Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2202.06090

Quantum Duality Principle for quantum continuous Kac-Moody algebras

Fabio Gavarini
View original source ↗Read source document ↗Export metadata ↓

Abstract

For the quantized universal enveloping algebra U_h(g_X) associated with a continuous Kac-Moody algebra g_X as in [A. Appel, F. Sala, "Quantization of continuum Kac-Moody algebras", Pure Appl. Math. Q. 16 (2020), no. 3, 439-493], we prove that a suitable formulation of the Quantum Duality Principle holds true, both in a "formal" version - i.e., applying to the original definition of U_h(g_X) as a formal QUEA over the algebra of formal series in h - and in a "polynomial" one - i.e., for a suitable polynomial form of U_h(g_X) over the algebra of Laurent polynomials in q. In both cases, the QDP states that a suitable subalgebra of the given quantization of the Lie bialgebra g_X is in fact a suitable quantization (in formal or in polynomial sense) of a connected Poisson group G^*_X dual to g_X .

Explore related subjects

math.QA

Keep this discovery

BibTeXRIS

Fabio Gavarini. 2022-02-12. Quantum Duality Principle for quantum continuous Kac-Moody algebras. https://arxiv.org/abs/2202.06090

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Diagrammatic Categorification of Verma Modules I: Braiding

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

math.QA↗
arXiv2026

Some finite dimensional representations of shifted quantum affine algebras of type A

In this paper, we study finite dimensional representations of shifted quantum affine algebras of type A. We give an explicit description of the tensor product of simple evaluation modules of the quantum loop algebra and a one-dimensional representation of the shifted quantum affine algebra under the separation condition. As a consequence, we give the q-characters of some finite dimensional simple modules of the shifted quantum affine algebra.

math.QA↗