Parabolic quantum affine algebras
Kudret Bostanci, Deniz Kus
TL;DR
This work constructs quantum analogues of parabolic subalgebras of untwisted affine Kac–Moody algebras, introducing parabolic quantum affine algebras and proving parallel structural results to the ordinary quantum affine case. It provides both the DJ and Drinfeld realizations, a PBW basis, and a second triangular decomposition, and extends the representation theory of equivariant map algebras to the quantum, parabolic setting. Under a repetition-free condition on the distinguished root $\gamma_0$ (and excluding type $G_2$), finite-dimensional irreducibles in category $\mathcal{C}_q$ are classified by Drinfeld polynomials with extra data, with polynomial degrees bounded by a highest weight rather than being entirely determined. Overall, the paper generalizes the loop-algebra picture to non-maximal parabolics, establishes foundational quantum structures, and lays groundwork for further study of parabolic quantum affine categories and related quantum symmetric constructions.
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 $γ_0$ 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 $γ_0$, 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.