On construction of unitary quantum group differential calculus

TL;DR

This paper constructs a unitary anti-involution for the quantum differential calculus over $GL_q(n)$ at $|q|=1$, using spectral extensions and automorphisms within a bicovariant algebraic framework.

Contribution

It introduces a novel spectral extension approach and automorphism-based method to define a unitary anti-involution in quantum group calculus.

Findings

Successfully constructs a unitary anti-involution for $GL_q(n)$ calculus at $|q|=1$

Develops a spectral extension with spectral variables of algebra generators

Defines a three-parametric family of automorphisms used in the construction

Abstract

We develop a construction of the unitary type anti-involution for the quantized differential calculus over $GL_q(n)$ in the case $|q|=1$. To this end, we consider a joint associative algebra of quantized functions, differential forms and Lie derivatives over $GL_q(n)/SL_q(n)$, which is bicovariant with respect to $GL_q(n)/SL_q(n)$ coactions. We define a specific non-central {\em spectral extension} of this algebra by the spectral variables of three matrices of the algebra generators. In the spectrally expended algebra we construct three-parametric family of its inner automorphisms. These automorphisms are used for construction of the unitary anti-involution for the (spectrally extended) calculus over $GL_q(n)$.