Related papers: A universal formula for the volume of compact Lie …
In this paper we initiate a study of the relation between weight modules for simple Lie algebras and unitary representations of the corresponding simply-connected Lie groups. In particular we consider in detail from this point of view the…
We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary…
The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special…
This is an overview article on compact Lie groups and their representations, written for the Encyclopedia of Mathematical Physics to be published by Elsevier.
For any finite-dimensional simple Lie algebra $\mathfrak{g}$ and commutative associative algebra $S$ of finite type, we give a complete classification of the simple weight modules of $\mathfrak{g}\otimes S$ with bounded weight…
We show that integration over a $G$-manifold $M$ can be reduced to integration over a minimal section $\Sigma$ with respect to an induced weighted measure and integration over a homogeneous space $G/N$. We relate our formula to integration…
We introduce some classical concepts in the representation theory of compact groups, in order to use them for a new generalization of the Peter-Weyl Theorem. We mostly deal with functions on locally compact groups possessing large…
Consider the space Hom(Z^n,G) of pairwise commuting n-tuples of elements in a compact Lie group G. This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingularization of the generic component of…
We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of…
We derive a formula for the spacetime volume of a small causal cone. We use this formula within the context of causal set theory to construct causal set expressions for certain geometric quantities relating to a spacetime with a spacelike…
We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of SO(4) and of U(n), endowed with suitable bi-invariant Riemannian metrics.
Let G be a special linear group over the real, the complex or the quaternion, or a special unitary group. In this note, we determine all special unipotent representations of G in the sense of Arthur and Barbasch-Vogan, and show in…
An unambiguous definition of Feynman amplitudes in the Wess-Zumino-Witten sigma model and the Chern-Simon gauge theory with a general Lie group is determined by a certain geometric structure on the group. For the WZW amplitudes, this is a…
We consider an extended version of Horn's problem: given two orbits $\mathcal{O}_\alpha$ and $\mathcal{O}_\beta$ of a linear representation of a compact Lie group, let $A\in \mathcal{O}_\alpha$, $B\in \mathcal{O}_\beta$ be independent and…
Intrinsic volumes are fundamental geometric invariants generalizing volume, surface area, and mean width for convex bodies. We establish a unified Laplace-Grassmannian representation for intrinsic and dual volumes of convex polynomial…
In this work we shall generalize the Selberg trace formula to a non-unitary finite-dimensional complex representation $\chi:\Gamma\rightarrow\operatorname{GL}(V)$ of a uniform lattice $\Gamma$ of a real Lie group $G$.
We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a…
For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters $\alpha, \beta, \gamma$ and…
Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of…
We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…