High powers of random elements of compact Lie groups

Summary. If a random unitary matrix is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of falling in a given arc, as the dimension of tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9]. Received: 15 October 1995 / In revised form: 7 March 1996     

Compact elements and Cartan subgroups of connected Lie groups

It is established that the number of compact Cartan subgroups of a connected Lie group is determined by the topological size of the set of its compact elements. This fact clarifies the structure of those connected Lie groups in which the set indicated is everywhere dense.Translated from Ukrainskii Matematicheskii Zhurnal, Vol 42, No. 2, pp. 164–168, February, 1990.     

Nilpotent commuting varieties of reductive Lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p0, and =LieG. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let =() denote the nilpotent variety of , and ^nil():={(x,y)×|[x,y]=0}, the nilpotent commuting variety of . Our main goal in this paper is to show that the variety ^nil() is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme _n Hilb ⁿ (²) is irreducible over any algebraically closed field. Mathematics Subject Classification (2000) 20G05     

On commuting automorphisms of groups

This paper examines group automorphisms for which each element commutes with its image. These automorphisms do not necessarily form a subgroup of the automorphism group, but have a number of interesting properties, close to those of central automorphisms. Research of the first named author was supported by Grant SM 177 from Kuwait University Research Administration.     

On multilinear operators commuting with Lie derivatives

LetE ₁, ...,E _k andE be natural vector bundles defined over the categoryMf _m ⁺ of smooth orientedm-dimensional manifolds and orientation preserving local diffeomorphisms, withm2. LetM be an object ofMf _m ⁺ which is connected. We give a complete classification of all separately continuousk-linear operatorsD : _c(E ₁ M) × ... × _c(E _k M) (EM) defined on sections with compact supports, which commute whith Lie derivatives, i.e., which satisfy

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple ${(G,\omega,\nabla)}$ such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ${\nabla}$ . In this paper starting from a special symplectic Lie group we show how to “deform” the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure ${\nabla}$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.     

Lie groups

The survey deals with the investigations reviewed inReferativnyi Zhurnal Matematika between 1977–1981. In the survey there are reflected the investigations on the structure of Lie groups and Lie algebras, on their finite-dimensional linear representations and universal enveloping algebras, on the theory of invariants and Lie groups of transformations, and also on continuous and discrete subgroups of Lie groups.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 20, No. 153–192, 1982.     

Non-closed Lie subgroups of Lie groups

We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that ₁(K) and ₁(H) have the same torsion subgroup, _n (K)= _n (H) for alln 2, and rank₁(K) — rank₁(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank₁(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).     

Spectra of elements in the range of strongly continuous unitary representations of Lie groups

The main result proved in this work is that, for a strongly continuous unitary representation of a Lie group, we have the following dichotomy: The representation is norm continuous or on an open neighborhood of the unit (which is the whole group for a weakly exponential Lie group), there is a residual set of elements whose range admits the entire torus as spectrum.     

Isometry groups of three-dimensional Lie groups

Annals of Global Analysis and Geometry - We compute the full isometry group of any left-invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we...     

On abelian groups with commuting monomorphisms

We find necessary and sufficient conditions under which an abelian group with commuting monomorphisms has the endomorphism ring normal. We describe divisible groups with left- or righ-tinvariant monomorphisms and prove that a reduced torsion-free group with left- or right-invariant monomorphisms has the endomorphism ring normal. Necessary and sufficient conditions are indicated for the left and right invariance of monomorphisms from some classes of splitting groups.     

Factorization of groups by means of commuting periodic subgroups with no elements of identical prime orders

A number of results are presented which establish properties of groups factorizable by means of pairwise commuting periodic subgroups with no elements of identical prime orders in terms of properties of the factors.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1429–1436, October, 1991.     

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.     

Extrapolation methods in Lie groups

Summary. The numerical solution of differential equations on Lie groups by extrapolation methods is investigated. The main principles of extrapolation for ordinary differential equations are extended on the general case of differential equations in noncommutative Lie groups. An asymptotic expansion of the global error is given. A symmetric method is given and quadratic asymptotic expansion of the global error is proved. The theoretical results are verified by numerical experiments. Received September 27, 1999 / Revised version received February 14, 2000 / Published online April 5, 2001     

Analytic density in Lie groups

A subgroupH of an analytic groupG is said to beanalytically dense if the only analytic subgroup ofG containingH isG itself. The main purpose of this paper is to give sufficient conditions onG (analogous to those of [8], [9], and [7] in the case of Zariski density) which guarantee the analytic density of cofinite volume subgroupsH. First we consider the case of arbitrary cofinite volume subgroups (Theorem 5 and its corollaries). Then we specialize to lattices, and prove the following result (Theorem 8):Let G be an analytic group whose radical is simply connected and whose Levi factor has no compact part and a finite center. Then any lattice in G is analytically dense. In proving this use is made of a result of Montgomery which also implies that for any simply connected solvable group, cocompactness of a closed subgroup implies analytic density. In the case of a solvable group with real roots this means analytic density and cocompactness are equivalent and thus completes a circle of ideas raised in Saito [13]. In Corollary 9 we deal with a related local condition. Finally in Theorem 10 and its corollaries we apply these considerations to prove a homomorphism extension theorem and an isomorphism theorem for 1-dimensional cohomology.     

On automorphism groups of connected Lie groups

We prove that ifG is a connected Lie group with no compact central subgroup of positive dimension then the automorphism group ofG is an almost algebraic subgroup of , where is the Lie algebra ofG. We also give another proof of a theorem of D. Wigner, on the connected component of the identity in the automorphism group of a general connected Lie group being almost algebraic, and strengthen a result of M.Goto on the subgroup consisting of all automorphisms fixing a given central element.