Distinguished orbits and the L–S category of simply connected compact Lie groups

We show that the Lusternik–Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the fundamental alcove for the action of the affine Weyl group on the Lie algebra of a maximal torus of G.     

Connected finite loop spaces with maximal tori

Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence for this conjecture by showing that the associated action of the Weyl group on the maximal torus always represents the Weyl group as a crystallographic group. We also develop the notion of normalizers of maximal tori for connected finite loop spaces, and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal torus is equivalent to a compact connected Lie group if it has the right normalizer of the maximal torus. Actually, in the cases under consideration the information about the Weyl group is sufficient to give the answer. All this is done by first studying the analogous local problems.

     

Curve selection Lemma for semianalytic sets and conjugacy classes of finite order in Lie groups

Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E_n(G)of all elements of order n in G is a conjugacy class in G. In particular, all conjugacy classes of finite order in G are closed. Some properties of connected components of E_n(G) are also given.     

Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras

A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern-Simons theories conjectured to give spin TQFTs by Dijkgraaf and Witten (1990, Commun. Math. Phys.129, 393-429). In particular, the spin invariants constructed by Kirby and Melvin (1991, Invent. Math.105, 473-545) are shown to be identical to the invariants associated to SO(3). Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finite-dimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg's noninvolutory invariant of framed manifolds associated to that Hopf algebra.     

Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups

Summary. The goal of this paper is to characterise certain probability laws on a class of quantum groups or braided groups that we will call nilpotent. First we introduce a braided analogue of the Heisenberg–Weyl group, which shall serve as standard example. We introduce Gaussian functionals on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein property, i.e. that the sum and difference of independent random variables are also independent. The corresponding functionals on the braided line, braided plane and a braided q-Heisenberg–Weyl group are determined. Section 5 deals with continuous convolution semigroups on nilpotent quantum groups and braided groups. We extend recent results proving the uniqueness of the embedding of an infinitely divisible probability law into a continuous convolution semigroup for simply connected nilpotent Lie groups to nilpotent quantum groups and braided groups. Finally, in Section 6 we give some indications how the semigroup approach of Heyer and Hazod to the Bernstein theorem on groups can be extended to quantum groups and braided groups. Received: 30 October 1996 / In revised form: 1 April 1997     

Product decompositions of solvable Lie groups

Summary The article investigate the structure of real solvable connected Lie groups. It is described how one can decompose a solvable Lie group in direct and semidirect products of closed connected subgroups. In particular, the commutator group, Cartan subgroups, the center, maximal compactly embedded subgroups and tori are considered. Furthermore, one can find special solvable Lie groups and their product decompositions, namely compactly generated solvable Lie groups and those Lie groups which are generated by maximal compactly embedded subgroups. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Lévy Processes in a Step 3 Nilpotent Lie Group

The infinitesimal generators of Lévy processes in Euclidean space are pseudodifferential operators with symbols given by the Lévy-Khintchine formula. This classical analysis relies heavily on Fourier analysis which, in the case when the state space is a Lie group, becomes much more subtle. Still the notion of pseudo-differential operators can be extended to connected, simply connected nilpotent Lie groups by employing the Weyl functional calculus. With respect to this definition, the generators of Lévy processes in the simplest step 3 nilpotent Lie group G are pseudodifferential operators which admit C _c (G) as its core.     

Quantum zonal spherical functions and Macdonald polynomials

A unified theory of quantum symmetric pairs is applied to q-special functions. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions. Here a distinguished family of such functions, invariant under the Weyl group associated to the restricted roots, is shown to be a family of Macdonald polynomials, as conjectured by Koornwinder and Macdonald. Our results place earlier work for Lie algebras of classical type in a general context and extend to the exceptional cases.     

