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Simple Novikov algebras with an idempotent. (English) Zbl 0772.17001

An algebra is called Novikov if it satisfies the identities (x,y,z)=(y,x,z) and (xy)z=(xz)y. Simple finite-dimensional Novikov algebras, over a field of characteristic 0, were shown to be 1- dimensional in [E. I. Zel’manov, Dokl. Akad. Nauk SSSR 292, 1294- 1297 (1987; Zbl 0629.17002)]. In this paper simple finite-dimensional Novikov algebras with idempotent are classified as follows: Let A be a simple finite-dimensional Novikov algebra, over a perfect field F of characteristic p>2, that contains an idempotent e.

(a) Then for some integer n and a, cF, A has a basis {x i }, where -1is=p n -2, with the products of the basis elements given by x i x j =i+j+1 jx i+j +δ i,-1 δ j,-1 ax s +δ i,-1 δ j,0 cx s .

(b) If e is contained in the maximal subalgebra S of A spanned by the x i ’s with nonnegative subscripts, and it acts as the right identity on S, then there is a basis for A which consists of generalized eigenvectors with respect to L e , and which multiplies as in (a). In this basis, x 0 =e.

(c) eS, then for some finite additive subgroup Δ of F and integer r, there is a basis {x αk } consisting of generalized eigenvectors for A, where αΔ, 0kp r -1, and x 00 =e. For these basis elements the products are given

x αk x β =(β+1)k+ kx α+β,k+ +k+-1 kx α+β,k+-1 ·

In addition, the algebras described in (a) and (c) are simple Novikov algebras. In particular, r=0 in (c) gives the simple algebras in [V. T. Filippov, Mat. Zametki 45, 101-105 (1989; Zbl 0659.17003)].


MSC:
17A30Nonassociative algebras satisfying other identities
17D99Other nonassociative rings and algebras