Kac-Moody Type Research Articles

SL2 representations and relative property (T)

We investigate relative property (T) of Kazhdan-Margulis for (SL2(R)⋉Rn,Rn), mainly where R is a discrete finitely generated commutative ring. The action of SL2(R) on Rn is defined through admissible lattices of the irreducible representations of sl2(C). Both positive and negative results are obtained. Applications of these results will be given, including the establishment of property (T) for certain Kac-Moody type groups over commutative rings.

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 from the universal ıquantum group of Kac-Moody type arising from quantum symmetric pairs to the ıHall algebra associated to a virtually acyclic ıquiver.

Relative braid group symmetries on $$\imath $$quantum groups of Kac–Moody type

Relative braid group symmetries on $$\imath $$quantum groups of Kac–Moody type

On the boundary conformal field theory approach to symmetry-resolved entanglement

We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved Rényi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with U(1)U(1), \mathbb{R}ℝ and \mathbb{Z}_2ℤ2 symmetry.

Hall Algebras and Quantum Symmetric Pairs of Kac–Moody Type II

Hall Algebras and Quantum Symmetric Pairs of Kac–Moody Type II

Serre-Lusztig relations for ıquantum groups III

Let U˜ı be a quasi-split universal ıquantum group associated to a quantum symmetric pair (U˜,U˜ı) of Kac-Moody type with a diagram involution τ. We establish the Serre-Lusztig relations for U˜ı associated to a simple root i such that i≠τi, complementary to the Serre-Lusztig relations associated to i=τi which we obtained earlier. A conjecture on braid group symmetries on U˜ı associated to i disjoint from τi is formulated.

Braid group symmetries on quasi-split $$\imath $$quantum groups via $$\imath $$Hall algebras

We establish automorphisms with closed formulas on quasi-split \(\imath \)quantum groups of symmetric Kac-Moody type associated to restricted Weyl groups. The proofs are carried out in the framework of \(\imath \)Hall algebras and reflection functors, thanks to the \(\imath \)Hall algebra realization of \(\imath \)quantum groups in our previous work. Several quantum binomial identities arising along the way are established.

Universal K-matrices for quantum Kac-Moody algebras

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H H endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U q L s l 2 U_qL\mathfrak {sl}_{2} .

Representations of Involutory Subalgebras of Affine Kac–Moody Algebras

We consider the subalgebras of split real, non-twisted affine Kac–Moody Lie algebras that are fixed by the Cartan–Chevalley involution. These infinite-dimensional Lie algebras are not of Kac–Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of {mathbb {N}}-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity and work out a detailed example in the case of the affine extension of {mathfrak {e}}_8.

Serre–Lusztig relations for $$\imath $$quantum groups II

The $\imath$Serre relations and the corresponding Serre-Lusztig relations are formulated and established for arbitrary $\imath$quantum groups arising from quantum symmetric pairs of Kac-Moody type.

Complexity and the Big Bang

After a brief review of current scenarios for the resolution and/or avoidance of the Big Bang, an alternative hypothesis is put forward implying an infinite increase in complexity towards the initial singularity. This may result in an effective non-calculability which would present an obstruction to actually reaching the beginning of time. This proposal is motivated by the appearance of certain infinite-dimensional duality symmetries of indefinite Kac–Moody type in attempts to unify gravity with the fundamental matter interactions, and deeply rooted in properties of Einstein’s theory.

Symmetry-resolved entanglement in AdS3/CFT2 coupled to U(1) Chern-Simons theory

We consider symmetry-resolved entanglement entropy in AdS3/CFT2 coupled to U(1) Chern-Simons theory. We identify the holographic dual of the charged moments in the two-dimensional conformal field theory as a charged Wilson line in the bulk of AdS3, namely the Ryu-Takayanagi geodesic minimally coupled to the U(1) Chern-Simons gauge field. We identify the holonomy around the Wilson line as the Aharonov-Bohm phases which, in the two-dimensional field theory, are generated by charged U(1) vertex operators inserted at the endpoints of the entangling interval. Furthermore, we devise a new method to calculate the symmetry resolved entanglement entropy by relating the generating function for the charged moments to the amount of charge in the entangling subregion. We calculate the subregion charge from the U(1) Chern-Simons gauge field sourced by the bulk Wilson line. We use our method to derive the symmetry-resolved entanglement entropy for Poincaré patch and global AdS3, as well as for the conical defect geometries. In all three cases, the symmetry resolved entanglement entropy is determined by the length of the Ryu-Takayanagi geodesic and the Chern-Simons level k, and fulfills equipartition of entanglement. The asymptotic symmetry algebra of the bulk theory is of hat{mathfrak{u}}{(1)}_k Kac-Moody type. Employing the hat{mathfrak{u}}{(1)}_k Kac-Moody symmetry, we confirm our holographic results by a calculation in the dual conformal field theory.

Canonical bases arising from quantum symmetric pairs of Kac–Moody type

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\textbf U$-modules and their tensor products regarded as $\textbf {U}^\imath$-modules, as well as an $\imath$-canonical basis for the modified form of the $\imath$-quantum group $\textbf {U}^\imath$. A key new ingredient is a family of explicit elements called $\imath$-divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$. We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-$K$-matrix and the constructions of $\imath$-canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

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 analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra U_q(\mathfrak{g}) . We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous q -Serre type and are valid without restrictions on the generalized Cartan matrix. We draw special attention to the split case, where the quantum symmetric pair coideal subalgebras are generalized q -Onsager algebras.

Serre–Lusztig Relations for $$\imath $$Quantum Groups

Let $(\bf U, \bf U^\imath)$ be a quantum symmetric pair of Kac-Moody type. The $\imath$quantum groups $\bf U^\imath$ and the universal $\imath$quantum groups $\widetilde{\bf U}^\imath$ can be viewed as a generalization of quantum groups and Drinfeld doubles $\widetilde{\bf U}$. In this paper we formulate and establish Serre-Lusztig relations for $\imath$quantum groups in terms of $\imath$divided powers, which are an $\imath$-analog of Lusztig's higher order Serre relations for quantum groups. This has applications to braid group symmetries on $\imath$quantum groups.

Hall Algebras and Quantum Symmetric Pairs II: Reflection Functors

We extend our $\imath$Hall algebra construction from acyclic to arbitrary $\imath$quivers, where the $\imath$quiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism from the universal $\imath$quantum group of Kac-Moody type arising from quantum symmetric pairs to the $\imath$Hall algebra associated to a virtually acyclic $\imath$quiver.

Defining relations of quantum symmetric pair coideal subalgebras

AbstractWe 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 noncommutative${\mathbb N}$-graded algebras. The resulting defining relations are expressed in terms of continuousq-Hermite polynomials and a new family of deformed Chebyshev polynomials.

Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-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 [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] 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.

A SERRE PRESENTATION FOR THE ıQUANTUM GROUPS

Let (U, ) be a quasi-split quantum symmetric pair of arbitrary Kac–Moody type, where “quasi-split” means the corresponding Satake diagram contains no black node. We give a presentation of the group with explicit relations. The verification of new relations is reduced to some new q-binomial identities. Consequently, is shown to admit a bar involution under suitable conditions on the parameters.

Davey–Stewartson equations in (3 + 1) dimensions with an infinite-dimensional symmetry algebra

This article is devoted to discovering Lie symmetry algebra of a (3+1)-dimensional Davey-Stewartson system which appears in the field of plasma physics. It is found that the algebra is an infinite dimensional one and of Kac-Moody type. Making use of these symmetries, some reduced equations to lower dimensions are also presented.