Combinatorial Polynomial Identity Theory

This dissertation consists of two parts. Part I examines certain Burnside-type conditions on the multiplicative semigroup of an (associative unital) algebra $A$. A semigroup $S$ is called $n$-collapsing if, for every $a_1,\ldots, a_n \in S$, there exist functions $f\neq g$ on the set $\{1,2,\ldots,n\}$ such that \begin{center} $s_{f(1)}\cdots s_{f(n)} = s_{g(1)}\cdots s_{g(n)}$. \end{center} If $f$ and $g$ can be chosen independently of the choice of $s_1,\ldots,s_n$, then $S$ satisfies a semigroup identity. A semigroup $S$ is called $n$-rewritable if $f$ and $g$ can be taken to be permutations. Semple and Shalev extended Zelmanov\u27s Fields Medal writing solution of the Restricted Burnside Problem by proving that every finitely generated residually finite collapsing group is virtually nilpotent. The primary result of Part I is that the following conditions are equivalent for every algebra $A$ over an infinite field: the multiplicative semigroup of $A$ is collapsing, $A$ satisfies a multiplicative semigroup identity, and $A$ satisfies an Engel identity: $[x,_my]=0$. Furthermore, in this case, $A$ is locally (upper) Lie nilpotent. It is also shown that, if the multiplicative semigroup of $A$ is rewritable, then $A$ must be commutative. In Part II of this dissertation, we study algebraic analogues to well-known problems of Philip Hall on the verbal and marginal subgroups of a group. We begin by proving two algebraic analogues of the Schur-Baer-Hall Theorem: if $G$ is a group such that $G/\centre_n(G)$ is finite, where $\centre_n(G)$ is the $n^{\text{th}}$ higher centre of $G$, then the $(n+1)^{\text{st}}$ term, $\g_{n+1}(G)$, of the lower central series of $G$ is also finite; conversely, if $\g_{n+1}(G)$ is finite, then so is $G/\centre_{2n}(G)$. Next, we prove results of a more general type. Given an algebra $A$ and a polynomial $f$, we define the verbal subspace, $\S_A(f)$, of $A$ to be spanned by the set of $f$-values in $A$, the verbal subalgebra, $\A_A(f)$, and the verbal ideal, $\I_A(f)$, of $A$ to be generated by the set of $f$-values in $A$. We also define the marginal subspace $\widehat{\S}_A(f)$ of $A$ to be the set of all elements $z \in A$ such that \begin{center} $f(b_1,\ldots,b_{i-1},b_i+\a z,b_{i+1},\ldots,b_n)=f(b_1,\ldots,b_{i-1},b_i,b_{i+1},\ldots,b_n)$, \end{center} for all $i=1,2,\ldots,n$, $b_1,\ldots,b_n \in A$, and $\a \in K$. Furthermore, we define the marginal subalgebra, $\widehat{\A}_A(f)$, and the marginal ideal, $\widehat{\I}_A(f)$, to be the largest subalgebra, respectively, largest ideal, of $A$ contained in $\widehat{\S}_A(f)$. We consider the following problems: \begin{enumerate} \item If $\widehat{\S}_A(f)$ is of finite codimension in $A$, is $\S_A(f)$ finite-dimensional? \item If $\S_A(f)$ is finite-dimensional, is $\widehat{\S}_A(f)$ of finite codimension in $A$? \item If $\S_A(f)$ is finite-dimensional, is $\A_A(f)$ or $\I_A(f)$ finite-dimensional? \item If $A/\widehat{\S}_A(f)$ is finite-dimensional, is $A/\widehat{\A}_A(f)$ or $A/\widehat{\I}_A(f)$ finite-dimensional? \end{enumerate