Hasil untuk "math.QA"

Alea Hofstetter, Christoph Schweigert

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 M\"uger center of $\mathcal{C}$.

Anton Alekseev, Florian Naef, Muze Ren

We derive explicit closed formulas for the Kirillov-Kostant-Souriau (KKS) coaction maps of open path regularized holonomies of the Knizhnik-Zamolodchikov (KZ) equation, and the corresponding Poisson brackets for the Lie algebra gl(N,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{gl}(N, \mathbb {C})$$\end{document}. Our main technical tool is a certain projection of the generalized pentagon equation of Alekseev et al. Generalized Pentagon equation, arXiv:2402.19138 [math.QA] (2025).

Oleg Chalykh, Bradley Ryan

This paper studies the spherical subalgebra of the double affine Hecke algebra of type $C^\vee C_n$ and relates it, at the classical level $q = 1$, to a certain character variety of the four-punctured Riemann sphere. This establishes a conjecture from math.QA/0504089. As a by-product, we find a completed phase space for the trigonometric van Diejen system, explicitly integrate its dynamics and explain how it can be obtained via Hamiltonian reduction.

Dražen Adamović, Kazuya Kawasetsu, David Ridout

The Bershadsky-Polyakov algebras are the subregular quantum Hamiltonian reductions of the affine vertex operator algebras associated associated with [Formula: see text]. In (D. Adamović, K. Kawasetsu and D. Ridout, A realisation of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys. 111 (2021) 38, arXiv:2007.00396 [math.QA]), we realized these algebras in terms of the regular reduction, Zamolodchikov’s W3-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky-Polyakov modules has the property that the result is 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 (Z. Fehily, K. Kawasetsu and D. Ridout, Classifying relaxed highest-weight modules for admissible-level Bershadsky–Polyakov algebras, Comm. Math. Phys. 385 (2021) 859–904, 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 k=-[Formula: see text], which is new.

D. Arnaudon, J. Avan, N. Crampé et al.

N. Hu, Rongchuan Xiong

Let k be an algebraically closed field of characteristic zero. We construct several families of finite-dimensional Hopf algebras over k without the dual Chevalley property via the generalized lifting method. In particular, we obtain 14 families of new Hopf algebras of dimension 128 with non-pointed duals which cover the eight families obtained in our unpublished version, arXiv:1701.01991 [math.QA].

Frédéric Chapoton

30 pages, 12 figures

A. Davydov, D. Nikshych, V. Ostrik

We analyze the structure of the Witt group $${\mathcal{W}}$$ of braided fusion categories introduced in Davydov et al. (Journal für die reine und angewandte Mathematik (Crelle’s Journal), eprint arXiv: 1009.2117 [math.QA], 2010). We define a “super” version of the categorical Witt group, namely, the group $${s\mathcal{W}}$$ of slightly degenerate braided fusion categories. We prove that $${s\mathcal{W}}$$ is a direct sum of the classical part, an elementary Abelian 2-group, and a free Abelian group. Furthermore, we show that the kernel of the canonical homomorphism $${S : \mathcal{W} \to s\mathcal{W}}$$ is generated by Ising categories and is isomorphic to $${{\mathbb{Z}}/16\mathbb{Z}}$$ . Finally, we give a complete description of étale algebras in tensor products of braided fusion categories.

Vyjayanthi Chari, S. Loktev

B. Feigin, E. Frenkel, L. Rybnikov

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singulairity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, we obtain the structure of a Gorenstein ring on any such module. This fact may have geometric significance related to the intersection cohomology of Schubert varieties in the affine Grassmannian.

A. Sevostyanov

In 1978 Kostant suggested the Whittaker model of the center of the universal enveloping algebra U(g) of a complex simple Lie algebra g. The main result is that the center of U(g) is isomorphic to a commutative subalgebra in U(b), where b is a Borel subalgebra in g. This observation is used in the theory of principal series representations of the corresponding Lie group G and in the proof of complete integrability of the quantum Toda lattice. In this paper we generalize the Kostant's construction to quantum groups. In our construction we use quantum analogues of regular nilpotent elements defined in math.QA/9812107. Using the Whittaker model of the center of 5the algebra U_h(g) we define quantum deformations of Whittaker modules. The new Whittaker model is also applied to the deformed quantum Toda lattice recently studied by Etingof in math.QA/9901053. We give new proofs of his results which resemble the original Kostant's proofs for the quantum Toda lattice.

J. Teschner

This paper has been withdrawn by the author(s). The material contained in the paper will be published in a subtantially reorganized form, part of it is now included in math.QA/0510174

V. Dolgushev

Boris Shoikhet noticed that the proof of lemma 1 in section 2.3 of math.QA/0504420 contains an error. In this note I give a correct proof of this lemma which was suggested to me by Dmitry Tamarkin. The correction does not change the results of math.QA/0504420.

Dražen Adamović

By using generalized vertex algebras associated to rational lattices, we construct explicitly the admissible modules for the affine Lie algebra $A_1 ^{(1)}$ of level $-{4/3}$. As an application, we show that the W(2,5) algebra with central charge c=-7 investigated in math.QA/0207155 is a subalgebra of the simple affine vertex operator algebra $L(-{4/3}\Lambda_0)$.

P. Etingof, Shlomo Gelaki

We explain that a new theorem of Deligne on symmetric tensor categories implies, in a straightforward manner, that any finite dimensional triangular Hopf algebra over an algebraically closed field of characteristic zero has Chevalley property, and in particular the list of finite dimensional triangular Hopf algebras over such a field given in math.QA/0008232, math.QA/0101049 is complete. We also use Deligne's theorem to settle a number of questions about triangular Hopf algebras, raised in our previous publications, and generalize Deligne's result to nondegenerate semisimple categories in characteristic $p$, by using lifting methods developed in math.QA/0203060.

V. Dolgushev

Abstract We prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold M using the Fedosov resolutions proposed in math.QA/0307212 and the formality quasi-isomorphism for Hochschild chains of R [ [ y 1 ,..., y d ] ] proposed in paper math.QA/0010321 by Shoikhet. This result allows us to describe traces on the quantum algebra of functions on an arbitrary Poisson manifold.

Tom H. Koornwinder

This paper of tutorial nature gives some further details of proofs of some theorems related to the quantum dynamical Yang-Baxter equation. This mainly expands proofs given in “Lectures on the dynamical Yang-Baxter equation” by Etingof and Schiffmann, math.QA/9908064. This concerns the intertwining operator, the fusion matrix, the exchange matrix and the difference operators. The last part expands proofs given in “Traces of intertwiners for quantum groups and difference equations, I” by Etingof and Varchenko, math.QA/9907181. This concerns the dual Macdonald-Ruijsenaars equations.

N. Atakishiyev, A. Klimyk