Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Central elements and evaluation map for the quantum queer superalgebras

Ming Liu, Alexander Molev, Jian Zhang

TL;DR

This work develops an $R$-matrix framework for the quantum queer superalgebras $U_q( rak q_n)$ and its affine counterpart $U_q(rak q_n)$, deriving crossing symmetries to construct central elements in both settings. It introduces an explicit evaluation homomorphism ${ m ev}:U_q(rak q_n) o U_q({ rak q}_n)$ via $L(u) o L+ar{L}u^{-1}$, and proves a natural embedding of the finite algebra into the affine one. Central elements are obtained from ${ m str}(DM^k)$ in the finite case and from a generating series $z(u)$ in the affine case, with coefficients shown to be central through crossing symmetry arguments. The results connect the finite and affine centers and provide tools for constructing evaluation modules, enriching the representation theory of quantum queer superalgebras.

Abstract

We consider the $R$-matrix presentations of the quantum queer superalgebra $U_q(q_n)$ and its affine counterpart $U_q(\widehat q_n)$. We derive crossing symmetry relations for the $R$-matrices and use them to construct central elements in both superalgebras. We also produce an epimorphism $ev:U_q(\widehat q_n)\to U_q(q_n)$ identical on the subalgebra isomorphic to $U_q(q_n)$.

Central elements and evaluation map for the quantum queer superalgebras

TL;DR

This work develops an -matrix framework for the quantum queer superalgebras and its affine counterpart , deriving crossing symmetries to construct central elements in both settings. It introduces an explicit evaluation homomorphism via , and proves a natural embedding of the finite algebra into the affine one. Central elements are obtained from in the finite case and from a generating series in the affine case, with coefficients shown to be central through crossing symmetry arguments. The results connect the finite and affine centers and provide tools for constructing evaluation modules, enriching the representation theory of quantum queer superalgebras.

Abstract

We consider the -matrix presentations of the quantum queer superalgebra and its affine counterpart . We derive crossing symmetry relations for the -matrices and use them to construct central elements in both superalgebras. We also produce an epimorphism identical on the subalgebra isomorphic to .

Paper Structure

This paper contains 5 sections, 8 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Evaluation homomorphism
  3. Reflection equation superalgebra
  4. Central elements in ${\rm U}_q(\mathfrak q_n)$
  5. Central elements in ${\rm U}_q(\widehat{\mathfrak q}_n)$

Key Result

Theorem 2.1

The mapping defines a homomorphism ${\rm U}_q(\widehat{\mathfrak q}_n)\to {\rm U}_q(\mathfrak q_n)$.

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Theorem 4.1
  • ...and 10 more