Abstract
The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in a countable dimension. The constructed algebras each contains a non-isotropic idempotent e2 = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A, the group Ortaut A of orthogonal automorphisms is specified.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Almost Associative Rings [in Russian], Nauka, Moscow (1978).
A. T. Gainov, “Monocomposition algebras,” Sib. Mat. Zh., 10, No. 1, 3–30 (1969).
H. P. Petersson, “The classification of two-dimensional nonassociative algebras,” Res. Math., 37, Nos. 1/2, 120–154 (2000).
Author information
Authors and Affiliations
Additional information
__________
Translated from Algebra i Logika, Vol. 46, No. 4, pp. 428–447, July–August, 2007.
Rights and permissions
About this article
Cite this article
Gainov, A.T. Composition algebras of the second kind. Algebr Logic 46, 231–243 (2007). https://doi.org/10.1007/s10469-007-0022-2
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10469-007-0022-2