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At least 55 records · Page 3Linked to original sources
arXiv2007

A Diagrammatic Category for the Representation Theory of U_q(sl_n)

This thesis provides a partial answer to a question posed by Greg Kuperberg in q-alg/9712003 and again by Justin Roberts as problem 12.18 in "Problems on invariants of knots and 3-manifolds", math.GT/0406190, essentially: "Can one describe the category of representations of the quantum group U_q(sl_n) (thought of as a spherical category) via generators and relations?" For each n \geq 0, I define a certain tensor category of trivalent graphs, modulo isotopy, and construct a functor from this category onto (a full subcategory of) the category of representations of the quantum group U_q(sl_n). One would like to describe completely the kernel of this functor, by providing generators. The resulting quotient of the diagrammatic category would then be a category equivalent to the representation category of U_q(sl_n). I make significant progress towards this, describing certain generators of the kernel, and some obstructions to further elements. It remains a conjecture that these relations generate the kernel. My results extend those of q-alg/9712003, MR1659228, math.QA/0310143 and math.GT/0506403. The argument is essentially by constructing a diagrammatic version of the forgetful functor coming from the inclusion of U_q(sl_{n-1}) in U_q(sl_n}. We know this functor is faithful, so a diagram is in the kernel for n exactly if its image under the diagrammatic forgetful functor is in the kernel for n-1. This allows us to perform inductive calculations, both establishing families of elements of the kernel, and finding obstructions.

math.QA↗
arXiv2007

The Kashiwara-Vergne conjecture

This is a Bourbaki's seminar text. We introduce the combinatorial Kashiwara-Vergne conjecture on the Baker-Campbell-Hausdorff serie. After recalling previous results and consequences, we explain the Alekseev-Meinrenken's proof [math.QA/0506499]

math.QA↗
arXiv2007

Twisted Quantum Deformations of Lorentz and Poincaré algebras

We discussed twisted quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S.Zakrzewski classification can be presented as a sum of subordinated r-matrices of Abelian and Jordanian types. Corresponding twists describing quantum deformations are obtained in explicit form. This work is an extended version of the paper \url{arXiv:0704.0081v1 [math.QA]}.

math.QA↗
arXiv2008

Structural properties of acyclic heaps of pieces with Kazhdan--Lusztig theory

We introduce the notions of boundary vertex, linear equivalence and effective boundary vertex in the context of Viennot's heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every boundary vertex is linearly equivalent to an effective boundary vertex. Using this result, we establish Property W (in the sense of math.QA/0509363) for star reducible Coxeter groups; this corrects a mistake in the latter paper.

math.CO↗
arXiv2008

A Quantum Canonical Embedding

This paper presents an English version of a chapter of the L.L. Vaksman book `Quantum Bounded Symmetric Domains', see arXiv:0803.3769 [math.QA]. This chapter deals with a quantum analog of a canonical embedding of a bounded symmetric domain.

math.QA↗
arXiv2010

On braided fusion categories I

This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also give a comprehensive and self-contained exposition of the known results on braided fusion categories without assuming them pre-modular or non-degenerate. The guiding heuristic principle of our work is an analogy between braided fusion categories and Casimir Lie algebras.

math.QA↗
arXiv2011

The global quantum duality principle: a survey through examples

Let R be a 1-dimensional integral domain, let h (non-zero) be a prime element, and let \HA be the category of torsionless Hopf algebras over R. We call H in \HA a "quantized function algebra" (=QFA), resp. "quantized restricted universal enveloping algebras" (=QrUEA), at h if H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. An "inner" Galois correspondence on \HA is established via the definition of two endofunctors, ( )^\vee and ( )', of \HA such that: (a) the image of ( )^\vee, resp. of ( )', is the full subcategory of all QrUEAs, resp. QFAs, at h; (b) if p := Char(R/hR) = 0, the restrictions of ( )^\vee to QFAs and of ( )' to QrUEAs yield equivalences inverse to each other; (c) if p = 0, then starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra g, the functor ( )^\vee, resp. ( )', gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. a dual Poisson group. In particular, (a) provides a machine to produce quantum groups of both types (either QFAs or QrUEAs), (b) gives a characterization of them among objects of \HA, and (c) gives a "global" version of the so-called "quantum duality principle" (after Drinfeld's, cf. [Dr]). These notes draw a sketch of the construction leading to the "global quantum duality principle". Besides, the principle itself, and in particular the above mentioned application, is illustrated by means of several examples. There are no proofs, but all (of them, and any other detail) can be found instead in arXiv:math/0303019v8 [math.QA].

math.QA↗
arXiv2021

Kontsevich's deformation quantization: from Dirac to multiple zeta values

One way of reconciling classical and quantum mechanics is deformation quantization, which involves deforming the commutative algebra of functions on a Poisson manifold to a non-commutative, associative algebra, reminiscent of the space of quantum observables. This depends on a formal parameter $\hbar$, so that the original pointwise product is recovered when $\hbar=0$. In 1997 Kontsevich showed that a deformation quantization exists for every Poisson manifold. He furthermore gave a simple, combinatorial formula for producing a quantization of any Poisson structure on $\mathbb{R}^n$. The primary aim of this essay, largely drawn from the author's MMath dissertation at Oxford, is to present and explain Kontsevich's results. Starting with the motivation, we discuss how the problem is solved by situating it in a richer mathematical structure, performing a few original calculations along the way. We hope to communicate a sense of the strange links between this subject and seemingly distant areas of mathematics, and also to describe some of the contemporary research in the field. To these ends, we consider a recent paper, arXiv:1812.11649 [math.QA], which connects deformation quantization to multiple zeta values.

