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en>Bibcode Bot: Adding 1 arxiv eprint(s), 4 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2014-07-01T01:46:15Z

Adding 1 arxiv eprint(s), 4 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

← Older revision		Revision as of 03:46, 1 July 2014
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en>Wavelength: inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

2014-02-25T03:00:12Z

inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

Show changes

en>Anrnusna: journal name, replaced: Phys. Stat. Sol. → Physica Status Solidi using AWB

2013-10-08T16:05:08Z

journal name, replaced: Phys. Stat. Sol. → Physica Status Solidi using AWB

← Older revision		Revision as of 18:05, 8 October 2013
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	I'~~m Sallie~~ (27) ~~from Neerwaasten~~, ~~Belgium~~. <br>~~I'm learning Turkish literature at~~ a local ~~high school~~ and ~~I'm just about to graduate~~.<br>~~I have~~ a ~~part time job in~~ a ~~backery~~.<br><br>~~Also visit my web site~~ ... [http://www.~~youtube~~.~~com~~/~~watch~~?v=~~4ncVZsbjmNE Mediacom~~]		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:

			1. The [[Antisymmetric\|skew-symmetry]] condition
			:<math>g (A, B) =-g (B, A) </math>
			for all <math>A,B \in M</math>.

			2. The Valya identity
			:<math> J (g (A_1, A_2), g (A_3, A_4), g (A_5, A_6)) =0 </math>
			for all <math>A_k \in M</math>, where k=1,2,...,6, and

			<math> J (A, B, C):= g (g (A, B), C)+g (g (B, C), A)+g (g (C, A), B). </math>

			3. The bilinear condition
			:<math> g(aA+bB,C)=ag(A,C)+bg(B,C) </math>
			for all <math>A,B,C \in M</math> and <math>a,b \in F</math>.

			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.

			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>.
			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]].

			==Examples==

			Let us give the following examples regarding Valya algebras.

			(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.

			(2) A bilinear operation for the [[differential form\|differential 1-forms]]
			:<math> \alpha=F_k(x)\, dx^k , \quad \beta=G_k(x)\, dx^k </math>
			on a symplectic manifold can be introduced by the rule
			: <math> (\alpha,\beta)_0=d \Psi(\alpha,\beta)+ \Psi(d\alpha,\beta)+\Psi(\alpha,d\beta), \, </math>
			where <math>(\alpha,\beta)</math> is 1-form. A set of all nonclosed 1-forms, together with this operation, is Lie algebra.

			If <math>\alpha</math> and <math>\beta</math> are closed 1-forms, then
			<math>d\alpha=d\beta=0</math> and
			: <math> (\alpha,\beta)=d \Psi(\alpha,\beta). \,</math>
			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]].

			==See also==
			* [[Malcev algebra]]
			* [[Alternative algebra]]
			* [[Commutant-associative algebra]]

			==References==
			* A. Elduque, H. C. Myung ''Mutations of alternative algebras'', Kluwer Academic Publishers, Boston, 1994, ISBN 0-7923-2735-7
			* {{springer\|id=M/m062170\|author=V.T. Filippov\|title=Mal'tsev algebra}}
			* M.V. Karasev, V.P. Maslov, ''Nonlinear Poisson Brackets: Geometry and Quantization''. American Mathematical Society, Providence, 1993.
			* [[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
			* [[Aleksandr Gennadievich Kurosh\|A.G. Kurosh]], ''General algebra. Lectures for the academic year 1969/70''. Nauka, Moscow,1974. (In Russian)
			* [[Anatoly Maltsev\|A.I. Mal'tsev]], ''Algebraic systems.'' Springer, 1973. (Translated from Russian)
			* [[Anatoly Maltsev\|A.I. Mal'tsev]], ''Analytic loops.'' Mat. Sb., 36 : 3 (1955) pp. 569–576 (In Russian)
			* {{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}}
			* 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
			* [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.]
			*{{eom\|id=A/a012090\|first=K.A.\|last= Zhevlakov\|title=Alternative rings and algebras}}

			[[Category:Non-associative algebras]]
			[[Category:Lie algebras]]

en>Yobot: WP:CHECKWIKI error fixes +genfixes using AWB (7105)

2010-09-09T15:58:28Z

WP:CHECKWIKI error fixes +genfixes using AWB (7105)

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