On selfadjoint functors satisfying polynomial relations

Agerholm, Troels; Mazorchuk, Volodymyr

doi:10.1016/j.jalgebra.2011.01.004

Cited by 13 publications

(7 citation statements)

References 29 publications

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“…Proof Since M and Z (1) commute, we can simultaneously upper-triangularize them and find a basis of common right (generalized) eigenvectors for them. Similarly, we can simultaneously lower-triangularize M and Z (1) and find a basis of common left (generalized) eigenvectors.…”

Section: Proposition 320 the Grothendieck Ringmentioning

confidence: 99%

McKay matrices for finite-dimensional Hopf algebras

Benkart

Biswal²,

Kirkman³

et al. 2021

Can. J. Math.-J. Can. Math.

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For a finite-dimensional Hopf algebra A, the McKay matrix M V of an A-module V encodes the relations for tensoring the simple A-modules with V. We prove results about the eigenvalues and the right and left (generalized) eigenvectors of M V by relating them to characters. We show how the projective McKay matrix Q V obtained by tensoring the projective indecomposable modules of A with V is related to the McKay matrix of the dual module of V. We illustrate these results for the Drinfeld double D n of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of M V and Q V in terms of several kinds of Chebyshev polynomials. For the matrix N V that encodes the fusion rules for tensoring V with a basis of projective indecomposable D n-modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

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Section: Proposition 320 the Grothendieck Ringmentioning

confidence: 99%

McKay matrices for finite-dimensional Hopf algebras

Benkart

Biswal²,

Kirkman³

et al. 2021

Can. J. Math.-J. Can. Math.

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“…where the limit is taken over all pairs of submodules M 0 ⊂ M and N 0 ⊂ N such that M M 0 , and N 0 are contained in S. For an irreducible A-module N let P (N ) denote the projective cover of N . Then the following is well-known (see for instance [AM,Prop. 33] for a detailed proof).…”

Section: Serre Subcategories and Quotient Functorsmentioning

confidence: 99%

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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“…We also have the matrix bookkeeping the composition multiplicities in θ L ½j and θ L θ . By [AM,Lemma 8], this matrix is 0 1 b a .…”

Section: Application To Soergel Bimodules Part Imentioning

confidence: 99%

Analogues of Centralizer Subalgebras for Fiat 2-Categories and Their 2-Representations

Mackaay

Mazorchuk²,

Miemietz³

et al. 2018

J. Inst. Math. Jussieu

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The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex J of a fiat 2-category C and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of C associated with a diagonal H-cell in J . As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types B 3 and B 4 .We also thank Tobias Kildetoft for his help regarding the computation of cells in types B 3 , B 4 and B 5 .2. Preliminaries in abstract 2-representation theory 2.1. 2-categories and 2-representations. We refer to [Le, McL] for generalities and terminology on 2-categories. We refer to [MM1, MM2, MM3, MM4, MM5, MM6] for

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On selfadjoint functors satisfying polynomial relations

Cited by 13 publications

References 29 publications

McKay matrices for finite-dimensional Hopf algebras

McKay matrices for finite-dimensional Hopf algebras

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

Analogues of Centralizer Subalgebras for Fiat 2-Categories and Their 2-Representations

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