Citation
Mohammed, Nadia Faiq
(2017)
Group actions and their applications in associative algebras and algebraic statistics.
Doctoral thesis, Universiti Putra Malaysia.
Abstract
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 lowdimensional
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 socalled inner
derivation mapping which can be roughly interpreted as 1coboundaries 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 2cocycles
and 2coboundaries 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 lowdimensional 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 twoway 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.
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