Super-tetrahedra and super-algebras

TL;DR

This paper classifies three-dimensional quantum zero curvature representations and tetrahedron equations, deriving R-matrices that connect Bose and Fermi oscillators, and relates them to super-algebras through compactification.

Contribution

It introduces a comprehensive classification scheme for 3D quantum zero curvature representations, including super-algebras and super-tetrahedron equations, with explicit R-matrices.

Findings

Derived R-matrices intertwining Bose and Fermi oscillators

Reproduced super-algebras U_q(gl(n|m)) via 3d->2d compactification

Classified algebraic structures with parity components

Abstract

In this paper we give a detailed classification scheme for three-dimensional quantum zero curvature representation and tetrahedron equations. This scheme includes both even and odd parity components, the resulting algebras of observables are either Bose q-oscillators or Fermi oscillators. Three-dimensional $R$-matrices intertwining variously oriented tensor products of Bose and Fermi oscillators and satisfying tetrahedron and super-tetrahedron equations are derived. The 3d->2d compactification reproduces U_q(gl(n|m)) super-algebras and their representation theory.