Valya algebra

In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra M over a field F whose multiplicative binary operation g satisfies the following axioms: 1. The skew-symmetry condition

[math]\displaystyle{ g (A, B) =-g (B, A) }[/math]

for all [math]\displaystyle{ A,B \in M }[/math].

2. The Valya identity

[math]\displaystyle{ J (g (A_1, A_2), g (A_3, A_4), g (A_5, A_6)) =0 }[/math]

for all [math]\displaystyle{ A_k \in M }[/math], where k=1,2,...,6, and

[math]\displaystyle{ 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]\displaystyle{ g(aA+bB,C)=ag(A,C)+bg(B,C) }[/math]

for all [math]\displaystyle{ A,B,C \in M }[/math] and [math]\displaystyle{ 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]\displaystyle{ M^{(-)} }[/math]. If M is a commutant-associative algebra, then [math]\displaystyle{ 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 algebra of an analytic local commutant-associative 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 algebras.

(2) A bilinear operation for the differential 1-forms

[math]\displaystyle{ \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]\displaystyle{ (\alpha,\beta)_0=d \Psi(\alpha,\beta)+ \Psi(d\alpha,\beta)+\Psi(\alpha,d\beta), \, }[/math]

where [math]\displaystyle{ (\alpha,\beta) }[/math] is 1-form. A set of all nonclosed 1-forms, together with this operation, is Lie algebra.

If [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are closed 1-forms, then [math]\displaystyle{ d\alpha=d\beta=0 }[/math] and

[math]\displaystyle{ (\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]\displaystyle{ (\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
• Hazewinkel, Michiel, ed. (2001), "Mal'tsev algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=M/m062170 
• M.V. Karasev, V.P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization. American Mathematical Society, Providence, 1993.
• 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
• A.G. Kurosh, General algebra. Lectures for the academic year 1969/70. Nauka, Moscow,1974. (In Russian)
• A.I. Mal'tsev, Algebraic systems. Springer, 1973. (Translated from Russian)
• A.I. Mal'tsev, Analytic loops. Mat. Sb., 36 : 3 (1955) pp. 569–576 (In Russian)
• Schafer, R.D. (1995). An Introduction to Nonassociative Algebras. New York: Dover Publications. ISBN 0-486-68813-5. https://archive.org/details/introductiontono0000scha. 
• V.E. Tarasov Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. Elsevier Science, Amsterdam, Boston, London, New York, 2008. ISBN:0-444-53091-6 ISBN:9780444530912
• 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.
• Hazewinkel, Michiel, ed. (2001), "Alternative rings and algebras", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=A/a012090