Novikov–Poisson Algebras and Superalgebras of Jordan Brackets

Abstract

We prove that for a generalized Novikov–Poisson algebra ⟨ A, ·, ○ ⟩ the bracket

is a Jordan bracket, and therefore, the Kantor superalgebra J(A, {,}) is a Jordan superalgebra. Also we prove that ⟨ A, ○ ⟩ is nontrivial simple Novikov algebra and A = AA if and only if J(A, {,}) is a simple Jordan superalgebra.

Key Words:

• Differentially simple algebra
• Jordan bracket
• Jordan superalgebra
• Novikov algebra
• Novikov–Poisson algebras
• Superalgebra of vector type

ACKNOWLEDGMENTS

The author is thankful to professors Viktor N. Zhelyabin and Maksim E. Goncharov for useful discussions. The author is also grateful to the referee for the comments that help to improve the article.

The author is Supported by RFBR 11-01-00938, Federal Program “Scientific and Pedagogical Staff of Innovative Russia” 2009–2013 (State Contract No. 14.740.11.0346.

Notes

Communicated by I. Shestakov.