Some remarks on the construction of quantum symmetric spaces

Some remarks on non-symmetric interpolation Macdonald polynomials

We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular for binomial formulas involving non-symmetric interpolation Macdonald polynomials.

Remarks on Quantum Symmetric Algebras

We examine the quantum symmetric and exterior algebras of finite-dimensional \uqg-modules first systematically studied by Berenstein and Zwicknagl, and resolve some questions that they raised. We show that the difference (in the Grothendieck group) between the quantum symmetric and exterior cubes of a finite-dimensional module is the same as it is classically. Furthermore, we show that quantum symmetric algebras are commutative in an appropriate sense, and are universal with this property. We make extensive use of the coboundary structure on the module category.

Some remarks on symmetric linear functions and pseudotrace maps

Let A be a finite-dimensional associative algebra and $ϕ$ a symmetric linear function on $A$. In this note, we will show that the pseudotrace maps are obtained as special cases of well-known symmetric linear functions on the endomorphism rings of projective modules. We also prove that modules are interlocked with $ϕ$ if and only if they are projective. As an application of our approach, we will give proofs of several propositions and theorems for pseudotrace maps for an arbitrary finite-dimensional associative algebra.

正特征低秩对称融合范畴的注记

Remarks on symmetric fusion categories of low rank in positive characteristic

本文给出正特征下对称融合范畴秩的下界，证明第二 Adams 运算非恒等且满足特定条件的范畴为超 Tannakian，并分类了秩 3 及部分秩 4 的范畴。

A remark on matrix product operator algebras, anyons and subfactors

We show that the mathematical structures in a recent work of Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete are the same as those of flat symmetric bi-unitary connections and the tube algebra in subfactor theory. More specifically, a system of flat symmetric bi-unitary connections arising from a subfactor with finite index and finite depth satisfies all their requirements for tensors and the tube algebra for such a subfactor and the anyon algebra for such tensors are isomorphic up to the normalization constants. Furthermore, the matrix product operator algebras arising from tensors…

关于边界、反常与不可逆对称性的评述

Remarks on Boundaries, Anomalies, and Noninvertible Symmetries

本文区分1+1维不可逆对称性下的弱、强对称边界，分析其与反常及重整化群流约束的关系，并在2+1维U(1)规范理论和TQFT中构造无穷多非Dirichlet、非Neumann边界。