David M. Riley

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σg∈GRg be any associative algebra over a field such that the subspacesRg satisfyRgRh⊆Rgh. We prove that ifR1 satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withRH satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.

Associative algebras satisfying a semigroup identity

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

Group algebras and enveloping algebras with nonmatrix and semigroup identities

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

Power closure and the Engel condition

A Liep-algebraL is calledn-power closed if, in every section ofL, any sum ofpi+nth powers is apith power (i>0). It is easy to see that ifL ispn-Engel then it isn-power closed. We establish a partial converse to this statement: ifL is residually nilpotent andn-power closed for somen≥0 thenL is (3pn+2+1)-Engel ifp>2 and (3 · 2n+3+1)-Engel ifp=2. In particular, thenL is locally nilpotent by a theorem of Zel’manov. We deduce that a finitely generated pro-p group is a Lie group over thep-adic field if and only if its associated Liep-algebra isn-power closed for somen. We also deduce that any associative algebraR generated by nilpotent elements satisfies an identity of the form (x+y)pn=xpn+ypn for somen≥1 if and only ifR satisfies the Engel condition.

ON THE GRASSMANN T-SPACE

We study the Grassmann T-space, S3, generated by the commutator [x1,x2,x3] in the free unital associative algebra K 〈x1,x2,… 〉 over a field of characteristic zero. We prove that S3 = S2 ∩ T3, where S2 is the commutator T-space generated by [x1,x2] and T3 is the Grassmann T-ideal generated by S3. We also construct an explicit basis for each vector space S3 ∩ Pn, where Pn represents the space of all multilinear polynomials of degree n in x1,…,xn, and deduce the recursive vector space decomposition T3 ∩ Pn = (S3 ∩ Pn) ⊕ (T3 ∩ Pn-1)xn.

Algebras with Collapsing Monomials

A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S , at least two distinct words of length n on these letters are equal in S . In particular, S is collapsing whenever it satisfies a law. Let [Uscr ]( A ) denote the group of units of a unitary associative algebra A over a field k of characteristic zero. If A is generated by its nilpotent elements, then the following conditions are equivalent: [Uscr ]( A ) is collapsing; [Uscr ]( A ) satisfies some semigroup law; [Uscr ]( A ) satisfies the Engel condition; [Uscr ]( A ) is nilpotent; A is nilpotent when considered as a Lie algebra.

Lie identities for Hopf algebras

Let R denote either a group algebra over a field of characteristic p > 3 or the restricted enveloping algebra of a restricted Lie algebra over a field of characteristic p > 2. Viewing R as a Lie Algebra in the natural way, our main result states that R satisfies a law of the form [[x1, x2, …, xn], [xn + 1, xn + 2, …, xn + m], xn + m + 1] = 0 if and only if R is Lie nilpotent. It is deduced that R is commutative provided p > 2 max m, n. Group algebras over fields of characteristic p = 3 are shown to be Lie nilpotent if they satisfy an identity of the form [[x1,x2,…,xn], [xn + 1, xn + 2, …, xn + m]] = 0 . It was previously known that Lie centre-by-metabelian group algebras are commutative provided p > 3, and that a Lie soluble group algebra of derived length n is commutative if its characteristic exceeds 2n.

ALGEBRAS WITH COLLAPSING MONOMIALS. II

A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S at least two distinct words of length n on these letters are equal in S. Let U(A) denote the group of units of an associative algebra A over an infinite field of characteristic p > 0. We show that if A is unitally generated by its nilpotent elements then the following conditions are equivalent: U(A) is collapsing; U(A) satisfies some semigroup identity; U(A) satisfies an Engel identity; A satisfies an Engel identity when viewed as a Lie algebra; and, A satisfies a Morse identity. The characteristic zero analogue of this result was proved by the author in a previous paper.

Infinitesimally PI radical algebras

The Golod-Shafarevich examples show that not every finitely generated nil algebraA is nilpotent. On the other hand, Kaplansky proved that every finitely generated nil PI-algebra is indeed nilpotent. We generalise Kaplansky’s result to include those algebras that are only infinitesimally PI. An associative algebraA is infinitesimally PI whenever the Lie subalgebra generated by the first homogeneous component of its graded algebra gr(A)=⊕t⩾1Ai/Ai+1 is a PI-algebra. We apply our results to a problem of Kaplansky’s concerning modular group algebras with radical augmentation ideal.

Group algebras with units satisfying an Engel identity

We classify group algebras of periodic groups over a field of positive characteristic with units satisfying an Engel identity.