Minimal polynomial identities for right-symmetric algebras

Askar Dzhumadil'daev

paper

Minimal polynomial identities for right-symmetric algebras

arXiv:math/9809082

Abstract

An algebra $A$ with multiplication $A \times A \to A, (a, b) \mapsto a \circ b$ , is called right-symmetric, if $a \circ (b \circ c) - (a \circ b) \circ a \circ (c \circ b) - (a \circ c) \circ b,$ for any $a, b, c \in A$ . The multiplication of right-symmetric Witt algebras $W_n=\{u\der_i: u\in U, U={\cal K}[x_1^{\pm 1},...,x_n^{\pm}$ or $= K [x_{1}, ..., x_{n}], i = 1, ..., n}, p = 0,$ or $W_n({\bf m)}=\{u\der_i: u\in U, U=O_n({\bf m})\}$, are given by $u\der_i\circ v\der_j=v\der_j(u)\der_i.$ An analogue of the Amitsur-Levitzki theorem for right-symmetric Witt algebras is established. Right-symmetric Witt algebras of $s a t i s f y t h es t an d a r d r i g h t - sy mm e t r i c i d e n t i t y o f d e g r ee$ 2n+1:\sum_{σ\in Sym_{2n}}sign(σ)a_{σ(1)}\circ(a_{σ(2)}\circ >...(a_{σ(2n)}\circ a_{2n+1})...)=0. $T h e minima l d e g$ left polynomial identities of $W_{n}^{r sy m}, W_{n}^{+ r sy m}, p = 0,$ i $T h e minima l d e g r eeo f m u l t i l in e a r l e f tp o l y n o mia l i d e n t i t y o f$ $i s a l so$ 2n+1. $A l l l e f tp o l y n o mia l (a l so m u l t i l in e a r, i f$ p>0 $) i d e n t i t i eso f r i g h t - sy mm e t r i c W i tt a l g e b r a so f minima l$ combinations of left polynomials obtained from standard ones by permutations of arguments.

20 pages, latex, no figures