Quantum Stiefel manifolds

TL;DR

This paper characterizes quantum Stiefel manifolds as universal $C^*$-algebras generated by specific relations derived from the $R$-matrix, providing a clear algebraic description of these quantum spaces.

Contribution

It establishes a universal $C^*$-algebra presentation of quantum Stiefel manifolds using generators and relations based on the $R$-matrix approach.

Findings

Describes $C(SU_q(n)/SU_q(n-m))$ as a universal $C^*$-algebra.

Identifies generators from the last $m$ rows of the fundamental matrix.

Derives relations using the $R$-matrix and existing mathematical results.

Abstract

Quantum analogs of Stiefel manifolds $SU_{q}(n)/SU_q(n-m)$ were introduced by Podkolzin \& Vainerman. The underlying $C^*$-algebra $C(SU_{q}(n)/SU_q(n-m))$ can be described as the $C^*$-subalgebra of $C(SU_q(n))$ generated by elements of last $m$ rows of the fundamental matrix of $SU_q(n)$. Using $R$-matrix of type $A_{n-1}$, one can find certain relations involving elements of last $m$ rows only. In this paper, by analyzing these relations and using a result of Neshveyev \& Tuset, we establish $C(SU_{q}(n)/SU_q(n-m))$ as a universal $C^*$-algbera given by finite sets of generators and relations.