Parabolic quantum affine algebras

TL;DR

This paper introduces parabolic quantum affine algebras as extensions of quantum affine algebras, providing their construction, basis, triangular decomposition, braid group action, and classifying finite-dimensional irreducible representations.

Contribution

It extends the theory of quantum affine algebras to non-maximal parabolic subalgebras, including their realization, basis, automorphisms, and representation classification.

Findings

Constructed PBW-type basis for parabolic quantum affine algebras

Established a second triangular decomposition

Classified finite-dimensional irreducible representations

Abstract

Maximal parabolic subalgebras of untwisted affine Kac-Moody algebras were studied in the context of Borel-de Siebenthal theory in [13], where they were realized as certain equivariant map algebras with a non-free abelian group action. In this paper, we show that this perspective naturally extends to non-maximal parabolic subalgebras and introduce their quantum analogues - called parabolic quantum affine algebras - in analogy with ordinary quantum affine algebras and their classical counterpart, the loop algebra. While the definition in the Drinfeld-Jimbo presentation is straightforward, the realization in Drinfeld's second presentation requires quantum root vectors associated not only to simple roots but also to certain non-simple roots. A distinguished positive root $\gamma_0$ plays a central role in all constructions. Along the way, we construct a PBW-type basis, establish a second triangular decomposition, and determine the action of the braid group on the Cartan part of the algebra via Lusztig's automorphisms. Finally, we classify the finite-dimensional irreducible representations under a technical condition on $\gamma_0$, referred to as repetition-free, in terms of Drinfeld polynomials with some additional data. The key difference from the ordinary quantum affine case is that the degrees of the polynomials are only bounded by a certain highest weight, rather than being uniquely determined by it. In the maximal parabolic case, the classification can alternatively be phrased in terms of Drinfeld polynomials satisfying certain divisibility conditions.