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| In [[abstract algebra]], a '''Valya algebra''' (or '''Valentina algebra''') is a [[Algebra over a field#Non-associative algebras|nonassociative algebra]] ''M'' over a field ''F'' whose [[product (mathematics)|multiplicative binary operation]] ''g'' satisfies the following axioms:
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| 1. The [[Antisymmetric|skew-symmetry]] condition
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| :<math>g (A, B) =-g (B, A) </math>
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| for all <math>A,B \in M</math>.
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| 2. The Valya identity
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| :<math> J (g (A_1, A_2), g (A_3, A_4), g (A_5, A_6)) =0 </math>
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| for all <math>A_k \in M</math>, where k=1,2,...,6, and
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| <math> J (A, B, C):= g (g (A, B), C)+g (g (B, C), A)+g (g (C, A), B). </math>
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| 3. The bilinear condition
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| :<math> g(aA+bB,C)=ag(A,C)+bg(B,C) </math>
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| for all <math>A,B,C \in M</math> and <math>a,b \in F</math>.
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| We say that M is a Valya algebra if the [[commutant]] of this algebra is a Lie subalgebra. Each [[Lie algebra]] is a Valya algebra.
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| There is the following relationship between the [[commutant-associative algebra]] and Valentina algebra. The replacement of the multiplication g(A,B) in an algebra M by the operation of commutation [A,B]=g(A,B)-g(B,A), makes it into the algebra <math>M^{(-)}</math>.
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| If M is a [[commutant-associative algebra]], then <math>M^{(-)}</math> is a Valya algebra. A Valya algebra is a generalization of a [[Lie algebra]].
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| ==Examples==
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| Let us give the following examples regarding Valya algebras.
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| (1) Every finite Valya algebra is the [[Tangent space|tangent algebra]] of an analytic local commutant-associative [[Quasigroup|loop]] (Valya loop) as each finite [[Lie algebra]] is the tangent algebra of an analytic local group ([[Lie group]]). This is the analog of the classical correspondence between analytic local groups ([[Lie groups]]) and [[Lie algebra]]s.
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| (2) A bilinear operation for the [[differential form|differential 1-forms]]
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| :<math> \alpha=F_k(x)\, dx^k , \quad \beta=G_k(x)\, dx^k </math>
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| on a symplectic manifold can be introduced by the rule
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| : <math> (\alpha,\beta)_0=d \Psi(\alpha,\beta)+ \Psi(d\alpha,\beta)+\Psi(\alpha,d\beta), \, </math>
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| where <math>(\alpha,\beta)</math> is 1-form. A set of all nonclosed 1-forms, together with this operation, is Lie algebra.
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| If <math>\alpha</math> and <math>\beta</math> are closed 1-forms, then
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| <math>d\alpha=d\beta=0</math> and
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| : <math> (\alpha,\beta)=d \Psi(\alpha,\beta). \,</math>
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| A set of all closed 1-forms, together with this bracket, form a [[Lie algebra]]. A set of all nonclosed 1-forms together with the bilinear operation <math>(\alpha,\beta)</math> is a Valya algebra, and it is not a [[Lie algebra]].
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| ==See also==
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| * [[Malcev algebra]]
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| * [[Alternative algebra]]
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| * [[Commutant-associative algebra]]
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| ==References==
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| * A. Elduque, H. C. Myung ''Mutations of alternative algebras'', Kluwer Academic Publishers, Boston, 1994, ISBN 0-7923-2735-7
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| * {{springer|id=M/m062170|author=V.T. Filippov|title=Mal'tsev algebra}}
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| * M.V. Karasev, V.P. Maslov, ''Nonlinear Poisson Brackets: Geometry and Quantization''. American Mathematical Society, Providence, 1993.
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| * [[Aleksandr Gennadievich Kurosh|A.G. Kurosh]], ''Lectures on general algebra.'' Translated from the Russian edition (Moscow, 1960) by K. A. Hirsch. Chelsea, New York, 1963. 335 pp. ISBN 0-8284-0168-3 ISBN 978-0-8284-0168-5
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| * [[Aleksandr Gennadievich Kurosh|A.G. Kurosh]], ''General algebra. Lectures for the academic year 1969/70''. Nauka, Moscow,1974. (In Russian)
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| * [[Anatoly Maltsev|A.I. Mal'tsev]], ''Algebraic systems.'' Springer, 1973. (Translated from Russian)
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| * [[Anatoly Maltsev|A.I. Mal'tsev]], ''Analytic loops.'' Mat. Sb., 36 : 3 (1955) pp. 569–576 (In Russian)
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| * {{cite book | first = R.D. | last = Schafer | title = An Introduction to Nonassociative Algebras | publisher = Dover Publications | location = New York | year = 1995 | isbn = 0-486-68813-5}}
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| * V.E. Tarasov [http://books.google.ru/books?id=pHK11tfdE3QC&dq=V.E.+Tarasov+Quantum+Mechanics+of+Non-Hamiltonian+and+Dissipative+Systems.&printsec=frontcover&source=bl&ots=qDERzjAJd9&sig=U8V7RUVd1SW8mx4GzE1T-2canhA&hl=ru&ei=pkvkSeycINiEsAbloKSfCw&sa=X&oi=book_result&ct=result&resnum=1 ''Quantum Mechanics of Non-Hamiltonian and Dissipative Systems.'' Elsevier Science, Amsterdam, Boston, London, New York, 2008.] ISBN 0-444-53091-6 ISBN 9780444530912
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| * [http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=962&option_lang=eng V.E. Tarasov, "Quantum dissipative systems: IV. Analogues of Lie algebras and groups" Theoretical and Mathematical Physics. Vol.110. No.2. (1997) pp.168-178.]
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| *{{eom|id=A/a012090|first=K.A.|last= Zhevlakov|title=Alternative rings and algebras}}
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| [[Category:Non-associative algebras]]
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| [[Category:Lie algebras]]
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