On Kac's Jordan superalgebra

TL;DR

This paper characterizes the automorphism group of Kac's ten-dimensional Jordan superalgebra, using it to classify its twisted forms and gradings more simply and comprehensively.

Contribution

It identifies the automorphism group as a specific semidirect product, extending and simplifying the classification of twisted forms and gradings of the superalgebra.

Findings

Automorphism group is isomorphic to a semidirect product of SL2×SL2 and C2.

Provides a unified framework for classifying twisted forms.

Simplifies previous classification results.

Abstract

The group-scheme of automorphisms of the ten-dimensional exceptional Kac's Jordan superalgebra is shown to be isomorphic to the semidirect product of the direct product of two copies of SL2 by the constant group scheme C2. This is used to revisit, extend, and simplify, known results on the classification of the twisted forms of this superalgebra and of its gradings.

Equations

$$a^2=a,av=\frac{1}{2}v=va,vw=(v|w)a,$$

$$(x\otimes y)(z\otimes t)=(-1)^{yz}\Bigl{(}xz\otimes yt-\frac{3}{4}(x\mid z)(y\mid t)1\Bigr{)}.$$

$$1x=x1=x,vw=(v\mid w)1,$$

$$G^D⟶\mathrm{Aut}(\mathcal{A}),$$

$$\begin{split}\Phi:\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}&\longrightarrow\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{10}\bigr{)}\\ (f,g)\qquad&\mapsto\quad\Phi_{(f,g)}:\begin{cases}\ 1\ \mapsto\ 1,\\ a\otimes a\mapsto a\otimes a,\\ v\otimes a\mapsto f(v)\otimes a,\\ a\otimes v\mapsto a\otimes g(v),\\ v\otimes w\mapsto f(v)\otimes g(w),\end{cases}\\ \text{generator of $\mathsf{C}_{2}$}&\mapsto\quad\tau:\begin{cases}\ 1\ \mapsto\ 1,\\ x\otimes y\mapsto(-1)^{xy}y\otimes x,\end{cases}\end{split}$$

$$\begin{split}\Psi^{1}:\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}&\longrightarrow\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{3}\times\mathsf{K}_{3}\bigr{)}\\ (f,g)\qquad&\mapsto\quad\Psi^{1}_{(f,g)}:\begin{cases}(a,0)\mapsto(a,0),\\ (0,a)\mapsto(0,a),\\ (v,w)\mapsto\bigl{(}f(v),g(w)\bigr{)},\end{cases}\\ \text{generator of $\mathsf{C}_{2}$}&\mapsto\quad\tau:(x,y)\mapsto(y,x),\end{split}$$

$$\begin{split}\Psi^{2}:\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}&\longrightarrow\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{J}(W)\times\mathsf{J}(W)\bigr{)}\\ (f,g)\qquad&\mapsto\quad\Psi^{2}_{(f,g)}:\begin{cases}(1,0)\mapsto(1,0),\\ (0,1)\mapsto(0,1),\\ (v,w)\mapsto\bigl{(}f(v),g(w)\bigr{)},\end{cases}\\ \text{generator of $\mathsf{C}_{2}$}&\mapsto\quad\tau:(x,y)\mapsto(y,x),\end{split}$$

$$1\longrightarrow\mathbf{SL}(W)\times\mathbf{SL}(W)\longrightarrow\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}\longrightarrow\mathsf{C}_{2}\longrightarrow 1$$

$$1\rightarrow\mathrm{SL}(W)\times\mathrm{SL}(W)\rightarrow\bigl{(}\mathrm{SL}(W)\times\mathrm{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}\rightarrow\mathsf{C}_{2}\\ \rightarrow H^{1}(\mathbb{F},\mathbf{SL}(W)\times\mathbf{SL}(W))\rightarrow H^{1}\bigl{(}\mathbb{F},\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}\bigr{)}\rightarrow H^{1}(\mathbb{F},\mathsf{C}_{2}).$$

