Poisson algebras in terms of non-associative algebras (Q947526)

Let \(K\) be a field of characteristic different from 2 and 3. A Poisson algebra is a \(K\)-vector space \(P\) equipped with two bilinear operations: (1) a Lie bracket, referred to as a Poisson bracket, usually denoted by \(\{\;,\;\}\); (2) an associative commutative multiplication denoted by the authors by \(\bullet\). These two operations are required to satisfy the Leibniz condition \[ \{X\bullet Y, Z\}=X\bullet \{Y,Z\}+\{X,Z\}\bullet Y, \] for all \(X,Y,Z\in P\). The Poisson algebra is denoted by \((P, \{\;,\;\},\bullet)\). Now let \((X,Y)\to X\cdot Y\) be a bilinear map on the \(K\)-vector space \(P\). The associator \(A\) of \(\cdot\) is the trilinear map on \(P\) given by \[ A(X,Y,Z)=(X\cdot Y)\cdot Z-X\cdot (Y\cdot Z). \] Algebras with one bilinear operation are referred to as nonassociative. Having such a non-associative algebra one defines two operations \[ \{X,Y\}=\frac 12 (X\cdot Y-Y\cdot X); \] \[ X\bullet Y=\frac 12 (X\cdot Y+Y\cdot X). \] Then \((P, \{\;,\;\},\bullet)\) is a Poisson algebra if and only if the operation \(X\cdot Y\) satisfies the following identity [\textit{M. Markl} and \textit{E. Remm}, J. Algebra 299, No. 1, 171--189 (2006; Zbl 1101.18004)]: \[ 3A(X,Y,Z)=(X\cdot Z)\cdot Y+(Y\cdot Z)\cdot X-(Y\cdot X)\cdot Z-(Z\cdot X)\cdot Y. \] If the associator of a nonassociative algebra \((P,\cdot)\) satisfies the above identity then \(P\) is called an \textit{admissible Poisson algebra}. The authors prove a one-to-one correspondence between admissible Poisson algebras and ordinary Poisson algebras. Thus, the authors describe Poisson algebras in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of nonassociative algebras. The authors study their algebraic and cohomological properties, their deformations as nonassociative algebras, and settle the classification problem in low dimensions.