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On right ideals of a free associative algebra, generated by free colour Lie superalgebras and \(p\)-superalgebras. (English. Russian original) Zbl 0785.17004
Russ. Math. Surv. 47, No. 5, 196-197 (1992); translation from Usp. Mat. Nauk 47, No. 5, 187-188 (1992).
Let \(k\) be a field, \(\text{char } k\neq 2\), and \(G\) be an abelian group. Fix a skew-bilinear function \(\varepsilon: G\times G\to k^*\). A \(G\)- graded (nonassociative) \(k\)-algebra \(A=\oplus_{g\in G} A_ g\) with multiplication \[ [x,y]=- \varepsilon (d(x),d(y))[y,x], \qquad [x,[y,z]]= [[x,y],z]+ \varepsilon (d(x),d(y)) [y,[x,z]], \] where \(x\), \(y\) are homogeneous elements in \(A\), \(z\in A\). Here \(d(x)=g\), if \(x\in A_ g\). It is also assumed that \([t,[t,t]]=0\) if \(t\in A_ g\), \(\varepsilon(g,g)=-1\).
Let \(L\) be a free colour Lie superalgebra. Assume that \(B\) is a homogeneous finitely generated subalgebra in \(L\). If \(a\in L\setminus B\) is a homogeneous element then there exists a homogeneous ideal \(A\in (L\setminus (B+I))\).
Let \(X=\{x_ 1, \dots, x_ n\}\) be a free generating set in \(L\). Note that a universal envelope of \(L\) is a free associative algebra \(A(X)\) on \(X\). Every element \(a\in A(X)\) has a unique representation \(a= \sum x_ i u_ i+ \alpha\), \(\alpha\in k\), \(u_ i\in A(X)\). Put \({{\partial a} \over {\partial x_ i}}=u_ i\) (Fox partial derivatives). Let \(f_ 1,\dots,f_ n\in L\) and let \(x_ i\), \(f_ i\) belong to the same homogeneous component \(A_{g_ i}\), \(i=1,\dots,n\). The set \(\{f_ 1,\dots,f_ n\}\) is a set of free generators in \(L\) if and only if a matrix \[ \left({{\partial {f_ i}}\over {\partial{x_ j}}}\right), \qquad 1\leq i,j\leq n, \] is invertible in \(\text{Mat}(n,A(X))\).
Let \(\text{char } k=p>2\). A colour Lie superalgebra with a partial unary operation \(x\to x^{[p]}\) defined on homogeneous elements \(x\) is a colour Lie superalgebra if the operation \(x^{[p]}\) satisfies standard identities of Lie \(p\)-algebras. It is shown that previous results hold for colour Lie \(p\)-superalgebras. Note that similar results for Lie algebras have been proven by U. Umirbaev [Uch. Zap. Tartu Univ. 878, 147-152 (1990)].

17B01 Identities, free Lie (super)algebras
17B70 Graded Lie (super)algebras
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