The bar involution for quantum symmetric pairs

  title={The bar involution for quantum symmetric pairs},
  author={Martina Balagovic and Stefan Kolb},
  journal={arXiv: Quantum Algebra},
We construct a bar involution for quantum symmetric pair coideal subalgebras $B_{\mathbf{c},\mathbf{s}}$ corresponding to involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras. To this end we give unified presentations of these algebras in terms of generators and relations extending previous results by G. Letzter and the second named author. We specify precisely the set of parameters $\mathbf{c}$ for which such an intrinsic bar involution exists. 

Defining relations for quantum symmetric pair coideals of Kac–Moody type

Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group

The bar involution for quantum symmetric pairs -- hidden in plain sight

We show that all quantum symmetric pair coideal subalgebras Bc of Kac-Moody type have a bar involution for a suitable choice of parameters c. The proof relies on a generalized notion of quasi

Defining relations of quantum symmetric pair coideal subalgebras

Abstract We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on

Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type

We present a generalization of the theory of quantum symmetric pairs as developed by Kolb and Letzter. We introduce a class of generalized Satake diagrams that give rise to (not necessarily

Quantum symmetric pairs at roots of 1

Hall algebras and quantum symmetric pairs of Kac-Moody type

We extend our ıHall algebra construction from acyclic to arbitrary ıquivers, where the ıquiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism

Quantum supersymmetric pairs and $\imath$Schur duality of type AIII

Expanding the classic work of Letzter and Kolb, we construct quantum supersymmetric pairs $({\mathbf U},{\mathbf U}^\imath)$ of type AIII and formulate their basic properties. We establish an

Canonical bases arising from quantum symmetric pairs

We develop a general theory of canonical bases for quantum symmetric pairs $$({\mathbf{U}}, {\mathbf{U}}^\imath )$$(U,Uı) with parameters of arbitrary finite type. We construct new canonical bases

Braided module categories via quantum symmetric pairs

  • S. Kolb
  • Mathematics
    Proceedings of the London Mathematical Society
  • 2019
Let g be a finite‐dimensional complex semisimple Lie algebra. The finite‐dimensional representations of the quantized enveloping algebra Uq(g) form a braided monoidal category Oint . We show that the



Quantum symmetric Kac–Moody pairs

Geometric Schur Duality of Classical Type

This is a generalization of the classic work of Beilinson, Lusztig and MacPherson. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in

Reflection equation algebras, coideal subalgebras, and their centres

Reflection equation algebras and related $${U{_q}(\mathfrak g)}$$ -comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or

Quantum Symmetric Pairs and Their Zonal Spherical Functions

We study the space of biinvariants and zonal spherical functions associated to quantum symmetric pairs in the maximally split case. Under the obvious restriction map, the space of biinvariants is

Symmetric Pairs for Quantized Enveloping Algebras

Abstract Let θ be an involution of a semisimple Lie algebra g , let g θ denote the fixed Lie subalgebra, and assume the Cartan subalgebra of g has been chosen in a suitable way. We construct a

Coideal Subalgebras and Quantum Symmetric Pairs

Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum

Real forms of Uq(g)

In this Letter we consider the real forms of quantum groups associated to generalized Cartan matrices. There are two main results. The first is a description of the Hopf algebra automorphisms and of

Lectures on quantum groups

Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as