arXiv subjects

Stefan Kolb

Publications and source records attributed to Stefan Kolb.

At least 19 recordsLinked to original sources

arXiv2026

Short star products for quantum symmetric pairs and applications

We prove that the star product for quantum symmetric pair coideal subalgebras is short. We apply this result to obtain new conceptual proofs, from first principles, of several fundamental facts about quantum symmetric pairs. In particular, we establish the existence of the algebra anti-automorphism $\sigma_\tau$ and of the bar involution, without making use of the quasi K-matrix. We give a new elementary proof of a conjecture by Balagovi\'c and Kolb, sometimes referred to as the fundamental lemma for quantum symmetric pairs. We obtain a conceptual formula expressing the tensor quasi K-matrix in terms of the much studied quasi R-matrix and the Letzter map. This also allows for a new independent proof of the intertwiner property of the quasi K-matrix.

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arXiv2024

Representation theory of very non-standard quantum $so(2N-1)$

We classify the finite dimensional representations of the quantum symmetric pair coideal subalgebra $B_{\mathbf c}$ of type $DII$ corresponding to the symmetric pair $(so(2N),so(2N-1))$. For $B_{\mathbf c}$ defined over an arbitrary field $k$ and $q\in k$ not a root of unity we establish a one-to-one correspondence between finite dimensional, simple $B_{\mathbf c}$-modules and dominant integral weights for $so(2N-1)$. We use specialisation to show that the category of finite dimensional $B_{\mathbf c}$-modules is semisimple if $\mathrm{char}(k)=0$ and $q$ is transcendental over ${\mathbb Q}$. In this case the characters of simple $B_{\mathbf c}$-modules are given by Weyl's character formula. This means in particular that the quantum symmetric pair of type $DII$ can be used to obtain Gelfand-Tsetlin bases for irreducible representations of the Drinfeld-Jimbo quantum group $U_q(so(2N))$.

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arXiv2021

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

We show that all quantum symmetric pair coideal subalgebras $B_\mathbf{c}$ of Kac-Moody type have a bar involution for a suitable choice of parameters $\mathbf{c}$. The proof relies on a generalized notion of quasi K-matrix. The proof does not involve an explicit presentation of $B_\mathbf{c}$ in terms of generators and relations.

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arXiv2021

Defining relations of quantum symmetric pair coideal subalgebras

We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantized enveloping algebras of Kac-Moody type. Our methods are based on star products on noncommutative $\mathbb{N}$-graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.

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arXiv2020

Bivariate Continuous q-Hermite Polynomials and Deformed Quantum Serre Relations

We introduce bivariate versions of the continuous q-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous q-Hermite polynomials and the star product method of [Kolb and Yakimov, Adv. Math. 2020] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac-Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

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arXiv2019

Braided module categories via quantum symmetric pairs

Let ${\mathfrak g}$ be a finite dimensional complex semisimple Lie algebra. The finite dimensional representations of the quantized enveloping algebra $U_q({\mathfrak g})$ form a braided monoidal category $O_{int}$. We show that the category of finite dimensional representations of a quantum symmetric pair coideal subalgebra $B_{c,s}$ of $U_q({\mathfrak g})$ is a braided module category over an equivariantization of $O_{int}$. The braiding for $B_{c,s}$ is realized by a universal K-matrix which lies in a completion of $B_{c,s}\otimes U_q({\mathfrak g})$. We apply these results to describe a distinguished basis of the center of $B_{c,s}$.

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arXiv2019

Braid group action and root vectors for the $q$-Onsager algebra

We define two algebra automorphisms $T_0$ and $T_1$ of the $q$-Onsager algebra $B_c$, which provide an analog of G. Lusztig's braid group action for quantum groups. These automorphisms are used to define root vectors which give rise to a PBW basis for $B_c$. We show that the root vectors satisfy $q$-analogs of Onsager's original commutation relations. The paper is much inspired by I. Damiani's construction and investigation of root vectors for the quantized enveloping algebra of $\widehat{\mathfrak{sl}}_2$.

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arXiv2019

On the Adjoint Representation of a Hopf Algebra

We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{\text{adfin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{\text{adfin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{\text{adfin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.

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arXiv2019

Symmetric pairs for Nichols algebras of diagonal type via star products

We construct symmetric pairs for Drinfeld doubles of pre-Nichols algebras of diagonal type and determine when they possess an Iwasawa decomposition. This extends G. Letzter's theory of quantum symmetric pairs. Our results can be uniformly applied to Kac-Moody quantum groups for a generic quantum parameter, for roots of unity in respect to both big and small quantum groups, to quantum supergroups and to exotic quantum groups of ufo type. We give a second construction of symmetric pairs for Heisenberg doubles in the above generality and prove that they always admit an Iwasawa decomposition. For symmetric pair coideal subalgebras with Iwasawa decomposition in the above generality we then address two problems which are fundamental already in the setting of quantum groups. Firstly, we show that the symmetric pair coideal subalgebras are isomorphic to intrinsically defined deformations of partial bosonizations of the corresponding pre-Nichols algebras. To this end we develop a general notion of star products on N-graded connected algebras which provides an efficient tool to prove that two deformations of the partial bosonization are isomorphic. The new perspective also provides an effective algorithm for determining the defining relations of the coideal subalgebras. Secondly, for Nichols algebras of diagonal type, we use the linear isomorphism between the coideal subalgebra and the partial bosonization to give an explicit construction of quasi K-matrices as sums over dual bases. We show that the resulting quasi K-matrices give rise to weakly universal K-matrices in the above generality.

