# Varieties of elementary Lie algebras

Friedlander and Parshall developed a theory of support varieties and rank varieties for finite-dimensional restricted Lie algebras over a feld \$k\$ of characteristic \$p>0\$. In particular, for any restricted Lie algebra \$\fg\$, the rank variety can be identified with its nullcone \$V(\fg)\$ which is a conical closed subvariety of \$\fg\$. By a result of Carlson, the projectivization of \$V(\fg)\$ is connected. Recently, Carlson, Friedlander and Pevtsova introduced the elementary subalgebras of restricted Lie algebras. The varieties of elementary subalgebras are natural varieties in which to define generalized support varieties for restricted representations of \$\fg\$. Farnsteiner defined an invariant for modules of restricted Lie algebras, called \$j\$-degrees. The invariant, together with the constant \$j\$-rank property can be linked by the two-dimensional elementary Lie subalgebras.\\ The aim of this thesis is to understand the geometric structure of the variety \$\EE(2,\fg)\$. We first investigate the non-empty property of \$\EE(2,\fg)\$. By using the sandwich elements, we give a complete description of the restricted Lie algebras which have no two-dimensional elementary subalgebras. Moreover, we show that the variety \$\EE(2,\fg)\$ is always connected whenever \$\fg\$ is centerless and \$p>5\$. This is a generalization of Carlson's result, which claims that \$\EE(1,\fg)\$ is connected.\\ We also study the degree functions \$\deg_M^j\$ for centerless restricted Lie algebra. We show that the function \$\deg_M^j\$ is constant on \$\EE(2,\fg)\$ if \$M\$ is a restricted \$\fg\$-module of constant \$j\$-rank.\\ In the last, we will endow the set \$\EE(2,\fg)\$ with a graph structure which plays an important role in the investigation of the equal images property. We give a sufficient condition for the connectedness of the graph \$\EE(2,\fg)\$. As a consequence, the graph \$\EE(2,W(n))\$ is connected, where \$W(n\$) is the \$n\$-th Witt-Jacobson algebra.

### Rights

Use and reproduction: