How can you find a complete set of inequivalent irreducible (ordinary) representations of a finite group? The theory is classical but, except when the group was very small or had a rather special structure, the actual computations were prohibitive before the advent of high-speed computers ; and there remain practical difficulties even for groups of relatively small orders ( á 100). The present ...

Among the various branches of algebra, linear algebra has the distinctions of being well understood and applicable to a wide variety of scientific and mathematical problems. In particular, it can be used to convert mathematical problems into a format where computers can easily be applied to perform calculations that would be impractical for humans to attempt by hand. This project deals with one...

In this article, we commence an investigation of the SU(N) representation space of Seifert bered homology spheres (a 1 ; : : : ; a n): Under mild assumptions (e.g. if N is prime), then Theorem 3.1 implies that any closed connected component of irreducible SU(N) representations of (a 1 ; : : : ; a n) is homeomorphic to a component of SU(N) representations of an associated genus zero Fuchsian gro...

We describe irreducible conformal subalgebras of CendN and build the structure theory of associative conformal algebras with finite faithful representation.

Polynomials of boson creation and annihilation operators which form irreducible tensor operators for Jordanian quantum algebra U h (sl(2)), called h-symplecton, are introduced and their properties are investigated. It is shown that many properties of symplecton for Lie algebra sl(2) are extended to h-symplecton. The h-symplecton is also a basis of irreducible representation of SL h (2) dual to ...

11. Assume that (ρ, V ) is an irreducible regular representation of the linear algebraic group G. Fix v∗ ∈ V ∗ with v∗ 6= 0. For v ∈ V let φv ∈ Aff(G) be the representative function φv(g) = 〈v∗, ρ(g)v〉. Let E = {φv : v ∈ V } and let T : V → E be the map Tv = φv. Prove that T is a bijective linear map and that Tρ(g) = R(g)T for all g ∈ G, where R(g)f(x) = f(xg) for f ∈ Aff(G). Thus every irreduc...

To recap our story so far: we began by identifying an important abelian subgroup of G, the maximal torus T . By restriction any G representation π is a T representation π|T . In general π|T is a reducible T representation and the irreducible representations of T (which are all one-dimensional) that occur in π|T are called the weights of π. Irreducible representations of T , and thus weights, ar...

Let k be a non-Archimedean local field of characteristic 0. Let D be a division algebra with center k and index n. The group D∗ is a locally compact group which is compact modulo the center. Its complex irreducible representations are finite dimensional. If π is an irreducible representation of D∗ which is self-dual, i.e., if π∨ denotes the dual of π, π∨ ∼= π, then there exists a D∗-invariant, ...

In this review we show that a Clifford algebra possesses a unique irreducible representation ; the spinor representation. We discuss what types of spinors can exist in Minkowski space-times and we explain how to construct all the supersymmetry algebras that contain a given space-time Lie algebra. After deriving the irreducible representations of the superymmetry algebras, we explain how to use ...

Hilbert space representations of the cross product ∗-algebras of the Hopf ∗-algebra Uq(su2) and its module ∗-algebras O(Sqr) of Podles’ spheres are investigated and classified by describing the action of generators. The representations are analyzed within two approaches. It is shown that the Hopf ∗-algebra O(SUq(2)) of the quantum group SUq(2) decomposes into an orthogonal sum of projective Hop...