Special identities for comtrans algebras

Abstract

Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gröbner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial identities for the special commutator [x, y, z] = xyz − yxz and special translator x, y, z = xyz − yzx in associative triple systems. In degree 3, the defining identities for comtrans algebras generate all identities. In degree 5, we simplify known identities for each operation and determine new identities relating the operations. In degree 7, we use representation theory of the symmetric group to show that each operation satisfies identities which do not follow from those of lower degree but there are no new identities relating the operations. We use noncommutative Gröbner bases to construct the universal associative envelope for the special comtrans algebra of 2 × 2 matrices.2010 Mathematics Subject Classification. Primary 17A40. Secondary 15-04, 15A21, 15A69, 15B36, 18D50, 20C30, 68W30.Key words and phrases. Comtrans algebras, trilinear operations, polynomial identities, lattice basis reduction, representation theory of the symmetric group, algebraic operads, Gröbner bases, universal associative enveloping algebras.The commutator alternates in the first two arguments (1), the translator satisfies the Jacobi identity (2), and the operations are related by the comtrans identity (3). Example 1.2. If T is a Lie triple system with bracket [x, y, z] then letting both commutator and translator equal [x, y, z] gives T the structure of a comtrans algebra T CT . If T is obtained from a Lie algebra L by [x, y, z] = [[x, y], z] then Shen & Smith [32, Theorem 3.2] have shown that L is simple if and only if T CT is simple. Example 1.3. Let A m,n denote the vector space of m × n matrices over F. Fix matrices p (n × n) and q (m × m) over F. Define a trilinear operation on A m,n by (x, y, z) = xpy t qz. Setting [x, y, z] = (x, y, z) − (y, x, z) and x, y, z = (x, y, z) − (y, z, x) gives A m,n the structure of a comtrans algebra [32, Example 2.1]. Definition 1.4. An associative triple system [25] is a vector space A with a trilinear operation xyz satisfying (vw(xyz)) = (v(wxy)z) = (vw(xyz)). The special commutator and special translator in A are [x, y, z] = xyz − yxz and x, y, z = xyz − yzx.