Separability of Nonassociative Algebras with Metagroup Relations
Department of Applied Mathematics, MIREA—Russian Technological University, av. Vernadsky 78, 119454 Moscow, Russia
Axioms 2019, 8(4), 139; https://doi.org/10.3390/axioms8040139
Received: 15 November 2019 / Revised: 6 December 2019 / Accepted: 10 December 2019 / Published: 12 December 2019
(This article belongs to the Special Issue Non-associative Structures and Other Related Structures)
This article is devoted to a class of nonassociative algebras with metagroup relations. This class includes, in particular, generalized Cayley–Dickson algebras. The separability of the nonassociative algebras with metagroup relations is investigated. For this purpose the cohomology theory is utilized. Conditions are found under which such algebras are separable. Algebras satisfying these conditions are described.
Associative separable algebras play an important role and have found many-sided application (see, for example, [1,2,3,4,5,6,7,8,9] and references therein). Studies of their structure are based on cohomology theory. On the other hand, cohomology theory of associative algebras was investigated by Hochschild and other authors [10,11,12,13], but it is not applicable to nonassociative algebras. Cohomology theory of group algebras is an important and great part of algebraic topology.
It is worth mentioning that nonassociative algebras with some identities in them found many-sided applications in physics, noncommutative geometry, quantum field theory, partial differential equations (PDEs) and other sciences (see [14,15,16,17,18,19,20,21,22,23,24,25] and references therein).
An extensive area of investigations of PDEs intersects with cohomologies and deformed cohomologies . Therefore, it is important to develop this area over octonions, generalized Cayley–Dickson algebras and more general metagroup algebras (see also Appendix A). Some results in this area are presented in . The structure of metagroups, their construction and examples, and smashed and twisted wreath products were studied and described in [26,27,28]. In particular, a class of metagroup algebras contains a family of generalized Cayley–Dickson algebras and nonassociative analogs of algebras.
For comparison it is worth noting that there are algebras with relations T induced by Jordan-type or Lie-type homomorphisms in the sense of . Their unified approach (UJLA) was studied in . In those works the unital universal envelope of a nonassociative algebra A with relations T was considered, where R denotes an associative commutative unital ring. The algebra is associative and may be noncommutative. This theory is applicable to Lie algebras, alternative algebras, Jordan algebras and UJLA fitting to algebras with relations T.
However, this technique is not applicable to the metagroup algebras studied in this article. Indeed, there are several obstacles. The algebra is associative and with it a lot of information about the metagroup algebras is lost. A derivation functor cannot serve as a starting point for a construction of a cohomology theory for the metagroup algebras. Moreover relations in metagroup algebras are external to them and do not fit to the nonassociative algebras with relations T considered in [22,29].
This article is devoted to a separability of nonassociative algebras with metagroup relations. Conditions are found under which they are separable. Algebras satisfying these conditions are described in Theorems 1–3.
All main results of this paper are obtained for the first time.
2. Separable Nonassociative Algebras
Nonassociative metagroups, their centers, metagroup algebras and modules over them were defined in [26,27,28] (see also Appendix A). To avoid misunderstandings we also give specific necessary definitions and notations.
Let Ψ be a (proper or improper) subgroup in the center of a metagroup G, let 1 denote a unit in , be a unit in G and let
A be a nonassociative metagroup algebra over a commutative associative unital ring such thatwhere , denotes a metagroup algebra.
A G-graded A-module P (also see Definition 3 in ) is called projective if it is isomorphic with a direct summand of a free G-graded A-module. The metagroup algebra A is called separable if it is a projective G-graded -module.
One puts for each , where A is considered as the G-graded right -module.
Suppose that A is a nonassociative algebra satisfying condition (1). Then the following conditions are equivalent:the exact sequencewhere is considered as the G-graded two-sided A-module.
The implication is evident.
. If the exact sequence splits, then as the -module is isomorphic with , where ⊕ denotes a direct sum. Therefore, A is separable. The sequence splits if and only if there exists such that . With this homomorphism p put . Then , hence and . Thus is valid.
. Suppose that condition is fulfilled. Then a mapping exists such that . The element b has the decomposition with , where and and for each j. Therefore, using condition above and conditions (1)–(3) in Definition 3 in  we infer thatandfor each x and , where and with and in G, and in for each k and l. Thus . Moreover, for each , consequently, the exact sequence splits. □
An element fulfilling condition in Proposition 1 is called a separating idempotent of an algebra A.
Let A be a nonassociative algebra satisfying condition . Let also M be a two-sided A-module.
. Since , then . By virtue of Theorem 1 in  is the derivation having also properties and . □
Suppose that A is a nontrivial nonassociative algebra satisfying condition . Then for each two-sided A-module M if and only if A is a separable -algebra.
In view of Proposition 1 the algebra A is separable if and only if the exact sequence splits. That is, a homomorphism h exists such that its restriction is the identity mapping. Therefore, if , then the algebra A is separable due to Lemma 1.
