On the representation theory of partition (easy) quantum groups

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations of representation theory of quantum groups at roots of unity which applied only to quantizations of the simplest groups, or to certain fractional levels, or only to the projective form of the group. The second half of this paper applies the representation theory to give a sequence of results crucial to applications in topology. In particular, for each compact, simple, simply-connected Lie group we show that at each integer level the quotient category is in fact modular (thus leading to a Topological Quantum Field Theory), we determine when at fractional levels the corresponding category is modular, and we give a quantum version of the Racah formula for the decomposition of the tensor product.

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GEOMETRY OF QUANTUM HOMOGENEOUS VECTOR BUNDLES AND REPRESENTATION THEORY OF QUANTUM GROUPS I

Reviews in Mathematical Physics ◽

10.1142/s0129055x99000209 ◽

1999 ◽

Vol 11 (05) ◽

pp. 533-552 ◽

Cited By ~ 7

Author(s):

A. R. GOVER ◽

R. B. ZHANG

Keyword(s):

Representation Theory ◽

Finite Type ◽

Quantum Groups ◽

Vector Bundles ◽

Homogeneous Spaces ◽

Projective Modules ◽

Geometrical Structures ◽

Induced Representations ◽

Compact Quantum Groups ◽

Homogeneous Vector

Quantum homogeneous vector bundles are introduced in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies, and their sections furnish finite type projective modules over algebras of functions on quantum homogeneous spaces. Further properties of the quantum homogeneous vector bundles are investigated, and applied to the study of the geometrical structures of induced representations of quantum groups.

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

The Electronic Journal of Combinatorics ◽

10.37236/10240 ◽

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 $G(l,l,n)$.

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