Abstract
A class of nonassociative algebras is investigated with mild relations induced from metagroup structures. Modules over nonassociative algebras are studied. For a class of modules over nonassociative algebras, their extensions and splitting extensions are scrutinized. For this purpose tensor products of modules and induced modules over nonassociative algebras are investigated. Moreover, a developed cohomology theory on them is used.
Highlights
Extensions and splitting extensions of associative algebras play a very important role and have found many-sided applications
A class of nonassociative algebras is investigated with mild relations induced from metagroup structures
Group algebras appearing over C in conjunction with Cayley–Dickson algebras lead to extensions that are generalized Cayley–Dickson algebras or even more general metagroup algebras
Summary
Extensions and splitting extensions of associative algebras play a very important role and have found many-sided applications (see, for example, refs. [1–5] and references therein). This article is devoted to extensions and splitting extensions of nonassociative algebras For this purpose cohomology theory on them was developed in [12]. Generic morphisms are scrutinized in Theorem 6 Isomorphisms of such nonassociative algebras are studied in Proposition 4 and Corollary 6. The exactness of isomorphisms related with extensions and metagroup algebras is studied in Lemma 6 and Propositions 6 and 7 (see Definitions 1 and 6). All main results of this paper are obtained for the first time They can be used for further studies of nonassociative algebras, their cohomologies, algebraic geometry, PDEs, their applications in the sciences, etc
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