Vice versa if a homomorphism exists with , then each has the form with . By virtue of Lemma 1 for each two-sided A-module M. □
Let a noncommutative algebra A fulfill condition andwhere denotes the radical of A.
Then a subalgebra D in A exists such that as -modules and is isomorphic with D as the algebra.
For we get .
For a natural projection exists, where , since . The algebra A is G-graded and , hence , where e is the unit element of G. In view of conditions (1)–(3) in Definition 3 in  J is the two-sided ideal in A and for each positive integer m, where , , and . Condition in Definition 1 in  and conditions (1)–(3) in Definition 3 in  imply that is also G-graded, since .
By condition the -module is projective, consequently, an exact splitting sequence of -modules existsThus a homomorphism of -modules exists such that on . For any two elements x and y in we putTherefore, we infer thatsince is the algebra homomorphism and . Thus . One has by the definition that
Then put and to be the right and left actions of on J. Since is the homomorphism of -modules and , then for each pure states x, y and u we infer:where . Then we deduce thatwhere . Thus J has the structure of the two-sided -module.
Evidently, is -bilinear. Then for every pure states x, y and z in :consequently, , where . Thus by the -linearity a homomorphism h in exists possessing the propertyfor each x and y in .
Let now , consequently, , since . This implies that for each x and y in , since and . Since , then . Therefore, p is the algebra homomorphism. This implies that is the subalgebra in A such that .
Let now and this theorem is proven for . Put , then is the two-sided ideal in and is isomorphic with , also . Thus and satisfies conditions (8)–(10) of this theorem and is G-graded, since A and J are G-graded and due to condition in Definition 1 in  and conditions (1)–(3) in Definition 3 in .
From the proof for we get that a subalgebra in exists such that . Consider a subalgebra E in D such that and . Then is isomorphic with . Moreover, , hence . Thus the algebra E fulfills conditions (8)–(10) of this theorem and is G-graded and .
By the induction supposition a subalgebra F in E exists such that ; consequently, and . Thus . □
Suppose that conditions of Theorem 2 are satisfied and condition takes the form . Then for any two G-graded subalgebras B and C in A such that and an element exists for which such that has a right inverse and a left inverse.
Let and be the canonical projections induced by the decompositions and , where . Then and , where is the quotient homomorphism, and are homomorphisms as in the proof of Theorem 2, since q and r are homomorphisms of algebras. We put for each , . Then we deduce thatsince . Therefore, , hence . Then we infer thatconsequently, w is the derivation of the algebra with values in the two-sided A-module (see also the proof of Theorem 2). Since , then w is the inner derivation by Theorem 1 in . Thus an element exists for which for each . This implies that for each . The element has a right inverse and a left inverse, since implies and , where , , and for each positive integer m. Therefore,□
Definition 1 is natural. For example, if is a commutative associative unital ring and S is a subgroup in , then is a commutative associative unital ring such that .
The results of this article can be used for further studies of nonassociative algebras, their structure, cohomologies, algebraic geometry, PDEs, their applications in the sciences, etc. They also can serve for investigations of extensions of nonassociative algebras, decompositions of algebras and modules, and their morphisms. In particular, they can be applied to cohomologies of PDEs and solutions of PDEs with boundary conditions which can have a practical importance [13,30].
Other applications are in mathematical coding theory, information flows analysis and their technological implementations [31,32,33,34]. Indeed, frequently codes are based on binary systems and algebras. On the other hand, metagroup relations are weaker than relations in groups. This means that a code complexity can increase by using nonassociative algebras with metagroup relations in comparison with group algebras or Lie algebras.
Besides applications of cohomologies outlined in the introduction they also can be used in mathematical physics and quantum field theory .
This research received no external funding.
Conflicts of Interest
The author declares no conflict of interest.
Appendix A. Metagroups
Let G be a set with a single-valued binary operation (multiplication) defined on G satisfying the conditions:which are denoted by
there exists a neutral (i.e., unit) element :
The set of all elements commuting and associating with G:is called the center of G.
We call G a metagroup if a set G possesses a single-valued binary operation and satisfies conditions (A1)–(A3) andfor each a, b and c in G, where , ; where shortens a notation , where denotes a (proper or improper) subgroup of .
Then G will be called a central metagroup if in addition to (A10) it satisfies the condition:for each a and b in G, where .
From conditions (1)–(3) in Definition 3 in  it follows that for each a and b in the metagroup algebra and x in a (smashly G-graded) two-sided A-module M there may exist a -homomorphism of right -modules and such that for chosen a, b and x. Similar homomorphisms and may exist on and , respectively. Generally these homomorphisms , and depend nontrivially on all variables a, b and x (see also Remark 1 in ). So they cannot be realized by identities of Jordan-type or Lie-type or UJLA-type (see also the introduction).
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