Thin subalgebras of Lie algebras of maximal class

Abstract: Joint work with M. Avitabile, A. Caranti, V. Monti, M. F. Newman and E. O'Brien

Let $L$ be an infinite dimensional Lie algebra which is graded over the positive integers and is generated by its first homogeneous component $L_1$. The algebra $L$ is of maximal class if $\dim(L_1)=2$ and $\dim(L_i)=1$ for $1$ larger than $1$. The algebra $L$ is thin if it is not of maximal class, $\dim(L_1)=2$ and $L_{i+1}=[x,L_1]$ for any nontrivial element $x$ in $L_i$.

Suppose that $E$ is a quadratic extension of a field $F$ and that $M$ is a Lie algebra of maximal class over $E$. We consider the Lie algebra $L$ generated over the field $F$ by an $F$-subspace $L_1$ of $M_1$ having dimension $2$ over $F$. We give necessary and sufficient conditions for the lie algebra $L$ to be a thin graded $F$-subalgebra of the $F$-algebra $M$. We show also that there are uncountably many such thin algebras that can be constructed by way of this “recipe”, attaining the maximum possible cardinality.

The authors started this project almost independently since 1999 and their partial results have been luckily and duly recorded by A. Caranti. Only recently we have been able to develop together thorough and concise results for this research.

group theory

Audience: researchers in the topic

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