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3 edited body

I think that the explanation

"Because the Cartan classification of isomorphism classes of semisimples is discrete (no continuous families), connected components of the space of semisimples are always contained within isomorphism classes"

is a bit simplistic. The real reason, as many people have already mentioned, is Weyl's theorem on complete reducibility, which of course fails in charatcteristic $p$. And it shouldn't come as a surpise that over an algebraically closed field $K$ of characteristic $p>3$ one encounters situations where there are finitely many isoclasses of simple Lie algebras of dimension $N$ and, at the same time, there exist algebraic families of simple $N$ dimensional N$-dimensional Lie algebras {$\mathfrak{g}_t|\ t\in K$} over$K$such that$\mathfrak{g}_t\cong L$for all$t\ne 0$and$\mathfrak{g}_0\not\cong L$for some simple Lie algebra$L\$.

Indeed, let $N=p^2-2$. Then it follows from the the classification theory that there are finitely many isoclasses of simple $N$-dimensional Lie algebras over $K$. Now consider the associative $K$-algebra $A$ generated by two elements $x,y$ such that $x^p=y^p=0$ and $[x,y]=1$. This is a fake modular version of the first Weyl algebra, and it is easy to see that it is simple and has dimension $p^2$. It has a finite increasing algebra filtration (with $x,y$ living in degree $1$) such that the corresponding graded algebra $P:={\rm gr}(A)$ is the truncated polinomial ring in $x,y$ with induced Poisson bracket satisfying {$x,y$} $=1.$ Then the Lie algebra $[A,A]/K1$ is isomorphic to $\mathfrak{psl}_p(K)$ whilst the Lie algebra {$P,P$}$/K1$ is nothing but the simple Cartan type Lie algebra $H(2;\underline{1})^{(2)}$. Both Lie algebras are simple of dimension $N$ and the latter is a contraction of the former. Moreover, they are not isomorphic when $p>3$.

2 deleted 12 characters in body

I think that the explanation

"Because the Cartan classification of isomorphism classes of semisimples is discrete (no continuous families), connected components of the space of semisimples are always contained within isomorphism classes"

is a bit simplistic. The real reason, as many people have already mentioned, is Weyl's theorem on complete reducibility, which of course fails in charatcteristic $p$. And it shouldn't come as a surpise that over an algebraically closed field $K$ of characteristic $p>3$ one encounters situations where there are finitely many isoclasses of simple Lie algebras of dimension $N$ and, at the same time, there exist algebraic families of simple $N$ dimensional Lie algebras {$\mathfrak{g}_t|\ t\in K$} over $K$ such that $\mathfrak{g}_t\cong L$ for all $t\ne 0$ and $\mathfrak{g}_0\not\cong L$ for some simple Lie algebra $L$.

Indeed, let $N=p^2-2$. Then it follows from the the classification theory that there are finitely many isoclasses of simple $N$-dimensional Lie algebras over $K$. Now consider the associative associative $K$-algebra $A$ generated by two elements $x,y$ such that $x^p=y^p=0$ and $[x,y]=1$. This is a fake modular version of the first Weyl algebra, and it is easy to see that it is simple and has dimension $p^2$. It has a finite increasing algebra filtration (with $x,y$ living in degree $1$) such that the corresponding graded algebra $P:={\rm gr}(A)$ is the truncated polinomial ring in $x,y$ with induced Poisson bracket satisfying {$x,y$} $=1.$ Then the Lie algebra $[A,A]/K1$ is isomorphic to $\mathfrak{psl}_p(K)$ whilst the Lie algebra {$P,P$}$/K1$ is nothing but the simple Cartan type Lie algebra $H(2;\underline{1})^{(2)}$. Both Lie algebras are simple of dimension $N$ and the latter is a contraction of the former. Moreover, they are not isomorphic when $p>3$.

1

I think that the explanation

"Because the Cartan classification of isomorphism classes of semisimples is discrete (no continuous families), connected components of the space of semisimples are always contained within isomorphism classes"

is a bit simplistic. The real reason, as many people have already mentioned, is Weyl's theorem on complete reducibility, which of course fails in charatcteristic $p$. And it shouldn't come as a surpise that over an algebraically closed field $K$ of characteristic $p>3$ one encounters situations where there are finitely many isoclasses of simple Lie algebras of dimension $N$ and, at the same time, there exist algebraic families of simple $N$ dimensional Lie algebras {$\mathfrak{g}_t|\ t\in K$} over $K$ such that $\mathfrak{g}_t\cong L$ for all $t\ne 0$ and $\mathfrak{g}_0\not\cong L$ for some simple Lie algebra $L$.

Indeed, let $N=p^2-2$. Then it follows from the the classification theory that there are finitely many isoclasses of simple $N$-dimensional Lie algebras over $K$. Now consider the associative associative $K$-algebra $A$ generated by two elements $x,y$ such that $x^p=y^p=0$ and $[x,y]=1$. This is a fake modular version of the first Weyl algebra, and it is easy to see that it is simple and has dimension $p^2$. It has a finite increasing algebra filtration (with $x,y$ living in degree $1$) such that the corresponding graded algebra $P:={\rm gr}(A)$ is the truncated polinomial ring in $x,y$ with induced Poisson bracket satisfying {$x,y$} $=1.$ Then the Lie algebra $[A,A]/K1$ is isomorphic to $\mathfrak{psl}_p(K)$ whilst the Lie algebra {$P,P$}$/K1$ is nothing but the simple Cartan type Lie algebra $H(2;\underline{1})^{(2)}$. Both Lie algebras are simple of dimension $N$ and the latter is a contraction of the former. Moreover, they are not isomorphic when $p>3$.
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