On generators of free color Lie superalgebras of rank two

AYDUN E., Ekici N.

JOURNAL OF LIE THEORY, vol.12, no.2, pp.529-534, 2002 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 12 Issue: 2
  • Publication Date: 2002
  • Title of Journal : JOURNAL OF LIE THEORY
  • Page Numbers: pp.529-534


P. M. Cohn proved in [3] that the t-automorphisms generate the group of all automorphisms of a free Lie algebra of finite rank. In [6], [7] Mikhalev obtained the following analogue of Cohn's theorem: The elementary automorphisms and linear changes generate a group of automorphisms of a free color Lie superalgebra of finite rank. The freedom of the subalgebras of free color Lie algebras [6], [7] gives rise to the following analogue of Nielsen's theorem: If n G-homogeneous elements generate a free color Lie superalgebra of rank n, then these elements are free generators of it. Now let X = {x, y} and L(X) be a free color Lie superalgebra freely generated by the set X. If two G-homogeneous elements h(1), h(2) generate L(X), then they freely generate L(X). So [h(1), h(2) is a linear combination of the elements [x, x], [y, y], [x, y]. The main assertion-of this note is the theorem that the subalgebra generated by h(1), h(2) is equal to the free color Lie superalgebra L(X) if and only if [h(1), h(2)] = alpha[x, x] + beta[x, y] + gamma[y, y], where alpha, beta, gamma epsilon K*. In [4] Dicks obtained a similar criterion for free associative algebras of rank two.