$$1\longrightarrow H^{1}\bigl{(}\mathbb{F},\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}\bigr{)}\rightarrow H^{1}(\mathbb{F},\mathsf{C}_{2}),$$

$$\begin{array}{rl}{\mathsf{K}}_{10,\mathbb{K}}& =\{X\in {\mathsf{K}}_{10}{\otimes }_{\mathbb{F}}\mathbb{K}:(\tau \otimes \mathsf{id})(X)=(\mathsf{id}\otimes \sigma )(X)\}\\ & =\{X\in {\mathsf{K}}_{10}{\otimes }_{\mathbb{F}}\mathbb{K}:(\tau \otimes \sigma )(X)=X\},\end{array}$$

$$\begin{split}\bigl{(}\mathsf{K}_{10,\mathbb{K}}\bigr{)}_{\bar{0}}&=\mathbb{F}\textup{-span}\Bigl{\langle}1\otimes 1,(a\otimes a)\otimes 1,(u\otimes u)\otimes\alpha,(v\otimes v)\otimes\alpha,\\[-3.0pt] &\hskip 86.72377pt(u\otimes v-v\otimes u)\otimes 1,(u\otimes v+v\otimes u)\otimes\alpha\Bigr{\rangle},\\[3.0pt] \bigl{(}\mathsf{K}_{10,\mathbb{K}}\bigr{)}_{\bar{1}}&=\mathbb{F}\textup{-span}\Bigl{\langle}(a\otimes x+x\otimes a)\otimes 1,(a\otimes x+x\otimes a)\otimes\alpha:x\in\{u,v\}\Bigr{\rangle}.\end{split}$$

$$\begin{split}&\deg(x)=e\ \text{for any $x\in\bigl{(}\mathsf{K}_{3}\times\mathsf{K}_{3})_{\bar{0}}$},\\ &\deg(u,0)=g_{1}=\deg(v,0)^{-1},\quad\deg(0,u)=g_{2}=\deg(0,v)^{-1}.\end{split}$$

$$\begin{array}{rl}& \mathrm{deg}(a,a)=e,\mathrm{deg}(a,-a)=h,\\ & \mathrm{deg}(u,u)=g=\mathrm{deg}(v,v{)}^{-1},\mathrm{deg}(u,-u)=gh=\mathrm{deg}(v,-v{)}^{-1}.\end{array}$$

$$\textstyle{\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{3}\times\mathsf{K}_{3}\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$

$$\mathrm{deg}(u\otimes a)=g_1=\mathrm{deg}(v\otimes a{)}^{-1},\mathrm{deg}(a\otimes u)=g_2=\mathrm{deg}(a\otimes v{)}^{-1}.$$

$$\begin{array}{rl}& \mathrm{deg}(u\otimes a+a\otimes u)=g=\mathrm{deg}(v\otimes a+a\otimes v{)}^{-1},\\ & \mathrm{deg}(u\otimes a-a\otimes u)=gh=\mathrm{deg}(v\otimes a-a\otimes v{)}^{-1}.\end{array}$$

Full text

This is used to revisit, extend, and simplify, known results on the classification of the twisted forms of this superalgebra and of its gradings.

⋆ Supported by the Spanish Ministerio de Economía y Competitividad—Fondo Europeo de Desarrollo Regional (FEDER) MTM2013-45588-C3-2-P, and by the Diputación General de Aragón—Fondo Social Europeo (Grupo de Investigación de Álgebra). A.S. Córdova-Martínez also acknowledges support from Consejo Nacional de Ciencia y Tecnología (CONACyT, México) through grant 420842/262964.

⋄ This research was in part supported by a grant from IPM (No. 94160221) and partially carried out in IPM-Isfahan Branch.

Kac’s ten-dimensional superalgebra $\mathsf{K}_{10}$ is an exceptional Jordan superalgebra which appeared for the first time in Kac’s classification [Kac77] of the finite dimensional simple Jordan superalgebras over an algebraically closed field of characteristic [math]. It was constructed by Lie theoretical terms from a $3$-grading of the exceptional simple Lie superalgebra $F(4)$.

