A note on quantum structure constants

TL;DR

This paper presents explicit Cartan-Maurer equations for various $q$-groups, clarifying their structure and connection to classical cases, which is crucial for developing $q$-deformed gauge theories.

Contribution

It provides a convenient form of Cartan-Maurer equations for $q$-groups in the $A_{n-1}, B_n, C_n, D_n$ series, facilitating their computation and understanding.

Findings

Explicit expressions for $q$-commutation relations of left-invariant forms.

Clarification of the connection between $q$-deformed and classical cases.

Foundational equations essential for $q$-deformed gauge theories.

Abstract

The Cartan-Maurer equations for any $q$-group of the $A_{n-1}, B_n, C_n, D_n$ series are given in a convenient form, which allows their direct computation and clarifies their connection with the $q=1$ case. These equations, defining the field strengths, are essential in the construction of $q$-deformed gauge theories. An explicit expression $\om ^i\we \om^j= -\Z {ij}{kl}\om ^k\we \om^l$ for the $q$-commutations of left-invariant one-forms is found, with $\Z{ij}{kl} \om^k \we \om^l \qonelim \om^j\we\om^i$.