D. M. Riley and Mark C. Wilson, Glasgow Mathematical Journal 41 (1999), 453-462.

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Abstract: Denote by $latex (R,cdot)$ the multiplicative semigroup of an associative algebra $latex R$ over an infinite field, and let $latex (R,circ)$ represent $latex R$ when viewed as a semigroup via the circle operation $latex xcirc y=x+y+xy$. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of $latex R$. Namely, we prove that the following conditions on $latex R$ are equivalent: the semigroup $latex (R,circ)$ satisfies an identity; the semigroup $latex (R,cdot)$ satisfies a reduced identity; and, the associated Lie algebra of $latex R$ satisfies the Engel condition. When $latex R$ is finitely generated these conditions are each equivalent to $latex R$ being upper Lie nilpotent.

Jeffrey Bergen and Mark C. Wilson, Journal of Algebra 220, (1999), 152-173.

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Abstract: Many important quantum algebras such as quantum symplectic space, quantum euclidean space, quantum matrices, $latex q$-analogs of the Heisenberg algebra and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras $latex U(L_+)$ of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the X-inner automorphisms of such algebras.

The theorems and examples in our paper show that for algebras $latex R$ of this type, the non-identity X-inner automorphisms of $latex R$ tend to have infinite order. Thus if $latex G$ is a finite group of automorphisms of $latex R$, then the action of $latex G$ will be X-outer and this immediately gives useful information about crossed products $latex R*_t G$.

D. M. Riley and Mark C. Wilson, Communications in Algebra 27 (1999), 3545-3556.

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Abstract: Let $latex K$ be a field of characteristic $latex p>0$. Denote by $latex omega(R)$ the augmentation ideal of either a group algebra $latex R=K[G]$ or a restricted enveloping algebra $latex R=u(L)$ over $latex K$. We first characterize those $latex R$ for which $latex omega(R)$ satisfies a polynomial identity not satisfied by the algebra of all $latex 2times 2$ matrices over $latex K$. Then, we examine those $latex R$ for which $latex omega(R)$ satisfies a semigroup identity (that is, a polynomial identity which can be written as the difference of two monomials).

D.M. Riley and Mark C. Wilson, Proceedings of the American Mathematical Society 127 (1999), 973-976.

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Abstract: Let $latex C$ be a commutative ring, and let $latex R$ be an associative $latex C$-algebra generated by elements $latex {x_1,ldots,x_d}$. We show that if $latex R$ satisfies the Engel condition of degree $latex n$ then $latex R$ is nilpotent as a Lie algebra of class bounded by a function that depends only on $latex d$ and $latex n$. We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.

Mark C. Wilson, Journal of Pure and Applied Algebra 133 (1998), 241-260.

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Abstract: We determine all finite-dimensional simple Lie superalgebras $latex L$ such that $latex U(L)$ satisfies a primeness criterion due to Bell. Some open problems related to primeness of enveloping algebras are listed.

Geoffrey Pritchard and Mark C. Wilson, Experimental Mathematics 6 (1997), 77-85.

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Abstract: On the basis of experimental work involving matrix computations, we conjecture that a criterion due to Bell for primeness of the universal enveloping algebra of a Lie superalgebra applies to the Cartan type Lie superalgebras $latex W(n)$ for $latex n=3$ but does not apply for odd $latex ngeq 5$. The experiments lead to a rigorous proof, which we present.

Mark C. Wilson, Bulletin of the Australian Mathematical Society 56 (1997), 483-488.

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Abstract: In 1990 Allen Bell presented a sufficient condition for the primeness of the universal enveloping algebra of a Lie superalgebra. Let $latex Q$ be a nonsingular bilinear form on a finite-dimensional vector space over a field of characteristic zero. In this paper we show that Bell’s criterion applies to the Hamiltonian Cartan type superalgebras determined by $latex Q$, and hence obtain some new prime noetherian rings.

Mark C. Wilson, Communications in Algebra 25 (1997), 487-496.

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Abstract:

Let $latex K$ be a field of characteristic $latex p>0$, let $latex L$ be a restricted Lie algebra and let $latex R$ be an associative $latex K$-algebra. It is shown that the various constructions in the literature of crossed product of $latex R$ with $latex u(L)$ are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.

Mark C. Wilson, Proceedings of the American Mathematical Society 124 (1996), 383-387.

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Abstract: We show that a primeness criterion for enveloping algebras of Lie superalgebras discovered by Bell is applicable to the Cartan type Lie superalgebras $latex W(n)$, $latex n$ even. Other algebras are considered but there are no definitive answers in these cases.