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arXiv2018

Factorisation of quasi K-matrices for quantum symmetric pairs

The theory of quantum symmetric pairs provides a universal K-matrix which is an analogue of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed recursively. In this paper we restrict to the cases where the underlying Lie algebra is sl(n) or the Satake diagram has no black dots. In these cases we give an explicit formula for the quasi K-matrix as a product of quasi K-matrices for Satake diagrams of rank one. This factorisation depends on the restricted Weyl group of the underlying symmetric Lie algebra in the same way as the factorisation of the quasi R-matrix depends on the Weyl group of the Lie algebra. We conjecture that our formula holds in general.

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arXiv2016

Universal K-matrix for quantum symmetric pairs

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra and let $U_q(\mathfrak{g})$ denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras $B_{c,s}$ of $U_q(\mathfrak{g})$ have a universal K-matrix if $\mathfrak{g}$ is of finite type. By a universal K-matrix for $B_{c,s}$ we mean an element in a completion of $U_q(\mathfrak{g})$ which commutes with $B_{c,s}$ and provides solutions of the reflection equation in all integrable $U_q(\mathfrak{g})$-modules in category $\mathcal{O}$. The construction of the universal K-matrix for $B_{c,s}$ bears significant resemblance to the construction of the universal R-matrix for $U_q(\mathfrak{g})$. Most steps in the construction of the universal K-matrix are performed in the general Kac-Moody setting. In the late nineties T. tom Dieck and R. Häring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.

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arXiv2016

The bar involution for quantum symmetric pairs

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.

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arXiv2014

Quantum symmetric Kac-Moody pairs

The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as q-Onsager algebras and the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba.

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arXiv2013

Radial part calculations for affine sl2 and the Heun KZB-heat equation

In the present paper we determine the radial part of the Casimir element for the Lie algebra affine sl2 with respect to the Chevalley involution. The resulting operator is identified with a blend of the Inozemtsev Hamiltonian and the KZB-heat equation in dimension one. Moreover, it is shown how the corresponding zonal spherical functions give rise to symmetric theta functions and convergence is discussed. The paper takes guidance from previous work by Etingof and Kirillov on the diagonal case.

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arXiv2011

Braid group actions on coideal subalgebras of quantized enveloping algebras

We construct braid group actions on coideal subalgebras of quantized enveloping algebras which appear in the theory of quantum symmetric pairs. In particular, we construct an action of the semidirect product of Z^n and the classical braid group in n strands on the coideal subalgebra corresponding to the symmetric pair (sl_{2n}(C), sp_{2n}(C)). This proves a conjecture by Molev and Ragoucy. We expect similar actions to exist for all symmetric Lie algebras. The given actions are inspired by Lusztig's braid group action on quantized enveloping algebras and are defined explicitly on generators. Braid group and algebra relations are verified with the help of the package Quagroup within the computer algebra program GAP.

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arXiv2008

Reflection equation algebras, coideal subalgebras, and their centres

Reflection equation algebras and related U_q(g)-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called `covariantized' algebras, in particular concerning their centres, invariants, and characters. Generalising M. Noumi's construction of quantum symmetric pairs we define a coideal subalgebra B_f of U_q(g) for each character f of a covariantized algebra. The locally finite part F_l(U_q(g)) of U_q(g) with respect to the left adjoint action is a special example of a covariantized algebra. We show that for each character f of F_l(U_q(g)) the centre Z(B_f) canonically contains the representation ring Rep(g) of the semisimple Lie algebra g. We show moreover that for g=sl_n(C) such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of Rep(sl_n(C)) inside U_q(sl_n(C)). As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in C^{2m}.

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arXiv2007

The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight

It is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS-Cohen-Macaulay. The main ingredient in the proof is the notion of a quantum graded algebra with a straightening law, introduced by T.H. Lenagan and L. Rigal [J. Algebra 301 (2006), 670-702]. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS-Gorenstein.

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arXiv2006

Quantum Symmetric Pairs and the Reflection Equation

It is shown that central elements in G. Letzter's quantum group analogs of symmetric pairs lead to solutions of the reflection equation. This clarifies the relation between Letzter's approach to quantum symmetric pairs and the approach taken by M. Noumi, T. Sugitani, and M. Dijkhuizen. We develop general tools to show that a Noumi-Sugitani-Dijkhuizen type construction of quantum symmetric pairs can be performed preserving spherical representations from the classical situation. These tools apply to the symmetric pair FII and to all symmetric pairs which correspond to an automorphism of the underlying Dynkin diagram. Hence Noumi-Sugitani-Dijkhuizen type constructions with desirable properties are possible for various symmetric pairs for exceptional Lie algebras.

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