On finite-dimensional simple Novikov algebras of characteristic \(p \) (Q6610191)

Novikov algebras appeared in the work by \textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman} in studying Hamiltonian operators [Funct. Anal. Appl. 13, 248--262 (1980; Zbl 0437.58009)]. Also these algebras played a key role in classifying the linear Poisson brackets of hydrodynamic type by \textit{A. A. Balinskiĭ} and \textit{S. P. Novikov} [Sov. Math., Dokl. 32, 228--231 (1985; Zbl 0606.58018); translation from Dokl. Akad. Nauk SSSR 283, 1036--1039 (1985)].\N\NIn 1987, Zelmanov proved that each finite-dimensional simple Novikov algebra over an algebraically closed field of characteristic zero is a field. A complete classification of finite-dimensional simple Novikov algebras over an algebraically closed fields of characteristic \(p > 2\) is given in [\textit{X. Xu}, J. Algebra 246, No. 2, 673--707 (2001; Zbl 1003.17003)].\N\NThe main construction of Novikov algebras (the Gelfand-Dorfman construction) arises in [\textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman}, Funct. Anal. Appl. 13, 248--262 (1980; Zbl 0437.58009)]. Let \((A, \cdot )\) be an associative commutative algebra with a derivation \(d\). Take \(\alpha \in A\). We define the new operation of multiplication \(\circ\) on the vector space \(A\) by \begin{center} \(a \circ b = a \cdot d(b) + \alpha \cdot a \cdot b\) for all \(a, b \in A.\) \end{center}\N\NIn this article the authors proved that a finite-dimensional simple nonassociative Novikov algebra over an algebraically closed field \(F\) of characteristic \(p > 0\) is the Gelfand-Dorfman construction of a truncated polynomial algebra in \(k\) variables.