Comtrans algebras and bilinear forms
type: Item
instance of: scholarly article
title: Comtrans algebras and bilinear forms
zbMATH Open document ID: 0739.17010
DOI: 10.1007/BF01197047
author: Xiarong Shen
author: J. D. H. Smith
published in: Archiv der Mathematik
publication date: 1992-01-01T00:00:00Z
review text: Comtrans algebras were introduced in answer to a problem from differential geometry: finding the algebraic structure in the tangent bundle corresponding to the coordinate \(n\)-ary loop of an \((n+1)\)-web. The current paper is part of a programme beginning a study of comtrans algebras from a purely algebraic point of view. It was noted earlier that a comtrans algebra arises from the repeated commutator \([[ ,\;],\;]\) of a Lie algebra. For the Lie algebra of the Euclidean space \(\mathbb{R}^ 3\) under the ``vector'' or ``cross'' product \(\times\), this repeated product is the ``vector triple product'' \((\tilde x\times\tilde y)\times\tilde z=\tilde y(\tilde x\cdot\tilde z)-\tilde x(\tilde y\cdot\tilde z)\). One could thus regard the vector triple product comtrans algebra as arising from the Euclidean inner product on \({\mathbb{R}}^ 3\) rather than from the repeated Lie algebra commutator. The main topic of the present paper is a generalization of this construction, producing a comtrans algebra \(CT(E,\beta)\) from a pair \((E,\beta)\) consisting of a unital module \(E\) over a commutative ring \(R\) with 1 and a bilinear form \(\beta: E^ 2\to R\). A ``transposed'' comtrans algebra \(CT(E,\beta)^ r\) is also given by the pair \((E,\beta)\). These constructions compare with the currently popular methods of making algebras out of spaces with forms, such as Jordan algebras or Clifford algebras. One major advantage of the comtrans algebras \(CT(E,\beta)\) and \(CT(E,\beta)^ r\) is that they do not require any extension of the underlying module \(E\) in order to achieve closure. By contrast, the underlying modules of Clifford algebras (for example) blow up exponentially in size. For tight connections between the form and the algebras, some restrictions on the underlying module \(E\) are required. Appropriate restrictions are encoded in the concept of ``formed space'', making the underlying module free of rank more than 1. Theorem 3.4 shows how simplicity of the comtran…
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 17B99
Mathematics Subject Classification ID: 53A60
zbMATH DE Number: 4216216
zbMATH Keywords: comtrans algebras
zbMATH Keywords: tangent bundle
zbMATH Keywords: web
zbMATH Keywords: vector triple product
zbMATH Keywords: Euclidean inner product
zbMATH Keywords: Lie algebra commutator
zbMATH Keywords: bilinear form
zbMATH Keywords: automorphism groups
zbMATH Keywords: formed space
zbMATH Keywords: form algebra
zbMATH Keywords: hyperbolic plane
MaRDI profile type: Publication
cites work: Q4039680
cites work: Q3994471
cites work: Lie's fundamental theorems for local analytical loops
cites work: Endliche Gruppen I
cites work: Simple multilinear algebras, rectangular matrices and Lie algebras
cites work: Representation theory of comtrans algebras
cites work: Multilinear algebras and Lie's theorem for formal \(n\)-loops
cites work: Q3739181
description: scientific article; zbMATH DE number 4216216