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On fixed points of a free chromatic Lie superalgebra under the action of a finite group of linear automorphisms. (English. Russian original) Zbl 0792.17005

Russ. Math. Surv. 47, No. 4, 220-221 (1992); translation from Usp. Mat. Nauk 47, No. 4(286), 205-206 (1992).
Let \(k\) be a commutative domain with a unit, \(\text{char } k\neq 2\). Suppose that \(G\) is a group with a skew-symmetric bilinear function \(\varepsilon: G\times G\to k^*\). If \(a\) is a homogeneous element of a \(G\)-graded \(k\)-module \(R=\oplus_{g\in G} R_ g\) and \(a\in R_ g\) then put \(d(a)=g\). A \(G\)-graded \(k\)-algebra \(R\) is a chromatic (colour) Lie superalgebra with multiplication \([x,y]\) if all homogeneous elements \(x,y,a,u,v\in R\), satisfying the identities \[ \begin{aligned} [x,y]&=- \varepsilon (d(x), d(y)) [y,x],\\ [x,[y,z]]&= [[x,y],z]+ \varepsilon (d(x),d(y)) [y,[x,z]],\\ [u,u]&=0, \quad \text{if} \quad \varepsilon (g,g)=-1,\\ [v,[v,v]]&=0, \quad \text{if}\quad \varepsilon (g,g)=1. \end{aligned} \] Let \(k\) be a PID and \(L\) be a free chromatic Lie superalgebra over \(k\) with homogeneous set of free generators \(X\), \(\text{card } X\geq 2\). Suppose that \(H\) is a finite nontrivial group of automorphisms of \(L\) such that each linear span of \(X\cap L_ g\), \(g\in G\), is \(H\)-invariant. Then the subalgebra \(L^ H\) of \(H\)-stable elements of \(L\) is a free chromatic Lie superalgebra of infinite rank.
In the case of ordinary Lie algebra this result has been proved by R. M. Bryant [J. Lond. Math. Soc., II. Ser. 43, No. 2, 215-224 (1991; Zbl 0685.20004)].

MSC:

17A70 Superalgebras
17B65 Infinite-dimensional Lie (super)algebras
17B70 Graded Lie (super)algebras
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