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On simple Novikov algebras and their irreducible modules. (English) Zbl 0863.17003
An algebra $A$ is called Novikov if $(x,y,z) =(y,x,z)$ and $(xy)z= (xz)y$ for all $x,y,z \in A$. The author gives a complete classification of finite-dimensional simple Novikov algebras and their irreducible modules over an algebraically closed field $F$ with prime characteristic $p>2$. (In particular, if $A$ is a finite-dimensional simple Novikov algebra over $F$, then for some positive integer $n$ there is a basis $\{y_{-1}, y_0, \dots, y_{p^n-2}\}$ for $A$ such that $$y_iy_j= {i+j+1 \choose j} y_{i+j} +\delta_{i,-1} \delta_{j,-1} ay_{p^n-2} +\delta_{i,-1} \delta_{j,0} by_{p^n -2},$$ where $a$, $b\in F$ are constants. (Here $y_k=0$ if $k\notin \{-1,0, \dots, p^n-2\}.))$ The author also introduces what he calls “Novikov-Poisson algebras” and their tensor theory. All of this builds on the results in a number of articles by J. M. Osborn.

17A30Nonassociative algebras satisfying other identities
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