## 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

### Citations:

Zbl 0724.20008; Zbl 0685.20004
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