Finite groups of Lie type
Here we give the known results on conjugacy classes in finite groups of Lie type. For a description of these groups, see for example [ACG1], [MT].
Let $\mathbb{G}$ be a simple algebraic group, $\mathbb{G}_{sc}$ its simply connected cover with $\bar{\pi}: \mathbb{G}_{sc} \to \mathbb{G}$ the natural projection, $F$ a Steinberg endomorphism of $\mathbb{G}$, $\mathbf{G} = \mathbb{G}_{sc}^F/Z(\mathbb{G}_{sc}^F)$, $\pi: \mathbb{G}_{sc}^F \to \mathbf{G}$ the natural projection. Often $F$ descends to $\mathbb{G}$, and then there is a projection $\pi: [\mathbb{G}^F, \mathbb{G}^F] \to [\mathbb{G}^F, \mathbb{G}^F]/ \bar{\pi}(Z(\mathbb{G}_{sc}^F)) \simeq \mathbf{G}$.
Every $x\in \mathbb{G}_{sc}$ has a Chevalley-Jordan decomposition $x = x_sx_u= x_ux_s$, with $x_{s}$ semisimple and $x_{u}$ unipotent.
This decomposition boils down to $\mathbb{G}_{sc}$ and to the finite groups $\mathbb{G}_{sc}^F$ and $\mathbf{G}$, where it agrees with the decomposition in the $p$-part, namely $x_u$, and the $p$-regular part, namely $x_s$.
Let $x\in \mathbf{G}$; pick $\mathbb{G}_{sc}^F$ such that $\pi(\mathbf{x}) = x$. If $\mathbf{x} = \mathbf{x}_s\mathbf{x}_u$ is its Chevalley-Jordan decomposition, then $x_s = \pi(\mathbf{x}_s)$, $x_u = \pi(\mathbf{x}_u)$ is the Chevalley-Jordan decomposition of $x$, with $x_{s}$ semisimple and $x_{u}$ unipotent. Now $\mathbf{x}_u$ belongs to $\mathcal{K} := C_{\mathbb{G}_{sc}^F}(\mathbf{x}_s)$, thus $x_u\in K := \pi(\mathcal{K})$ and there are morphisms of racks
$$\mathcal{O}^{\mathcal{K}}_{\mathbf{x}_u}\simeq \mathcal{O}^{K}_{x_u} \hookrightarrow \mathcal{O}_x^{\mathbf{G}}.$$
Now the centralizer $C_{\mathbb{G}_{sc}}(\mathbf{x}_s)$ is a reductive subgroup of $\mathbb{G}_{sc}$, and $\mathcal{K} = C_{\mathbb{G}_{sc}}(\mathbf{x}_s)\cap \mathbb{G}_{sc}^F$. Strictly, $C_{\mathbb{G}_{sc}^F}(\mathbf{x}_s)$ is not of Lie type in the sense above, but close enough to allow some inductive procedure. So, we are reduced to investigate the conjugacy classes
Unipotent classes in Chevalley and Steinberg groups
Theorem. [ACG1, ACG2, ACG3]
Let $\mathbf{G}$ be a Chevalley or a Steinberg group. A unipotent conjugacy class in $\mathbf{G}$ that does not appear in the Table below is not kthulhu, hence collapses.
Mixed classes in Chevalley groups
We say that a conjugacy class is mixed if it is a conjugacy class of an element which is neither unipotent nor semisimple.
Theorem. [ACG1, ACG5]
Let $\mathbb{G}^F$ be a Chevalley group and $x = x_sx_u \in \mathbb{G}$ with $x_s, x_u \neq 1$. Then $\mathcal{O}^{\mathbf{G}}_x$ is not kthulhu, and consequently $\mathcal{O}^{\mathbf{G}}_x$ collapses.
Semisimple classes in $\mathbf{PSL}_n(q)$
Theorem. [ACG3, Theorem 1.1]
Let $\mathcal{O}$ be a semisimple conjugacy class in $\mathbf{PSL}_n(q)$. If either $n>2$ and $\mathcal{O}$ is not irreducible or $n=2$, $q\neq 2,3,4,5,9$ and $\mathcal{O}$ is not listed in the table below, then it collapses.
Suzuki-Ree groups
Theorem. [CC, Theorem 1.1, Theorem 1.2]
In simple Suzuki groups $^{2}B_{2}(q)$, $q>2$ and Ree groups $^{2}F_{4}(q)$, $q>2$ and $^{2}G_{2}(q)$, $q>3$, the class of an element $x$ is kthulhu if and only if
$\mathbf{G} =\ ^{2}B_{2}(q)$ and $|x|=2$ or divides $|x|$ divides $q^2+1$;
$\mathbf{G} =\ ^{2}F_{4}(q)$ and $|x|=13$ or divides $|x|$ divides $q^4-q^2+1$;
$\mathbf{G} =\ ^{2}G_{2}(q)$ and $|x|=3$ or $|x|=7$ and divides $|x|$ divides $q^2-q+1$.
However, dim $\mathfrak{B}(V) = \infty$ for every complex Yetter-Drinfeld module $V$ of $\mathbf{G}$ a simple Suzuki or Ree group. Thus, the only finite-dimensional complex pointed Hopf algebras whose group of grouplikes is $\mathbf{G}$ are their group algebras $\mathbb{C}\mathbf{G}$.
Let $\mathbb{G}$ be a simple algebraic group, $\mathbb{G}_{sc}$ its simply connected cover with $\bar{\pi}: \mathbb{G}_{sc} \to \mathbb{G}$ the natural projection, $F$ a Steinberg endomorphism of $\mathbb{G}$, $\mathbf{G} = \mathbb{G}_{sc}^F/Z(\mathbb{G}_{sc}^F)$, $\pi: \mathbb{G}_{sc}^F \to \mathbf{G}$ the natural projection. Often $F$ descends to $\mathbb{G}$, and then there is a projection $\pi: [\mathbb{G}^F, \mathbb{G}^F] \to [\mathbb{G}^F, \mathbb{G}^F]/ \bar{\pi}(Z(\mathbb{G}_{sc}^F)) \simeq \mathbf{G}$.
Every $x\in \mathbb{G}_{sc}$ has a Chevalley-Jordan decomposition $x = x_sx_u= x_ux_s$, with $x_{s}$ semisimple and $x_{u}$ unipotent.
This decomposition boils down to $\mathbb{G}_{sc}$ and to the finite groups $\mathbb{G}_{sc}^F$ and $\mathbf{G}$, where it agrees with the decomposition in the $p$-part, namely $x_u$, and the $p$-regular part, namely $x_s$.
Let $x\in \mathbf{G}$; pick $\mathbb{G}_{sc}^F$ such that $\pi(\mathbf{x}) = x$. If $\mathbf{x} = \mathbf{x}_s\mathbf{x}_u$ is its Chevalley-Jordan decomposition, then $x_s = \pi(\mathbf{x}_s)$, $x_u = \pi(\mathbf{x}_u)$ is the Chevalley-Jordan decomposition of $x$, with $x_{s}$ semisimple and $x_{u}$ unipotent. Now $\mathbf{x}_u$ belongs to $\mathcal{K} := C_{\mathbb{G}_{sc}^F}(\mathbf{x}_s)$, thus $x_u\in K := \pi(\mathcal{K})$ and there are morphisms of racks
$$\mathcal{O}^{\mathcal{K}}_{\mathbf{x}_u}\simeq \mathcal{O}^{K}_{x_u} \hookrightarrow \mathcal{O}_x^{\mathbf{G}}.$$
Now the centralizer $C_{\mathbb{G}_{sc}}(\mathbf{x}_s)$ is a reductive subgroup of $\mathbb{G}_{sc}$, and $\mathcal{K} = C_{\mathbb{G}_{sc}}(\mathbf{x}_s)\cap \mathbb{G}_{sc}^F$. Strictly, $C_{\mathbb{G}_{sc}^F}(\mathbf{x}_s)$ is not of Lie type in the sense above, but close enough to allow some inductive procedure. So, we are reduced to investigate the conjugacy classes
- $x$ semisimple (the case $x = x_s$), or
- $x$ unipotent, and from this try to catch the general case.
Theorem. [ACG1, ACG2, ACG3]
Let $\mathbf{G}$ be a Chevalley or a Steinberg group. A unipotent conjugacy class in $\mathbf{G}$ that does not appear in the Table below is not kthulhu, hence collapses.
$\mathbf{G}$ | class | $q$ |
$\mathbf{PSL}_2(q)$ | $(2)$ | even, or 9, or odd not a square |
$\mathbf{PSL}_3(2)$ | $(3)$ | $2$ |
$\mathbf{PSp}_{2n}(q)$, $n \ge 2$ | $W(1)^{n-1} \oplus V(2)$ | even |
$\mathbf{PSp}_{2n}(q)$, $n \ge 2$ | $(2, 1^{n-2})$ | 9 or odd not a square |
$\mathbf{PSp}_{4}(q)$ | $W(2)$ | even |
$\mathbf{PSU}_n(q)$, $n \ge 3$ | $(2, 1^{n-2})$ | even |
Mixed classes in Chevalley groups
We say that a conjugacy class is mixed if it is a conjugacy class of an element which is neither unipotent nor semisimple.
Theorem. [ACG1, ACG5]
Let $\mathbb{G}^F$ be a Chevalley group and $x = x_sx_u \in \mathbb{G}$ with $x_s, x_u \neq 1$. Then $\mathcal{O}^{\mathbf{G}}_x$ is not kthulhu, and consequently $\mathcal{O}^{\mathbf{G}}_x$ collapses.
Semisimple classes in $\mathbf{PSL}_n(q)$
Theorem. [ACG3, Theorem 1.1]
Let $\mathcal{O}$ be a semisimple conjugacy class in $\mathbf{PSL}_n(q)$. If either $n>2$ and $\mathcal{O}$ is not irreducible or $n=2$, $q\neq 2,3,4,5,9$ and $\mathcal{O}$ is not listed in the table below, then it collapses.
$q$ | class | Remark |
|---|---|---|
even and not a square | irreducible, order 3 | sober |
all | irreducible, order > 3 | sober |
Suzuki-Ree groups
Theorem. [CC, Theorem 1.1, Theorem 1.2]
In simple Suzuki groups $^{2}B_{2}(q)$, $q>2$ and Ree groups $^{2}F_{4}(q)$, $q>2$ and $^{2}G_{2}(q)$, $q>3$, the class of an element $x$ is kthulhu if and only if
$\mathbf{G} =\ ^{2}B_{2}(q)$ and $|x|=2$ or divides $|x|$ divides $q^2+1$;
$\mathbf{G} =\ ^{2}F_{4}(q)$ and $|x|=13$ or divides $|x|$ divides $q^4-q^2+1$;
$\mathbf{G} =\ ^{2}G_{2}(q)$ and $|x|=3$ or $|x|=7$ and divides $|x|$ divides $q^2-q+1$.
However, dim $\mathfrak{B}(V) = \infty$ for every complex Yetter-Drinfeld module $V$ of $\mathbf{G}$ a simple Suzuki or Ree group. Thus, the only finite-dimensional complex pointed Hopf algebras whose group of grouplikes is $\mathbf{G}$ are their group algebras $\mathbb{C}\mathbf{G}$.
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