A more conceptual definition was given in [BE02] over an arbitrary field $\mathbb{F}$ of characteristic not two (these assumptions on the ground field will be kept throughout the paper), using the three-dimensional Kaplansky superalgebra $\mathsf{K}_{3}$. The even part $(\mathsf{K}_{3})_{\bar{0}}$ is a copy of the ground field $\mathbb{F}$: $(\mathsf{K}_{3})_{\bar{0}}=\mathbb{F}a$, with $a^{2}=a$; while the odd part is a two-dimensional vector space $W$ endowed with a nonzero skew-symmetric bilinear form $(\,\mid\,)$. The multiplication is determined as follows:

[TABLE]

for any $v,w\in W$. We extend $(\,\mid\,)$ to a supersymmetric bilinear form on $\mathsf{K}_{3}$: $(\mathsf{K}_{3})_{\bar{0}}$ and $(\mathsf{K}_{3})_{\bar{1}}$ are orthogonal, with $(a|a)=\frac{1}{2}$.

Then \mathsf{K}_{10}=\mathbb{F}1\oplus\bigl{(}\mathsf{K}_{3}\otimes_{\mathbb{F}}\mathsf{K}_{3}\bigr{)}, with (\mathsf{K}_{10})_{\bar{0}}=\mathbb{F}1\oplus\bigl{(}\mathsf{K}_{3}\otimes_{\mathbb{F}}\mathsf{K}_{3}\bigr{)}_{\bar{0}}=\mathbb{F}1\oplus\mathbb{F}(a\otimes a)\oplus(W\otimes W), and (\mathsf{K}_{10})_{\bar{1}}=\bigl{(}\mathsf{K}_{3}\otimes_{\mathbb{F}}\mathsf{K}_{3}\bigr{)}_{\bar{1}}=(W\otimes a)\oplus(a\otimes W). The multiplication is given by imposing that $1$ is the unity, and for homogeneous elements $x,y,z,t\in\mathsf{K}_{3}$,

[TABLE]

If the characteristic is $3$, then $\mathsf{K}_{9}:=\mathsf{K}_{3}\otimes_{\mathbb{F}}\mathsf{K}_{3}$ is a simple ideal in $\mathsf{K}_{10}$. Otherwise $\mathsf{K}_{10}$ is simple.

It must be remarked that an ‘octonionic’ construction of $\mathsf{K}_{10}$ has been given in [RZ15].

Using the construction above of $\mathsf{K}_{10}$ in terms of $\mathsf{K}_{3}$, the group of automorphisms \mathsf{Aut}\bigl{(}\mathsf{K}_{10}\bigr{)} was computed in [ELS07]. This was used in [CDM10] to classify gradings on $\mathsf{K}_{10}$ over algebraically closed fields of characteristic zero.

Our goal here is to compute the group-scheme of automorphisms of $\mathsf{K}_{10}$, and to use it to revisit, extend, and simplify drastically, the known results on twisted forms and on gradings on $\mathsf{K}_{10}$.

Here we follow the functorial approach (see, for instance, [Wat79]). An affine group-scheme over $\mathbb{F}$ is a representable group-valued functor defined on the category of unital, commutative, associative algebras over $\mathbb{F}$: $\operatorname{\mathrm{Alg_{\mathbb{F}}}}$. Thus the group-scheme \operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{10}\bigr{)} is the functor $\operatorname{\mathrm{Alg_{\mathbb{F}}}}\rightarrow\operatorname{\mathrm{Grp}}$ that takes any object $R$ in $\operatorname{\mathrm{Alg_{\mathbb{F}}}}$ to the group \mathsf{Aut}_{R}\bigl{(}\mathsf{K}_{10}\otimes_{\mathbb{F}}R) (the group of automorphisms of the $R$-superalgebra $\mathsf{K}_{10}\otimes_{\mathbb{F}}R$, i.e., the group of $R$-linear isomorphisms preserving the multiplication and the grading), with the natural definition on morphisms.

