Thin Lie algebras are graded Lie algebras L = circle plus(infinity)(i=1)L(i) with dimL(i) <= 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L(1), are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if L(k) is the second diamond of L, then the quotient L/L(k) is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/L(k) is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/L(k) need not be metabelian in characteristic two. We describe here all the possibilities for L/L(k) up to isomorphism. In particular, we prove that k + 1 equals a power of two.
THE STRUCTURE OF THIN LIE ALGEBRAS WITH CHARACTERISTIC TWO
Jurman G;
2010-01-01
Abstract
Thin Lie algebras are graded Lie algebras L = circle plus(infinity)(i=1)L(i) with dimL(i) <= 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L(1), are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if L(k) is the second diamond of L, then the quotient L/L(k) is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/L(k) is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/L(k) need not be metabelian in characteristic two. We describe here all the possibilities for L/L(k) up to isomorphism. In particular, we prove that k + 1 equals a power of two.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.