Noncommutative pfaffians and representations

TL;DR

This paper explores the role of noncommutative pfaffians in orthogonal algebra representations, calculating their images in Mickelson-Zhelobenko algebra and analyzing their action on Gelfand-Tsetlin-Molev bases.

Contribution

It introduces the use of noncommutative pfaffians as raising operators and determines their placement among other such operators in representation theory.

Findings

Calculated images of pfaffians in Mickelson-Zhelobenko algebra

Determined the action of pfaffians on Gelfand-Tsetlin-Molev bases

Analyzed pfaffians in tensor realization of representations

Abstract

Noncommutative pfaffians associated with an orthogonal algebra are some special elements of the universal enveloping algebra. In the paper it is suggested to use some pfaffians as raising operators. The images of these pfaffians in the Mickelson-Zhelobenko algebra are calculated. It allows to find a place of pfaffians among other raising operators. As a byproduct the action of the pfaffians on the Gelfand-Tsetlin-Molev bases is found. The action of pfaffians in the tensor realization of representation is considered in the appendix.