The Homotopy Type of Lie Semigroups in Semi-Simple Lie Groups

 Let G be a noncompact semi-simple Lie group and a Lie semigroup with nonempty interior. We study the homotopy groups , , of S. Generalizing a well known fact for G, it is proved that there exists a compact and connected subgroup such that is isomorphic to . Furthermore, there exists a coset z contained in int S which is a deformation retract of S. Received 6 December 2000; in revised form 23 November 2001     

Connected Components of Open Semigroups in Semi-Simple Lie Groups

This paper studies connected components of open subsemigroups of non-compact semi-simple Lie groups by using control sets in the flag manifolds and their coverings. The concept of recurrent component is introduced and a method is given by which their number can be computed. It is shown that the union of all recurrent components is a semigroup. The idea of mid-reversibility comes up to show that an open semigroup has just one semigroup component if the identity belongs to its closure. A necessary and sufficient condition for mid-reversibility is proved showing that e.g. in a complex group any open semigroup is mid-reversible.     

一类无限维半单李代数

研究了秩为2的Witt型的李代数W2的子代数g1,讨论其同态,同构,核的性质.证明了g1为无限维半单李代数.     

Homotopy Groups of Spheres and Lipschitz Homotopy Groups of Heisenberg Groups

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, ${\pi_m^{\rm Lip}(\mathbb{H}_n)}$ , in terms of properties of the classical homotopy group of the sphere, ${\pi_m(\mathbb{S}^n)}$ . As an application we provide a new simplified proof of the fact that ${\pi_n^{\rm Lip}(\mathbb{H}_n)\neq \{0\}, n=1,2,\ldots}$ , and we prove a new result that ${\pi_{4n-1}^{\rm Lip}(\mathbb{H}_{2n})\neq \{0\}}$ for n = 1,2,… The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space ${W^{1,p}(\mathcal{M},\mathbb{H}_{2n})}$ when ${\dim \mathcal{M} \geq 4n}$ and 4n?1 ≤  p < 4n.     

On a Class of Non-Translation Invariant Feller Semigroups on Lie Groups

We consider a class of Feller semigroups on Lie groups which fail to commute with left translation due to the existence of a cocycle h which is identically one for Lévy processes. Under certain conditions, we are able to show that the infinitesimal generator of such a semigroup has the Lévy–Khintchine–Hunt form but with variable characteristics, thus we obtain an extension of classical work in Euclidean space by Courrège.     

A Group Theoretical Characterisation of S-Arithmetic Groups in Higher Rank Semi-Simple Groups

We give a characterisations for an abstract group to be an S-arithmetic group of a higher rank semi-simple group.     

Semigroups in symmetric Lie groups

Let G be a Lie group and L C G a Lie subgroup. We give necessary and sufficient conditions for a family of cosets of L to generate a subsemigroup with nonempty interior in G. We apply these conditions to symmetric pairs (G, L) where L is a subgroup of G such that Go C L C Gi and r is an involutive automorphism of G. As a consequence we prove that for several r the fixed point group GI is a maximal semigroup.     

Semisimple Modules for Finite Groups of Lie Type

Let k be an algebraically closed field of characteristic p >0, and let G be a connected, reductive algebraic group overk. In [8] and [11], conditions on the dimension of rationalG modules were seen to imply semisimplicity of these modules.In [8], certain of these conditions were extended to cover thefinite groups of Lie type. In this paper, we extend some ofthe results of [11] to cover these finite Lie type groups. Themain such extension is the following result.     

Positive Semicharacters of Lie Semigroups

We study positive semicharacters of generating Lie subsemigroup of a connected Lie group . These semicharacters are important for positive representations of in Hilbert space and for completely monotonic functions in . We describe the tangent map for a positive semicharacter and then obtain a necessary and sufficient condition for nontriviality of the wedge consisting of all bounded positive semicharacters of . In particular is nontrivial for a solvable simply connected and invariant without nontrivial subgroups, but it is trivial for a semisimple .     

Lipschitz Homotopy Groups of the Heisenberg Groups

Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group ${\mathbb{H}^n}$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Haj?asz-Lukyanenko-Tyson constructed horizontal maps from S ^k to ${\mathbb{H}^n}$ which factor through n-spheres and showed that these maps have no smooth horizontal fillings. In this paper, however, we build on an example of Kaufman to show that these maps sometimes have Lipschitz fillings. This shows that the Lipschitz and the smooth horizontal homotopy groups of a space may differ. Conversely, we show that any Lipschitz map ${S^k \to \mathbb{H}^1}$ factors through a tree and is thus Lipschitz null-homotopic if ${k \geq 2}$ .     

Degree Graphs of Simple Groups of Exceptional Lie Type

Abstract

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type.     

Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type

Let G = G(q) be a finite almost simple exceptional group ofLie type over the field of q elements, where q = p^a and p isprime. The main result of the paper determines all maximal subgroupsM of G(q) such that M is an almost simple group which is alsoof Lie type in characteristic p, under the condition that rank(M)> rank(G). The conclusion is that either M is a subgroupof maximal rank, or it is of the same type as G over a subfieldof F_q, or (G, M) is one of (, F₄(q)), (, C₄(q)), (E₇(q),³D₄(q)). This completes work of the first author with Saxl andTesterman, in which the same conclusion was obtained under someextra assumptions.     

Intermediate Semigroups are Groups

noindent We consider the lattice of subsemigroups of the general linear group over an Artinian ring containing the group of diagonal matrices and show that every such semigroup is actually a group. August 25, 1999     

Finite Sylow Subgroups in Simple Locally Finite Groups of Lie Type

The main result of the paper is the followingTheorem. Let S = {r₀, r₁, ..., r_n} be a finite nonempty set of primes and let L be a Lie type of Chevalley groups. Then there exists a locally finite field F of characteristic r₀ such that Sylow r-subgroups of the simple group L(F) of type L over F are finite if and only if r ? S.     

Simply Connected, Adjoint and Universal Groups of Lie Type

The special linear group is the simply connected group and theprojective linear group is the adjoint group of Lie type A_n.They are distinguished sections of the (reductive) general lineargroup which certainly is of this type as well (root system).We shall characterize the general linear group as the universalgroup of type A_n. Indeed we shall introduce corresponding algebraicgroups and finite groups for each Lie type (to indecomposableroot systems). Knowledge of the universal group implies knowledgeof the related simply connected and adjoint groups; in certainrespects the universal group even appears to be better behaved(automorphisms, Schur multiplier, character table).     

拓扑偶的自同伦等价群

研究拓扑偶范畴中的自同伦等价群.对于同伦可结合与同伦可逆的Co-H空间X与Y,我们将在一定的条件下得到一条可裂的短正合序列:1→(?)(X)→(?)(X(?)Y)→(?)(Y)→1.     

The Newton Iteration on Lie Groups

We define the Newton iteration for solving the equation f(y) = 0, where f is a map from a Lie group to its corresponding Lie algebra. Two versions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian manifolds. In particular, we show that, under classical assumptions on f, the proposed method converges quadratically. We illustrate the techniques by solving a fixed-point problem arising from the numerical integration of a Lie-type initial value problem via implicit Euler.