Research Areas of Murray R. Bremner

My primary research interests focus on connections between associative and nonassociative structures. In particular, I am interested in obtaining nonassociative structures from associative structures by means of multilinear operations. In the familiar binary case, one obtains Lie algebras (respectively Jordan algebras) from the polynomial identities of low degree satisfied by the Lie bracket [x,y] = xy - yx (respectively the Jordan product x o y = xy + yx) in an associative algebra.

One generalization of this construction is to replace the associative algebra with an alternative algebra; from the identities satisfied by [x,y] = xy - yx we obtain Malcev algebras. The next step is to replace the alternative algebra with a right alternative algebra; from the identities satisfied by [x,y] = xy - yx and [x,y,z] = (x o y) o z - x o (y o z) we obtain Bol algebras - which have two operations, one binary and one ternary. Continuing in this direction, we replace the right alternative algebra with an arbitrary nonassociative algebra; we obtain an infinite family of multilinear operations which satisfy an infinite family of polynomial identities - these structures are called Sabinin algebras.

Another generalization is to replace the binary associative algebra by an n-ary generalization. In the simplest case, one starts with a totally associative triple system, and introduces a new trilinear operation. In this way one obtains the familiar varieties of Lie, Jordan and anti-Jordan triple systems, together with a number of new varieties. Beyond this, one considers quadrilinear operations. This leads to the topic of n-ary generalizations of Lie and Jordan algebras.

A third generalization is to replace the associative algebra by an associative dialgebra which has two associative operations < and > related by the identities x < (y < z) = x < (y > z), (x < y) > z = (x > y) > z and (x > y) < z = x > (y < z). In this case the operation x < y - y > x gives rise to Leibniz algebras, and the operation x < y + y > x gives rise to quasi-Jordan algebras. A general theory has been developed for obtaining varieties of dialgebras corresponding to varieties of algebras; from this point of view, Leibniz algebras are Lie dialgebras, and quasi-Jordan algebras are Jordan dialgebras.

One general question that can be asked in connection with all these structures is the problem of the existence of special identities, or s-identities for short. An s-identity is a polynomial identity that is satisfied by all those nonassociative structures in a given variety that can be embedded into the corresponding associative structure but which is not satisfied by all structures in the variety. In the case of Lie algebras, the PBW theorem implies that there are no s-identities. In the case of Jordan algebras, it is known that s-identities exist but that the simplest example has degree 8. The question of the existence of s-identities is closely related to the problem of constructing universal enveloping algebras for nonassociative systems; that is, generalizing the Poincare-Birkhoff-Witt theorem to other nonassociative structures closely related to Lie algebras.

Most of my research involves the use of computer algebra systems, especially Maple. In particular, I use algorithms for linear algebra on large matrices over various domains (such as the rational numbers, finite fields, and the ring of integers, as well as rings of polynomials in one or more variables). I am particularly interested in using lattice basis reduction to find polynomial identities with few terms and small coefficients. The representation theory of the symmetric group is also very useful for breaking a large computational problem on multilinear identities into much smaller pieces. Many of the nonassociative structures that I am interested in are closely related to representations of Lie algebras.

For a detailed summary of my current research program, see my complete NSERC grant application for 2006-2011.