q-Deformed Clifford algebra and level zero fundamental representations of quantum affine algebras

TL;DR

This paper constructs a new realization of level zero fundamental representations of quantum affine algebras using q-deformed Clifford algebras, revealing their structure via tensor products and crystal bases.

Contribution

It introduces a semisimple module structure on tensor powers of exterior algebras with q-deformed Clifford generators, identifying fundamental representations as summands.

Findings

Each fundamental representation appears as an irreducible summand in the tensor product.

Provides a simple combinatorial description of the affine crystal structure.

Connects the representation theory with binary matrix combinatorics.

Abstract

We give a realization of the level zero fundamental weight representation $W(\varpi_k)$ of the quantum affine algebra $U_q'(\mf{g})$, when $\mf{g}$ has a maximal parabolic subalgebra of type $C_n$. We define a semisimple $U'_q({\mf g})$-module structure on $\E^{\otimes 2}$ in terms of q-deformed Clifford generators, where $\E$ is the exterior algebra generated by a dual natural representation $V$ of $U_q(\mf{sl}_{n})$. We show that each $W(\varpi_k)$ appears as an irreducible summand (not necessarily multiplicity free) in $\E^{\otimes 2}$. As a byproduct, we obtain a simple description of the affine crystal structure of $W(\varpi_k)$ in terms of $n\times 2$ binary matrices and their $(\mf{sl}_n,\mf{sl}_2)$-bicrystal structure.