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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)].

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