Non-semisimple planar algebras from the representation theory of Ūq(𝔰𝔩2)

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group [Formula: see text]. This is a diagrammatic description of [Formula: see text], where [Formula: see text] is a two-dimensional [Formula: see text] module.

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POISSON–LIE GROUPS AND CLASSICAL W-ALGEBRAS

International Journal of Modern Physics A ◽

10.1142/s0217751x92000405 ◽

1992 ◽

Vol 07 (05) ◽

pp. 853-876 ◽

Cited By ~ 33

Author(s):

V. A. FATEEV ◽

S. L. LUKYANOV

Keyword(s):

Field Theory ◽

Conformal Field Theory ◽

Quantum Group ◽

Lie Groups ◽

Group Structure ◽

Two Dimensional ◽

Poisson Structures ◽

Additional Symmetries ◽

Conformal Field ◽

Poisson Lie Groups

This is the first part of a paper studying the quantum group structure of two-dimensional conformal field theory with additional symmetries. We discuss the properties of the Poisson structures possessing classical W-invariance. The Darboux variables for these Poisson structures are constructed.

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q-ANALOGUE OF BOSON COMMUTATOR AND THE QUANTUM GROUPS SUq(2) AND SUq(1, 1)

Modern Physics Letters A ◽

10.1142/s0217732390000287 ◽

1990 ◽

Vol 05 (04) ◽

pp. 237-242 ◽

Cited By ~ 36

Author(s):

HARUO UI ◽

N. AIZAWA

Keyword(s):

Harmonic Oscillator ◽

Quantum Group ◽

Symmetry Group ◽

Quantum Groups ◽

Heisenberg Algebra ◽

Positive Definiteness ◽

Two Dimensional ◽

Number Operator ◽

Boson Operator ◽

Dynamical Group

We propose a defining set of commutation relations to a q-analogue of boson operator; [Formula: see text], [Formula: see text] and [N, aq]=−aq, which contracts to the Heisenberg algebra of boson operators in the limit of q=1. Here, N is the number operator, [N]q being its q-analogue operator. By making use of this set, we construct a new realization of the “noncompact” quantum group SUq(1, 1) in addition to that of the SUq(2) recently proposed by Biedenharn. The explicit form of the number operator is given in terms of aq and [Formula: see text] and its positive definiteness is proved. A uniqueness of our commutators is also discussed. It is shown that the quantum group SUq(2) appears as a true symmetry group of a q-analogue of the two-dimensional harmonic oscillator and the SUq(1, 1) as its dynamical group.

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Planar algebras, quantum information theory and subfactors

International Journal of Mathematics ◽

10.1142/s0129167x20501244 ◽

2020 ◽

pp. 2050124

Author(s):

Vijay Kodiyalam ◽

Sruthymurali ◽

V. S. Sunder

Keyword(s):

Information Theory ◽

Quantum Information ◽

Hadamard Matrices ◽

Quantum Information Theory ◽

Latin Squares ◽

Planar Algebras ◽

Planar Algebra

We define generalized notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum Latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.

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Affine Modules and the Drinfeld Center

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-23301 ◽

2016 ◽

Vol 118 (1) ◽

pp. 119 ◽

Cited By ~ 2

Author(s):

Paramita Das ◽

Shamindra Kumar Ghosh ◽

Ved Prakash Gupta

Keyword(s):

Finite Index ◽

Planar Algebras ◽

Infinite Depth ◽

Zero Level ◽

Fusion Algebra ◽

Planar Algebra ◽

The Way

Given a finite index subfactor, we show that the affine morphisms at zero level in the affine category over the planar algebra associated to the subfactor is isomorphic to the fusion algebra of the subfactor as a $*$-algebra. This identification paves the way to analyze the structure of affine $P$-modules with weight zero for any subfactor planar algebra $P$ (possibly having infinite depth). Further, for irreducible depth two subfactor planar algebras, we establish an additive equivalence between the category of affine $P$-modules and the center of the category of $N$-$N$-bimodules generated by $L^2(M)$; this partially verifies a conjecture of Jones and Walker.

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The Robinson―Schensted Correspondence and $A_2$-webs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2349 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Matthew Housley ◽

Heather M. Russell ◽

Julianna Tymoczko

Keyword(s):

Representation Theory ◽

Symmetric Group ◽

Quantum Group ◽

Classical Limit ◽

Young Tableaux ◽

Tensor Power ◽

Recent Interest ◽

International Audience ◽

Standard Young Tableaux ◽

Graphical Algorithm

International audience The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. A great deal of recent interest has focused on the combinatorics of invariant webs for tensors powers of $V^+$, the standard representation of the quantum group. In particular, the invariant webs for the 3$n$th tensor power of $V^+$ correspond bijectively to $[n,n,n]$ standard Young tableaux. Kuperberg originally defined this map in terms of a graphical algorithm, and subsequent papers of Khovanov–Kuperberg (1999) and Tymoczko (2012) introduce algorithms for computing the inverse. The main result of this paper is a redefinition of Kuperberg's map through the representation theory of the symmetric group. In the classical limit, the space of invariant webs carries a symmetric group action. We use this structure in conjunction with Vogan's generalized tau-invariant and Kazhdan–Lusztig theory to show that Kuperberg's map is a direct analogue of the Robinson–Schensted correspondence.

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Free oriented extensions of subfactor planar algebras

International Journal of Mathematics ◽

10.1142/s0129167x18500933 ◽

2018 ◽

Vol 29 (13) ◽

pp. 1850093 ◽

Cited By ~ 1

Author(s):

Shamindra Kumar Ghosh ◽

Corey Jones ◽

B. Madhav Reddy

Keyword(s):

Left Adjoint ◽

Planar Algebras ◽

Planar Algebra ◽

Restriction Functor

We show that the restriction functor from oriented factor planar algebras to subfactor planar algebras admits a left adjoint, which we call the free oriented extension functor. We show that for any subfactor planar algebra realized as the standard invariant of a hyperfinite [Formula: see text] subfactor, the projection category of the free oriented extension admits a realization as bimodules of the hyperfinite [Formula: see text] factor.

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THE MONOID OF FAMILIES OF QUIVER REPRESENTATIONS

Proceedings of the London Mathematical Society ◽

10.1112/s0024611502013497 ◽

2002 ◽

Vol 84 (3) ◽

pp. 663-685 ◽

Cited By ~ 6

Author(s):

MARCUS REINEKE

Keyword(s):

Algebraic Geometry ◽

Group Theory ◽

Representation Theory ◽

Quantum Group ◽

Subject Classification ◽

Quiver Representations ◽

Enveloping Algebras ◽

Geometric Methods ◽

Quantized Enveloping Algebras ◽

Monoid Structure

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, that is, subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analysed by representation-theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.

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