Group actions and their applications in associative algebras and algebraic statistics - Mohammed, Nadia Faiq (2017)
Mathematics and Physics. There can be different ways for a group to act on different kinds of objects. This dissertation is mostly concerned with group actions on vector spaces and affine algebraic varieties. It is mainly comprised of three parts. In the first part, we consider an action of associative algebra A on a vector space to study low-dimensional cohomology groups of associative algebras. The origin of cohomology groups is found in algebraic topology. The dimension of the cohomology groups is considered one of the important invariant to study properties of algebras. Particularly, this invariant plays a rigorous role in geometric classification of associative algebras. We focus on the applications of low dimensional cohomology groups Hi(A;A): We start with the zero order cohomology groups of two and three dimensional complex associative algebras. Then, we consider an important special case of derivations so-called inner derivation mapping which can be roughly interpreted as 1-coboundaries on vector space where the algebra A acting. We study some of their properties and give an algorithm to obtain the inner derivations of associative algebras. Another main result of this part is precisely formulated two algorithms for describing the 2-cocycles and 2-coboundaries of A. Using an existing classification result of low dimensional associative algebras, we apply all these algorithms to complex associative algebras up to dimension three. In the second part, general linear group acts on an affine algebraic variety over an algebraically closed field. This part is devoted to the study of the rigidity of associative algebras. We give necessary invariance arguments for the existence of degenerations which is helpful in finding out for associative algebras to be rigid. Subsequently, applications of the invariance arguments to the varieties of low-dimensional complex associative algebras are described. In the last part of the thesis, we consider an action of dihedral group Dp on the rational vector space Qp to study an invariance group of Markov basis for some contingency tables. Using this action, we find the invariance subgroup H of Markov basis B with respect to the dihedral group Dp: Moreover, an algorithm to obtain the set of all independence models of two-way contingency tables with the same row sums and column sums is proposed. Finally, we introduce a new class of algebras which is closely related to the Markov basis in algebraic statistics.