It turns out that \operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{10}\bigr{)} is isomorphic to a semidirect product \bigl{(}\mathbf{SL}_{2}\times\mathbf{SL}_{2}\bigr{)}\rtimes\mathsf{C}_{2} (Theorem 1.1), where $\mathsf{C}_{2}$ is the constant group scheme attached to the cyclic group of two elements (also denoted by $\mathsf{C}_{2}$). This extends the result in [ELS07].

Moreover, the canonical projection \pi:\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{10}\bigr{)}\rightarrow\mathsf{C}_{2} induces an isomorphism of pointed sets H^{1}\bigl{(}\mathbb{F},\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{10}\bigr{)}\bigr{)}\rightarrow H^{1}(\mathbb{F},\mathsf{C}_{2}) (Theorem 2.1). This associates to the isomorphism class of any twisted form of $\mathsf{K}_{10}$, the isomorphism class of a quadratic étale algebra over $\mathbb{F}$, which allows to get easily the classification of twisted forms of $\mathsf{K}_{10}$ (Corollary 2.2). Twisted forms of $\mathsf{K}_{10}$ were classified, in a completely different way, in [RZ03] and [EO00]. We believe that our approach is more natural.

A simple observation shows that \bigl{(}\mathbf{SL}_{2}\times\mathbf{SL}_{2}\bigr{)}\rtimes\mathsf{C}_{2} is also the automorphism group-scheme of $\mathsf{K}_{3}\times\mathsf{K}_{3}$, or of $\mathsf{J}(W)\times\mathsf{J}(W)$, where $\mathsf{J}(W)$ is the Jordan superalgebra of the ‘superform’ $(\,\mid\,)$ on $W=W_{\bar{1}}$ above. That is, $\mathsf{J}(W)_{\bar{0}}=\mathbb{F}1$, $\mathsf{J}(W)_{\bar{1}}=W$ and the multiplication is given by

[TABLE]

for any $x\in\mathsf{J}(W)$, $v,w\in W$. This observation will be used in the proof of the bijection H^{1}\bigl{(}\mathbb{F},\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{10}\bigr{)}\bigr{)}\simeq H^{1}(\mathbb{F},\mathsf{C}_{2}).

Finally, given an abelian group $G$, a $G$-grading on a superalgebra $\mathcal{A}$ corresponds to a homomorphism of group schemes

[TABLE]

where $G^{D}$ is the Cartier dual to the constant group scheme given by $G$. (See [EK13] for the basic definitions and facts on gradings.)

The classification of $G$-gradings up to isomorphism on $\mathsf{K}_{3}\times\mathsf{K}_{3}$ (or $\mathsf{J}(W)\times\mathsf{J}(W)$) is an easy exercise (Proposition 3.2), and it follows at once from this the classification of $G$-gradings, up to isomorphism, on $\mathsf{K}_{10}$ (Theorem 3.5), thus extending and simplifying widely the results in [CDM10].

1. The group-scheme of automorphisms

Consider the two-dimensional vector space $W=(\mathsf{K}_{3})_{\bar{1}}$, which is endowed with the nonzero skew-symmetric bilinear form $(\,\mid\,):W\times W\rightarrow\mathbb{F}$. The special linear group-scheme $\mathbf{SL}(W)$ coincides with the symplectic group scheme $\mathbf{Sp}(W)$, which is, up to isomorphism, the group-scheme of automorphisms of $\mathsf{K}_{3}$ (or of $\mathsf{J}(W)$).

