Commuting quantum traces for quadratic algebras

Commuting quantum traces: the case of reflection algebras

We formulate a systematic construction of commuting quantum traces for reflection algebras. This is achieved by introducing two sets of generalized reflection equations with associated consistent fusion procedures. Products of their solutions yield commuting quantum traces.

The Reciprocal of $\sum_{n\geq 0}a^nb^n$ for non-commuting $a$ and $b$, Catalan numbers and non-commutative quadratic equations

The aim of this paper is to describe the inversion of the sum $\sum_{n\geq 0}a^nb^n$ where $a$ and $b$ are non-commuting variables as a formal series in $a$ and $b$. We show that the inversion satisfies a non-commutative quadratic equation and that the number of certain monomials in its homogeneous components equals to a Catalan number. We also study general solutions of similar quadratic equations.

交换的 Clifford 作用

Commuting Clifford actions

本文研究两个 Clifford 代数在同一向量空间上的交换作用，证明其二次单项式给出正交李代数的旋量表示，并据此构造带 Spin(10) 作用的 Pati-Salam 谱三元组。

Traces of intertwiners for quantum affine algebras and difference equations (after Etingof-Schiffmann-Varchenko)

We modify and give complete proofs for the results of Etingof-Schiffmann-Varchenko on traces of intertwiners of untwisted quantum affine algebras in the opposite coproduct and the standard grading. More precisely, we show that certain normalized generalized traces for $U_q(\hat{\mathfrak{g}})$ solve four commuting systems of q-difference equations: the Macdonald-Ruijsenaars, dual Macdonald-Ruijsenaars, q-KZB, and dual q-KZB equations. In addition, we show a symmetry property for these renormalized trace functions. Our modifications are motivated by their appearance in recent work of the author…

Braided differential structure on Weyl groups, quadratic algebras and elliptic functions

We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type $A,$ we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided difference operators. The relations satisfied by the mutually commuting elements called Dunkl elements in the deformed Fomin-Kirillov algebra are determined. The Dunkl elements correspond to the truncated…

Quantization and dynamisation of Trace-Poisson brackets

The quantization problem for the trace-bracket algebra, derived from double Poisson brackets, is discussed. We obtain a generalization of the boundary YBE (or so-called ABCD-algebra) for the quantization of quadratic trace-brackets. A dynamical deformation is proposed on the lines of Gervais-Neveu-Felder dynamical quantum algebras.