#### Date of Award

1995

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Abstract

The thesis is on the Putcha-Renner theory of algebraic monoids over (an algebraically closed field) K, founded by M. Putcha and L. Renner in 1980s, and its approaches to linear associative K-algebras with 1 (LAAs). It splits into 4 subtopics: nilpotent algebraic monoids, reductive algebraic monoids, Cartan submonoids of algebraic monoids and algebraic monoid approaches to LAAs.;The structure of a nilpotent algebraic monoid is generally much more complicated than that of a nilpotent algebraic group. We find a faithful matrix representation for nilpotent algebraic monoids. With a variation of the proof of it, we characterize the Lie algebra of a LAA to be nilpotent by the finiteness of its idempotent set, and in turn by the nilpotency of all irreducible algebraic submonoids of codimension 1.;The study of regular algebraic monoids dominates Putcha-Renner theory. We find a useful characterization for algebraic monoids with group kernel to be regular. We characterize a regular irreducible algebraic monoid to be nilpotent by the finiteness of its idempotent set and nilpotency of its kernel and to be solvable by the self-normality of a Cartan submonoid.;We characterize an irreducible monoid to be reductive by the regularity of it and the reductivity of its kernel, extending the Putcha-Renner Theorem, the most powerful result in Putcha-Renner theory, to a new level.;Regarding the Cartan submonoids, we show that, given a maximal torus of an irreducible monoid, the centralizer of the torus in the monoid is irreducible and the normalizer of the torus in the monoid is closed, as long as the algebraic monoid is completely regular.;Regarding the algebraic monoid approaches to LAAs, we find a surprising unification of the concepts of algebraic monoid and Lie algebra in LAAs. We prove that for a LAA A, A with respect to multiplication is a solvable (respectively, nilpotent) algebraic monoid iff A with respect to the Lie bracket is so as a Lie algebra. Thus a linear associative algebra with its Lie algebra solvable is triangularizable.;Low dimensional algebraic monoids and LAAs are also extensively studied. We prove that all 4-dimensional irreducible nonsolvable algebraic monoids are generalized semisimple and that a 5-dimensional irreducible nonsolvable algebraic monoid is reductive iff the dimension of a maximal d-submonoid is other than 2. All 4-dimensional LAAs with infinitely many idempotents and all 3-dimensional LAAs are completely determined.

#### Recommended Citation

Huang, Wenxue, "Algebraic Monoids With Approaches To Linear Associative Algebras" (1995). *Digitized Theses*. 2494.

https://ir.lib.uwo.ca/digitizedtheses/2494