Weyl Groups are Finite — and Other Finiteness Properties of Cartan Subalgebras

For each pair (??,??) consisting of a real Lie algebra ?? and a subalgebra a of some Cartan subalgebra ?? of ?? such that [??, ??]∪ [??, ??] we define a Weyl group W(??, ??) and show that it is finite. In particular, W(??, ??,) is finite for any Cartan subalgebra h. The proof involves the embedding of 0 into the Lie algebra of a complex algebraic linear Lie group to which the structure theory of Lie algebras and algebraic groups is applied. If G is a real connected Lie group with Lie algebra ??, the normalizer N(??, G) acts on the finite set Λ of roots of the complexification ??c with respect to hc, giving a representation π : N(??, G)→ S(Λ) into the symmetric group on the set Λ. We call the kernel of this map the Cartan subgroup C(??) of G with respect to h; the image is isomorphic to W(??, ??), and C(??)= {g G : Ad(g)(h)— h ε [h,h] for all h ε h }. All concepts introduced and discussed reduce in special situations to the familiar ones. The information on the finiteness of the Weyl groups is applied to show that under very general circumstance, for b ∪ ?? the set ??? ?(b) remains finite as ? ranges through the full group of inner automorphisms of ??.     

李群表示论和Schubert条件

本文将Grassmann流形上的Schubert子簇所满足的经典的Schubert条件推广到一般的复半单李群G的广义旗流形.利用复半单李群的表示理论,我们首先在李群的权空间上引入自然的Ehresman偏序.这一偏序可以导出李群的最高权表示的权系、Weyl群及其陪集空间上的Ehresman偏序.然后我们对一般的复表示定义了相应的射影空间,Grassmann流形和旗流形.这使得能够像经典的情形一样来分析广义旗流形的Schubert子簇满足的Schubert条件.在讨论中,我们还给出了李群G的Weyl群及其陪集空间中的Bruhat-Chevalley偏序的简单判别条件.我们的结果应用到例外群,给出了Fulton提出的关于例外群的Schubert分析的问题的部分回答.     

Spherical orbits and representations of Uε(g)

Let U_ε(g) be the simply connected quantized enveloping algebra at roots of one associated to a finite dimensional complex simple Lie algebra g. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of U_ε(g) is proved for the representations corresponding to the spherical conjugacy classes of the simply connected algebraic group G with Lie algebra g. We achieve this result by means of a new characterization of the spherical conjugacy classes of G in terms of elements of the Weyl group.     

Proper actions on cohomology manifolds

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most , then the group is effectively a Lie group.

     

Multilinear Quantum Lie Operations

It is proved that under an existence condition, the dimension of all n-linear quantum Lie operations lies between (n - 2)! and (n - 1)!; moreover, the low bound is attained if the intersection of all conforming (i.e., satisfying the existence condition) subsets of a given set of quantum variables is nonempty. The upper bound is attained if all subsets are conforming. The space of multilinear quantum Lie operations is almost always generated by symmetric operations. All exceptional cases are given. In particular, the space of general n-linear Lie operations is always generated by general symmetric quantum Lie operations. Bibliography: 24 titles.     

Trace formula for almost lie group of operators and cyclic one-cocycles

In this paper cyclic one-cocycles of Heisenberg groups and some other Lie group are determined. The concept of almost Lie group of operators is introduced, and the trace formula is given by cyclicone cocyle on the Lie group. The Von Neumann theorem on Weyl commutation relation is generalized in certain case.     

A combinatorial formula for Macdonald polynomials

In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for Macdonald polynomials for all Lie types. These formulas resemble the formulas of Haglund, Haiman and Loehr for Macdonald polynomials of type GLn. At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent and Littelmann).     

Metric properties on Lie groups of polynomial volume growth

It is well known that we have an algebraic characterization of connected Lie groups of polynomial volume growth: a Lie group G has polynomial volume growth if and only if it is of type R. In this paper, we shall give a geometric characterization of connected Lie groups of polynomial volume growth in terms of the distance distortion of the subgroups. More precisely, we prove that a connected Lie group G has polynomial volume growth if and only if every closed subgroup has a polynomial distortion in G.     

Integrality and gauge dependence of Hennings TQFTs

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbb{k}$, starting from a finite Hopf algebra over $\mathbb{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds.We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotomic integers for surfaces with one boundary component.We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element.These results are combined and applied to show that the Hennings invariant associated to quantum-${\mathfrak{sl}}_2$ takes values in the cyclotomic integers. Using prior results of Chen et al. we infer integrality also of the Witten–Reshetikhin–Turaev $SO(3)$ invariant for rational homology spheres.As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras.