Comtrans algebras and bilinear forms
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 comtrans algebras is equivalent to non-degeneracy of the form and simplicity of the ring of scalars. In general, the radical of a bilinear form on a formed space may be described in comtrans algebra terms (Proposition 3.5). Theorem 3.6 and its corollary show that the automorphism groups of the formed space \((E,\beta)\) and of the comtrans algebras \(CT(E,\beta)\) and \(CT(E,\beta)^ \tau\) coincide. The final section is concerned with the problem of recognizing when a comtrans algebra is a `` form algebra'', i.e. \(CT(E,\beta)\) or \(CT(E,\beta)^ \tau\) for a formed space \((E,\beta)\). The answer is given by Theorem 4.1. Consideration of the hyperbolic plane (Example 4.2) shows that the two- dimensional case is anomalous.
scientific article; zbMATH DE number 4216216