The notion of the 'classical' multilinear maps such as the symmetric and skew-symmetric maps, has the following generalization: given a vector-space V and a pair (H[subscript n],X[subscript n]) where is a subgroup of the symmetric group S[subscript n] and X[subscript n] is a character of H[subscript n], we consider multilinear maps from V[superscript n] (n-fold cartesian product of V ) into any other vector space, which are ‘symmetric with respect to (H[subscript n],X[subscript n])’, i.e., which have a certain symmetry in their values on permuted tuples of vectors, where the permutations are in H[subscript n]. Given a pair (H[subscript n],X[subscript n]) and a vector-space V , we can construct a space V[superscript (n)] over V through which the maps 'symmetric with respect to (H[subscript n],X[subscript n])’ linearize. The space V[superscript (n)] is usually defined abstractly by means of a certain universal mapping property and gives the tensor, symmetric and Grassmarm spaces for the 'classical' maps. Given a sequence of pairs [formula omitted]and the corresponding spaces V[superscript (n)], we let [formula omitted] (where V[superscript b]) is the ground field). In the classical cases, A has a natural multiplicative operation which makes A an algebra, i.e., the Tensor, Symmetric and Grassmann algebras. This presentation has been motivated by the attempt to generalize the construction of an algebra A to a wider family of 'admissible' sequences of pairs [formula omitted]. This consideration has led us to investigate permutation groups on the numbers 1,2,3,… which are closed under a certain 'shift' of the permutations, i.e., if [formula omitted] is a permutation, we define [formula omitted] and we call a permutation group H 'translation-invariant' if for every [formula omitted] is also in H . We begin our presentation by characterising the 'translation-invariant' groups. We show that the study of these (infinite) groups can be reduced to the study of certain finite groups. Then, we proceed to discuss the lattice of the translation-invariant groups. Finally, we show that a translation-invariant group H , together, with an appropriate character X of H , represents an equivalence class of 'admissible' sequences of pairs [formula omitted]. For a particular choice of representatives of the equivalence class, we can construct an algebra of 'symmetry classes of tensors' which generalizes the Tensor, Symmetric and Grassmann algebras. / Science, Faculty of / Mathematics, Department of / Graduate

Certain non-associative algebras have important applications in theoretical Mendelian Genetics. In this thesis we will give definitions to these algebras and study their properties. Some examples will also be given. / Science, Faculty of / Mathematics, Department of / Graduate

Neste trabalho discutimos a semissimplicidade de álgebras de Hopf finito-dimensionais e construímos o Duplo de Drinfeld D(H) de uma tal álgebra H. Além disso, apresentamos um resultado mostrando a equivalência entre as categorias de representações dos módulos sobre D(H) e dos módulos de Yetter-Drinfeld sobre Hcop. Como consequência deste estudo, apresentamos um resultado que caracteriza uma álgebra de Hopf quase triangular. / In this work we discuss the semisimplicity of some finite-dimensional Hopf Algebras and we set up the Drinfel’d double D(H) of such an algebra H. In addiction, we present a result showing the equivalence between the representation category of modules over D(H) and the Yetter-Drinfeld modules over Hcop. As a consequence of this, we present a result that features a quasitriangular Hopf algebra.

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The present thesis mainly concerns B*-algebras, A*-algebras, and minimal ideals in topological rings.

For the algebraic group SLl+1(C) we describe a system of positive roots associated to conjugacy classes in its Weyl group Sl+1. Using this we explicitly describe the algebra of regular functions on certain transverse slices to conjugacy classes in SLl+1(C) as a polynomial algebra of invariants. These may be viewed as an algebraic group analogue of certain parabolic invariants that generate the W-algebra in type A found by Brundan and Kleshchev.