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arXiv2023

Weight module classifications for Bershadsky--Polyakov algebras

The Bershadsky--Polyakov algebras are the subregular quantum hamiltonian reductions of the affine vertex operator algebras associated with $\mathfrak{sl}_3$. In arXiv:2007.00396 [math.QA], we realised these algebras in terms of the regular reduction, Zamolodchikov's W$_3$-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky--Polyakov modules gives modules that are generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of arXiv:2007.03917 [math.RT] for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level $\mathsf{k}=-\frac{7}{3}$, which is new.

math.QA↗
arXiv2026

The 2-categorical S-matrix of a braided fusion 1-category is a character table

The semisimple module categories over a braided fusion category $\mathcal{C}$ form a connected fusion 2-category $\text{Mod}(\mathcal{C})$. Its Drinfeld center $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ is a braided fusion 2-category. To any braided fusion 2-category, Johnson-Freyd and Reutter arXiv:2105.15167v3 [math.QA] have associated a matrix-valued invariant, the 2-categorical $S$-matrix. In this short note we investigate this matrix of $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ as an invariant for the braided fusion 1-category $\mathcal{C}$ and show that it reduces to the character table of the Müger center of $\mathcal{C}$.

math.QA↗
arXiv2000

QDYBE: some explicit formulas for exchange matrix and related objects in case of sl(2), q=1

This mainly tutorial paper is intended as a somewhat larger example for parts of the theory exposed in the Lectures on the dynamical Yang-Baxter equations by P. Etingof and O. Schiffmann, math.QA/9908064. We explicitly compute the matrix entries of the intertwining operator, fusion matrix and exchange matrix associated to sl(2,R) for q=1. We also consider the weighted trace function and the dual Macdonald-Ruijsenaars equation for this particular case. As expected, the matrix entries of the exchange matrix turn out to be Racah polynomials. However, the road to their explicit formula via the fusion matrix is quick, and it also yields an interesting way to derive their orthogonality relations.

math.QA↗
arXiv2000

On the Classification of Finite-Dimensional Triangular Hopf Algebras

A fundamental problem in the theory of Hopf algebras is the classification and explicit construction of finite-dimensional quasitriangular Hopf algebras over C. These Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld (e.g. the quantum double of a Hopf algebra). However, this intriguing problem turns out to be extremely hard and it is still widely open. One can hope that resolving the problem in the triangular case would contribute to the understanding of the general case. There is no complte classification of finite-dimensional triangular Hopf algebras yet. However, the problem was solved in the semisimple case by Etingof and the author, in the minimal triangular pointed case by the author, and more generally for triangular Hopf algebras with the Chevalley property by Andruskiewitsch, Etingof and the author. The purpose of this paper is to report on all of this progress, and to explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne's theorem on Tannakian categories and on Movshev's theory in an essential way. We explian Movshev's theory in details, and refer the reader to math.QA/0007147 for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of related open problems.

math.QA↗
arXiv2000

Noncommutative algebras associated to complexes and graphs

This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The algebra $Q_n$ is closely related to factorizations of a generic polynomial of degree $n$ over a division algebra into linear factors.

math.QA↗
arXiv2001

Modular categories and orbifold models II

This is a continuation of the paper "Modular tensor categories and orbifold theories", arXiv:math.QA/0104242. It discusses orbifold models of conformal filed theory, or, in mathematical language, question of constructing the category of representations of the fixed point algebra $V^G$ for a given vertex operator algebra $V$ with an action of a finite group $G$. The previous paper gave a proof of well-known conjecture of Dijkgraaf-Vafa-Verlinde-Verlinde giving a complete answer to this question in the holomorphic case (when $V$ has a unique simple module, $V$ itself) under the assumption that categories of rrepresentations of $V$, $V^G$ are modular tensor categories. In the current paper, we give a partial answer in non-holomorphic case. In particular, we show that the category of representations of $V^G$ is completely determined by the category of twisted $V$-modules together with the action of $G$ on this category. Our approach is based on describing representations of $V$, $V^G$ and relation between them in terms of tensor categories and avoids using the technique of VOAs as much as possible.

math.QA↗
arXiv2001

A multiplicative property of quantum flag minors

We study multiplicative properties of the (quantum) dual canonical basis B* associated to a semi-simple complex Lie group G. We provide a subset D of B* such that the following property holds : if two elements b, b' in B* q-commute and if one of these elements is in D, then the product bb' is in B* up to a power of q, where q the quantum parameter. If G is SL_n, then D is the set of so-called quantum flag minors and we obtain a generalization of a result of Leclerc-Nazarov-Thibon, see ArXiv:Math.QA/0011074.

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