scholarly journals Coboundary Categories and Local Rules

10.37236/7019 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Bruce W. Westbury
Keyword(s):  
Quantum Groups ◽  
Hecke Algebras ◽  
Highest Weight

First we develop the theory of local rules for coboundary categories. Then we describe the local rules in two main cases. First for the quantum groups in general and in the seminormal representations of the Hecke algebras. Then for crystals in general and specifically for crystals of minuscule representations. Finally we show how growth diagrams can be extended to construct the action of the cactus group on highest weight words.  

scholarly journals A remark on convolution products for quiver Hecke algebras

2020 ◽  
Vol 31 (11) ◽  
pp. 2050092
Author(s):  
Myungho Kim ◽  
Euiyong Park
Keyword(s):  
Tensor Product ◽  
Quantum Groups ◽  
Hecke Algebras ◽  
Weight Vector ◽  
Base Field ◽  
Projection Formula ◽  
Positivity Condition ◽  
Highest Weight ◽  
Convolution Products ◽  
Quiver Hecke Algebra

In this paper, we investigate a connection between convolution products for quiver Hecke algebras and tensor products for quantum groups. We give a categorification of the natural projection [Formula: see text] sending the tensor product of the highest weight vectors to the highest weight vector in terms of convolution products. When the quiver Hecke algebra is symmetric and the base field is of characteristic [Formula: see text], we obtain a positivity condition on some coefficients associated with the projection [Formula: see text] and the upper global basis, and prove several results related to the crystal bases. We then apply our results to finite type [Formula: see text] using the homogeneous simple modules [Formula: see text] indexed by one-column tableaux [Formula: see text].

scholarly journals Two Maps on Affine Type $A$ Crystals and Hecke Algebras

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Nicolas Jacon
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Fock Space ◽  
Hecke Algebras ◽  
Type A ◽  
Crystal Basis ◽  
Affine Type

We use the crystal isomorphisms of the Fock space to describe two maps on partitions and multipartitions which naturally appear in the crystal basis theory for quantum groups in affine type $A$ and in the representation theory of Hecke algebras of type $G(l,l,n)$. 

UNITARY BRAID REPRESENTATIONS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 985-1006
Author(s):  
HANS WENZL
Keyword(s):  
Quantum Groups ◽  
Braid Group ◽  
Deformation Parameter ◽  
Hecke Algebras ◽  
Tensor Products ◽  
Roots Of Unity ◽  
Von Neumann ◽  
Root Of Unity ◽  
Brauer Algebras ◽  
Original Motivation

The original motivation for studying unitarizable representations of the braid group was to construct interesting examples of subfactors of II1 von Neumann factors. This is possible for representations factoring through Hecke algebras or q-Brauer algebras only if the deformation parameter is a root of unity. Information gained from this work can be applied to studying tensor products of representations of quantum groups at roots of unity and invariants of 3-manifolds.

scholarly journals UNITARY REPRESENTATIONS OF NONCOMPACT QUANTUM GROUPS AT ROOTS OF UNITY

2001 ◽  
Vol 13 (08) ◽  
pp. 1035-1054 ◽  
Author(s):  
H. STEINACKER
Keyword(s):  
Quantum Groups ◽  
Classical Case ◽  
Unitary Representations ◽  
Roots Of Unity ◽  
Classical Symmetry ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Frobenius Map ◽  
Highest Weight Modules ◽  
Weight Modules

Noncompact forms of the Drinfeld–Jimbo quantum groups [Formula: see text] with [Formula: see text], [Formula: see text] for si=±1 are studied at roots of unity. This covers [Formula: see text], su(n,p), so*(2l), sp(n,p), sp(l,ℝ), and exceptional cases. Finite dimensional unitary representations are found for all these forms, for even roots of unity. Their classical symmetry induced by the Frobenius map is determined, and the meaning of the extra quasi-classical generators appearing at even roots of unity is clarified. The unitary highest weight modules of the classical case are recovered in the limit q→1.

Quantum Groups: Singular Vectors and BGG Resolution

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 623-643 ◽  
Author(s):  
Fyodor Malikov
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Weight Module ◽  
Verma Modules ◽  
Highest Weight Module ◽  
Singular Vectors ◽  
Type Formula ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Dimensional Module

We prove existence of BGG resolution of an irreducible highest weight module over a quantum group, classify morphisms of Verma modules over a quantum group and find formulas for singular vectors in Verma modules. As an application we find cohomology of the quantum group of the type [Formula: see text] with coefficients in a finite-dimensional module.

DEGENERATE AFFINE HECKE ALGEBRAS AND TWO-DIMENSIONAL PARTICLES

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 109-140 ◽  
Author(s):  
IVAN CHEREDNIK
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Scattering Amplitudes ◽  
Quantum Groups ◽  
Hecke Algebras ◽  
Two Dimensional ◽  
Conformal Field ◽  
Affine Hecke Algebras ◽  
The Difference ◽  
Zamolodchikov Equation

We demonstrate that the quantization of momenta in different two dimensional theories of elementary particles with factorizable scattering amplitudes gives the so-called degenerate affine Hecke algebras and their versions. Some connections with quantum groups(Yangians), the two-dimensional conformal field theory and representation theory am discussed. In particular, an interpretation and generalizations of the difference counterpart of the Knizhnik-Zamolodchikov equation are found by means of the particles on a segment.

scholarly journals Crystals from categorified quantum groups

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Aaron D. Lauda ◽  
Monica Vazirani
Keyword(s):  
Crystal Structure ◽  
Quantum Groups ◽  
Grothendieck Group ◽  
Moody Algebra ◽  
Graded Modules ◽  
Highest Weight ◽  
Nous Montrons ◽  
International Audience

International audience We study the crystal structure on categories of graded modules over algebras which categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. We identify this crystal with Kashiwara's crystal for the corresponding negative half of the quantum Kac-Moody algebra. As a consequence, we show the simple graded modules for certain cyclotomic quotients carry the structure of highest weight crystals, and hence compute the rank of the corresponding Grothendieck group. Nous étudions la structure cristalline sur les catégories de modules gradués sur algèbres qui catégorifient la moitié négative du quantum de Kac-Moody algèbre associée à un ensemble de données symétrisables Cartan. Nous identifions ce cristal avec des cristaux de Kashiwara pour le négatif correspondant la moitié de l'algèbre de Kac-Moody quantum. En conséquence, nous montrons que les simples modules classés pour certains quotients cyclotomiques portent la structure des cristaux de poids le plus élevé, et donc calculons le rang du groupe correspondant Grothendieck.

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