Modular Lie algebra

Remarks

Example: $\mathfrak{sl}_2(\F_p)$, the $2 \times 2$ matrices of trace zero over $\F_p$ with bracket $[x, y] = xy - yx$, is a modular Lie algebra, since $\F_p$ has characteristic $p$. Non-example: $\mathfrak{sl}_2(\C)$ is a Lie algebra over a field of characteristic $0$, so it is not modular. Many authors also require the Lie algebra to be finite-dimensional. Lie's theorem fails in characteristic $p$. Let $V = F[t]/(t^p)$ and let the Heisenberg algebra with basis $x, y, z$, $[x, y] = z$, act by $x \mapsto d/dt$, $y \mapsto$ multiplication by $t$, $z \mapsto \id$. This is well defined because the derivative of $t^p$ is $0$, and $V$ is an irreducible module of dimension $p$ of a nilpotent Lie algebra. Over an algebraically closed field of characteristic $0$, finite-dimensional irreducible modules of solvable Lie algebras are one-dimensional. Let $F$ be algebraically closed of characteristic $p > 3$. By work of Block, Wilson, Premet and Strade, each finite-dimensional simple Lie algebra over $F$ is of classical type, of Cartan type, or of Melikian type. The Melikian type occurs only for $p = 5$.