Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2508.01795

Canonical bases in Lie theory and total positivity

G. Lusztig
View original source ↗Read source document ↗Export metadata ↓

Abstract

This represents a talk given at the International Conference for Basic Science, July 2025. We review the theory of canonical bases of quantum groups and its relation with the theory of total positivity.

Explore related subjects

math.QAmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

G. Lusztig. 2025-09-23. Canonical bases in Lie theory and total positivity. https://arxiv.org/abs/2508.01795

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

KEEP EXPLORING

Related papers

arXiv2026

Categorification of quasi-split iquantum groups

We introduce a new family of graded 2-categories generalizing the 2-quantum groups introduced by Khovanov, Lauda and Rouquier. We use them to categorify quasi-split iquantum groups in all symmetric types.

math.QA↗
arXiv2026

Coset representatives corresponding to Yetter-Drinfeld modules of modular group and continued fraction

We give complete conjugacy classes of modular group SL(2,Z). Particularly, the conjugacy classes of hyperbolic elements are decided by the proper equivalence classes of indefinite forms, and we give an example. Finally, we describe the coset representatives of centralizer of S, ST, T and hyperbolic elements of SL(2,Z) by regular continued fraction. In conclusion, most Nichols algebras over modular group are infinite-dimensional except Proposition 4.10.

math.QA↗
arXiv2026

Factorization envelopes and enveloping vertex algebras

We develop a bornological version of Costello and Gwilliam's procedure for extracting vertex algebras from suitable prefactorization algebras on the complex plane. Using bornological complex analysis, we remove the discreteness condition imposed in their extraction theorem. We then construct, from a suitable Lie conformal algebra, a prefactorization algebra to which this extraction procedure applies, and prove that the resulting vertex algebra is isomorphic to the enveloping vertex algebra of the original Lie conformal algebra. Our construction uses a factorization envelope and extends the construction of Costello--Gwilliam in the affine vertex algebra case and that of Williams in the Virasoro vertex algebra case. Moreover, a super analogue yields new prefactorization algebras corresponding to vertex superalgebras, such as the Neveu--Schwarz vertex superalgebra, the $N=2$ vertex superalgebra, and the $N=4$ vertex superalgebra.

math.QA↗