projectors and $3j$-symbols

Kostant's Problem and Parabolic Subgroups

Parabolic category O for classical Lie superalgebras

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