Mathematics

Let V be a vertex operator algebra, k a positive integer and σ a permutation automorphism of the vertex operator algebra V ⊗k . In this paper, we determine the fusion product of any V ⊗k -module with any σ -twisted V ⊗k -module.

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra K(s l 2 ,k) associated to the integrable highest weight modules for the affine Kac-Moody algebra A (1) 1 is the building block of the general parafermion vertex operator K(g,k) for any finite dimensional simple Lie algebra g and any positive integer k . We first classify the irreducible modules of Z 2 -orbifold of the simple affine vertex operator algebra of type A (1) 1 and determine their fusion rules. Then we study the representations of the Z 2 -orbifold of the parafermion vertex operator algebra K(s l 2 ,k) , we give the quantum dimensions, and more technically, fusion rules for the Z 2 -orbifold of the parafermion vertex operator algebra K(s l 2 ,k) are completely determined.

We give a construction and algorithmic description of the fusion ring of permutation extensions of an arbitrary modular tensor category using a combinatorial approach inspired by the physics of anyons and symmetry defects in bosonic topological phases of matter. The definition is illustrated with examples, namely bilayer symmetry defects and S 3 -extensions of small modular tensor categories like the Ising and Fibonacci theories. An implementation of the fusion algorithm is provided in the form of a Mathematica package. We introduce the notions of confinement and deconfinement of anyons and defects, respectively, which develop the tools to generalize our approach to more general fusion rings of G -crossed extensions.

In this letter we construct GL NM -valued dynamical R -matrix by means of unitary skew-symmetric solution of the associative Yang-Baxter equation in the fundamental representation of GL N . In N=1 case the obtained answer reproduces the GL M -valued Felder's R -matrix, while in the M=1 case it provides the GL N R -matrix of vertex type including the Baxter-Belavin's elliptic one and its degenerations.

Galilean W 3 vertex operator algebra G W 3 ( c L , c M ) is constructed as a universal enveloping vertex algebra of certain non-linear Lie conformal algebra. It is proved that this algebra is simple by using determinant formula of the vacuum module. Reducibility criterion for Verma modules is given, and the existence of subsingular vectors demonstrated. Free field realisation of G W 3 ( c L , c M ) and its highest weight modules is obtained within a rank 4 lattice VOA.

We prove a theorem in 3-dimensional topological field theory: a Reshetikhin-Turaev theory admits a nonzero boundary theory iff it is a Turaev-Viro theory. The proof immediately implies a characterization of fusion categories in terms of dualizability. The main theorem applies to physics, where it implies an obstruction to a gapped 3-dimensional quantum system admitting a gapped boundary theory. Appendices on bordism multicategories and on internal duals may be of independent interest.; v2 extensive revision: added theorem on dualizable 2-categories, material on natural transformations, reworked theorems and several proofs, and more.

A q -Gaussian measure is a generalization of a Gaussian measure. This generalization is obtained by replacing the exponential function with the power function of exponent 1/(1−q) ( q≠1 ). The limit case q=1 recovers a Gaussian measure. For 1≤q<3 , the set of all q -Gaussian densities over the real line satisfies a certain regularity condition to define information geometric structures such as an entropy and a relative entropy via escort expectations. The ordinary expectation of a random variable is the integral of the random variable with respect to its law. Escort expectations admit us to replace the law to any other measures. A choice of escort expectations on the set of all q -Gaussian densities determines an entropy and a relative entropy. One of most important escort expectations on the set of all q -Gaussian densities is the q -escort expectation since this escort expectation determines the Tsallis entropy and the Tsallis relative entropy. The phenomenon gauge freedom of entropies is that different escort expectations determine the same entropy, but different relative entropies. In this note, we first introduce a refinement of the q -logarithmic function. Then we demonstrate the phenomenon on an open set of all q -Gaussian densities over the real line by using the refined q -logarithmic functions. We write down the corresponding Riemannian metric.

A class of finite-dimensional Hopf algebras which generalise the notion of Taft algebras is studied. We give necessary and sufficient conditions for these Hopf algebras to omit a pair in involution, that is, to not have a group-like and a character implementing the square of the antipode. As a consequence we prove the existence of an infinite set of examples of finite-dimensional Hopf algebras without such pairs. This has implications for the theory of anti-Yetter-Drinfeld modules as well as biduality of representations of Hopf algebras.

Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.

The aim of this paper is to generalise the construction of n -ary Hom-Lie bracket by means of an (n−2) -cochain of given Hom-Lie algebra to super case inducing a n -Hom-Lie superalgebras. We study the notion of generalized derivation and Rota-Baxter operators of n -ary Hom-Nambu and n -Hom-Lie superalgebras and their relation with generalized derivation and Rota-Baxter operators of Hom-Lie superalgebras. We also introduce the notion of 3 -Hom-pre-Lie superalgebras which is the generalization of 3 -Hom-pre-Lie algebras.