Identities of the left-symmetric Witt algebras. (Q2799128)

Assume \(k\) is a field of characteristic 0, and let \(W_n\) be the vector space of all derivations of the polynomial algebra \(P_n=k[x_1,\ldots,x_n]\) in \(n\) variables over \(k\). Then \(W_n\) has a basis as a vector space consisting of the elements \(u\partial_i\) where \(\partial_i={{\partial}\over{\partial x_i}}\) where \(u\) runs over the monomials in \(P_n\) and \(1\leq i\leq n\). The vector space \(W_n\) together with the Lie bracket gives rise to the well known Witt algebra of index \(n\). On the other hand one may define another product on the vector space \(W_n\) namely \(a\partial_i\cdot b\partial_j=(a\partial_i(b))\partial_j\) for \(a,b\in W_n\). The vector space \(W_n\) with this product is denoted by \(L_n\). It satisfies the left-symmetric identity \((xy)z-x(yz)=(yx)z-y(xz)\) (that is the associator in \(L_n\) is symmetric with respect to the first two elements). It is well known that every left symmetric algebra becomes a Lie algebra if one substitutes its product \(\cdot\) by the Lie bracket. Thus \(L_n\) is called the left symmetric Witt algebra of index \(n\). The polynomial identities satisfied by the algebras \(L_n\) are quite important. The algebra \(L_1\) satisfies the identity \((x\cdot y)\cdot z=(x\cdot z)\cdot y\), that is it is a Novikov algebra. It is known that the left-symmetric identity together with the Novikov identity form a basis of the identities of \(L_1\); on the other hand, finding a basis of the identities of the Lie algebra \(W_1\) is still an open problem.NEWLINENEWLINE The authors of the paper under review describe the right operator identities for \(L_n\), together with the subalgebras of triangular derivations and of strongly triangular derivations. Recall that the right multiplications of \(L_n\) form an associative algebra isomorphic to the \(n\times n\) matrix algebra over \(P_n\). As a consequence the authors obtain that the varieties of algebras generated by the algebras \(L_n\) form a strictly ascending chain. Furthermore they prove that the algebras \(L_n\), \(n\geq 1\), generate the variety of all left-symmetric algebras. It follows that every identity satisfied by all \(L_n\) follows from the left-symmetric identity.NEWLINENEWLINE The authors observe that studying left operator identities for \(L_n\) is more important than their right counterparts but nonetheless much more difficult. They apply methods due to Razmyslov in order to obtain wider classes of identities for \(L_n\).