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.
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