On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

Open subgroups of locally compact Kac–Moody groups

Let $G$ be a complete Kac–Moody group over a finite field. It is known that $G$ possesses a BN-pair structure, all of whose parabolic subgroups are open in $G$ . We show that, conversely, every open subgroup of $G$ is contained with finite index in some parabolic subgroup; moreover there are only finitely many such parabolic subgroups. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.     

Character formula of Kac–Wakimoto type for generalized Kac–Moody algebras

We prove a character formula of Kac–Wakimoto type for generalized Kac–Moody algebras. A character formula of this type is a generalization of the Weyl–Kac character formula, and is proved by Kac–Wakimoto in the case of Kac–Moody algebras. We remark that the formula is a generalization of that of Kac–Wakimoto even in the case of Kac–Moody algebras of indefinite type.     

A Brylinski filtration for affine Kac–Moody algebras

Braverman and Finkelberg have recently proposed a conjectural analogue of the geometric Satake isomorphism for untwisted affine Kac–Moody groups. As part of their model, they conjecture that (at dominant weights) Lusztig's q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space, as occurs in the finite-dimensional case. We show that the conjectured equality holds for all affine Kac–Moody algebras if the principal nilpotent filtration is replaced by the principal Heisenberg filtration. The main body of the proof is a Lie algebra cohomology vanishing result. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity. Finally, we give some partial results for indefinite Kac–Moody algebras.     

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.     

Quantizations of Kac–Moody algebras

We analyze the extent to which a quantum universal enveloping algebra of a Kac–Moody algebra g$\mathfrak{g}$ is defined by multidegrees of its defining relations. To this end, we consider a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the Kac–Moody algebra g$\mathfrak{g}$. We demonstrate that if the generalized Cartan matrix A$A$ of g$\mathfrak{g}$ is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one “continuous” parameter q$q$ related to a symmetrization of A$A$, and one “discrete” parameter m$\mathfrak{m}$ related to the modular symmetrizations of A$A$. The Hopf algebra structure is defined by n(n−1)/2$n(n-1)/2$ additional “continuous” parameters. We also consider the exceptional cases for Cartan matrices of finite or affine types in more detail, establishing the number of exceptional parameter values in terms of the Fibonacci sequence.     

A Formal Chevalley Restriction Theorem for Kac–Moody Groups

Let G be a symmetrizable Kac–Moody group over a field of characteristic zero, let T be a split maximal torus of G. By using a completion of the algebra of strongly regular functions on G, and its restriction on T, we give a formal Chevalley restriction theorem. Specializing to the affine case, and to the field of complex numbers, we obtain a convergent Chevalley restriction theorem, by choosing the formal functions, which are convergent on the semi-groups of trace class elements G ^trG resp. T ^trT.     

Supercategorification of quantum Kac–Moody algebras

We show that the quiver Hecke superalgebras and their cyclotomic quotients provide a supercategorification of quantum Kac–Moody algebras and their integrable highest weight modules.     

s-Representations for involutions¶on affine Kac–Moody algebras are polar

Let denote the eigenspace decomposition of a twisted affine Kac–Moody algebra with respect to an involution , where is a twisted loop algebra, is the center and d is the scaling element of . We endow with the standard bilinear symmetrical form.Then with and carries a Lorentzian signature. Let denote the group that corresponds to , then the adjoint representation of on can be restricted to and this submanifold is isometrical to the Hilbert space E _ε, where is the decomposition of the twisted loop algebra with respect to the induced involutionρ₀.We thus obtain an affine representation on E _ε and we show that this representation is polar, i. e., there exists a submanifold that intersects all orbits, and intersects them orthogonally. Received: 16 February 2000 RID=" ID="Supported by a DFG grant.     

Hyperbolic subalgebras of hyperbolic Kac–Moody algebras

We investigate regular hyperbolic subalgebras of hyperbolic Kac–Moody algebras via their Weyl groups. We classify all subgroup relations between Weyl groups of hyperbolic Kac–Moody algebras, and show that for every pair of a group and subgroup there exists at least one corresponding pair of algebra and subalgebra. We find all types of regular hyperbolic subalgebras for a given hyperbolic Kac–Moody algebra, and present a finite algorithm classifying all embeddings.     

A study of saturated tensor cone for symmetrizable Kac–Moody algebras

Let \(\mathfrak {g}\) be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra \(\mathfrak {h}\) and the Weyl group \(W\) . Let \(P_+\) be the set of dominant integral weights. For \(\lambda \in P_+\) , let \(L(\lambda )\) be the integrable, highest weight (irreducible) representation of \(\mathfrak {g}\) with highest weight \(\lambda \) . For a positive integer \(s\) , define the saturated tensor semigroup as $$\begin{aligned} \Gamma _s:= \{(\lambda _1, \dots , \lambda _s,\mu )\in P_+^{s+1}: \exists \, N\ge 1 \,\text {with}\,L(N\mu )\subset L(N\lambda _1)\otimes \dots \otimes L(N\lambda _s)\}. \end{aligned}$$ The aim of this paper is to begin a systematic study of \(\Gamma _s\) in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by \(\Gamma _s\) . These inequalities are indexed by products in \(H^*(G^{\mathrm{min }}/B; \mathbb {Z})\) for \(B\) the standard Borel subgroup, where \(G^{\mathrm{min }}\) is the ‘minimal’ Kac-Moody group with Lie algebra \(\mathfrak {g}\) . The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that \(\mathfrak {g}\) is affine of rank 2, we show that these inequalities are necessary and sufficient. We further prove that any integer \(d>0\) is a saturation factor for \(A^{(1)}_1\) and 4 is a saturation factor for \(A^{(2)}_2\) .     

Linear variance bounds for particle approximations of time-homogeneous Feynman–Kac formulae

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance for particle approximations of time-homogeneous Feynman–Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman–Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state space: (1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and (2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.     

Weyl Modules Associated to Kac–Moody Lie Algebras

Local Weyl modules were originally defined for affine Lie algebras by Chari and Pressley in [5 Chari, V., Pressley, A. (2001). Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191–223 (electronic).[Crossref] , [Google Scholar]]. In this paper we extend the notion of local Weyl modules for a Lie algebra 𝔤 ?A, where 𝔤 is any Kac–Moody algebra and A is any finitely generated commutative associative algebra with unit over ?, and prove a tensor product decomposition theorem which generalizes result in [2 Chari, V., Fourier, G., Khandai, T. (2010). A categorical approach to Weyl modules. Transform. Groups 15(3):517–549.[Crossref], [Web of Science ®] , [Google Scholar], 5 Chari, V., Pressley, A. (2001). Weyl modules for classical and quantum affine algebras. Represent. Theory 5:191–223 (electronic).[Crossref] , [Google Scholar]].