Central measures on compact simple Lie groups

Proceedings of the American Mathematical Society, 1972

In this paper it is shown that for a connected semisimple Lie group with no nontrivial compact quotient any finite central measure is a discrete measure concentrated on the center of the group. More generally, the largest possible support set for a central measure on any semisimple Lie group is determined. From these results it follows that the center of the algebra L 1 ( H ) {L_1}(H) is trivial for any locally compact group H which has a noncompact connected simple Lie group as a homomorphic image.

Archiv der Mathematik, 1991

2000

We show that, for any connected semi-simple Lie group G, there is a natural isomorphism between the Galois cohomology H 2 (G, T) (with respect to the trivial action of G on the circle group T) and the Pontryagin dual of the homology group H 1 (G) (with integer coefficients) of G as a manifold. As an application, we find that there is a natural correspondence between the projective representations of any such group and a class of ordinary representations of its universal cover. We illustrate these ideas with the example of the group of bi-holomorphic automorphisms of the unit disc.

Transactions of the American Mathematical Society, 1983

Let g R = f R + p R {\mathfrak {g}_{\mathbf {R}}} = {\mathfrak {f}_{\mathbf {R}}} + {\mathfrak {p}_{\mathbf {R}}} be a Cartan decomposition of a real semisimple Lie algebra g R {\mathfrak {g}_{\mathbf {R}}} and let g = f + p \mathfrak {g} = \mathfrak {f} + \mathfrak {p} be the corresponding complexification. Also let a R {\mathfrak {a}_{\mathbf {R}}} be a maximal abelian subspace of p R {\mathfrak {p}_{\mathbf {R}}} and let a \mathfrak {a} be the complex subspace of p \mathfrak {p} generated by a R {\mathfrak {a}_{\mathbf {R}}} . We assume dim ⁡ a R = 1 \dim {\mathfrak {a}_{\mathbf {R}}} = 1 . Now let G G be the adjoint group of g \mathfrak {g} and let K K be the analytic subgroup of G G with Lie algebra ad g ( f ) {\text {ad}}_\mathfrak {g}(\mathfrak {f}) . If S ′ ( g ) S^\prime (\mathfrak {g}) denotes the ring of all polynomial functions on g \mathfrak {g} then clearly S ′ ( g ) S^\prime (\mathfrak {g}) is a G G -module and a fortiori a K K -module. In this paper, we determine the...

Advances in Linear Algebra & Matrix Theory

This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was more superficial in that book with some remarks about the examples given in relation to the left group. I decided to try a little harder and collect as much literature as possible, both for the needs of me and the others who will take after me. Searching for literature in my mother tongue I could not find anything, in English as someone who comes from a small country like Montenegro, all I could find was through the internet. I decided to gather what I could find from the literature in my own way and to my observation and make this kind of work. The main content of this paper is to present the Lie group in the simplest way. Before and before I started writing or collecting about Lie groups, it was necessary to say something about groups and subgroups that are taught in basic studies in algebra. In them I cited several deficits and an example. The following content of the paper is related to Lie groups primarily concerning the definition of examples such as The General Linear Group GL(n, R), The Complex General Linear Group GL(n, C), The Special Linear Group

2016

Mathematical Proceedings of the Cambridge Philosophical Society

In noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups ar...

Ukrainian Mathematical Journal, 1990

libdspace.uwaterloo.ca

Abstract: This thesis is an expository account of three central theorems in the representation theory of semisimple Lie groups, namely the theorems of Borel-Weil-Bott, Casselman-Osborne and Kostant. The first of these realizes all the irreducible holomorphic ...

2011

In this PhD thesis, we consider two problems that are related to finite simple groups of Lie type. First of them is a problem mentioned in the Kourovka notebook: describe the finite simple groups in which every element is a product of two involutions. We consider the simple orthogonal groups in even characteristic, and solve the problem for them. Since other groups have been dealt with elsewhere, the problem is then solved completely. The second part of the thesis is related to Lie algebras. Every complex simple Lie algebra has a compact real form that is associated with a compact Lie group. In this thesis, we consider the Lie algebra of type E8, and give a new construction of its compact real form. The Lie product is defined using the irreducible subgroup of shape 25+10 ·GL5(2) of the automorphism group.