The constant group scheme $\mathsf{C}_{2}$ acts on $\mathbf{SL}(W)\times\mathbf{SL}(W)$ by swapping the arguments, and hence we get a natural semidirect product \bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}. The group isomorphism $\Phi$ in [ELS07, p. 3809] extends naturally to a homomorphism of affine group-schemes:

[TABLE]

for any $R$ in $\operatorname{\mathrm{Alg_{\mathbb{F}}}}$, $f,g\in\mathbf{SL}(W)(R)$ (i.e., $f,g\in\mathsf{End}_{R}(W\otimes_{\mathbb{F}}R)\simeq M_{2}(R)$ of determinant $1$), $v,w\in W_{R}:=W\otimes_{\mathbb{F}}R$, $x,y\in(\mathsf{K}_{3})_{R}$. (Note that a representation of a constant group scheme $\rho:\mathsf{G}\rightarrow\mathbf{GL}(V)$ is determined by its behavior over $\mathbb{F}$: $\rho_{\mathbb{F}}:\mathsf{G}\rightarrow GL(V)$.)

Theorem 1.1**.**

$\Phi$* is an isomorphism of affine group-schemes.*

Proof.

If $\overline{\mathbb{F}}$ denotes an algebraic closure of $\mathbb{F}$, then $\Phi_{\overline{\mathbb{F}}}$ is a group isomorphism [ELS07, Theorem 3.3]. Since \bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}\simeq\bigl{(}\mathbf{SL}_{2}\times\mathbf{SL}_{2}\bigr{)}\rtimes\mathsf{C}_{2} is smooth, it is enough to prove that the differential $\textup{d}\Phi$ is bijective (see, for instance [EK13, Theorem A.50]). The Lie algebra of \bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2} is ${\mathfrak{sl}}(W)\times{\mathfrak{sl}}(W)$, while the Lie algebra of \operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{10}\bigr{)} is the even part of its Lie superalgebra of derivations, which is again, up to isomorphism, ${\mathfrak{sl}}(W)\times{\mathfrak{sl}}(W)$ identified naturally with a subalgebra of \mathsf{End}_{\mathbb{F}}\bigl{(}\mathsf{K}_{10}\bigr{)} (see [BE02, Theorem 2.8]). Moreover, with the natural identifications, $\textup{d}\Phi$ is the identity map. ∎

The last result in this section is the simple observation that \bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2} is also the group-scheme of automorphisms of the Jordan superalgebra $\mathsf{K}_{3}\times\mathsf{K}_{3}$ and $\mathsf{J}(W)\times\mathsf{J}(W)$. This will be instrumental in the next section. Its proof is straightforward.

Proposition 1.2**.**

The natural transformations defined by:

[TABLE]

for $v,w\in W$ and $x,y\in\mathsf{K}_{3}$, and

[TABLE]

for $v,w\in W$ and $x,y\in\mathsf{J}(W)$, are isomorphisms of group-schemes.

2. Twisted forms

The set of the isomorphism classes of twisted forms of $\mathsf{K}_{10}$ is identified with the pointed set H^{1}\bigl{(}\mathbb{F},\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{K}_{10}\bigr{)}\bigr{)}. (This is H^{1}\bigl{(}\overline{\mathbb{F}}/\mathbb{F},\operatorname{\mathbf{Aut}}(\mathsf{K}_{10}\bigr{)}\bigr{)} in the notation of [Wat79].)

The key to the classification of twisted forms of $\mathsf{K}_{10}$ is the following result:

Theorem 2.1**.**

The canonical projection \pi:\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}\rightarrow\mathsf{C}_{2} induces a bijection of pointed spaces \pi_{*}:H^{1}\bigl{(}\mathbb{F},\operatorname{\mathbf{Aut}}(\mathsf{K}_{10}\bigr{)}\bigr{)}\rightarrow H^{1}(\mathbb{F},\mathsf{C}_{2}).

Proof.

First, the natural section \iota:\mathsf{C}_{2}\rightarrow\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2} satisfies $\pi\circ\iota=\mathsf{id}$, so $\pi_{*}\circ\iota_{*}=\mathsf{id}$, and $\pi_{*}$ is surjective.

Second, the short exact sequence

[TABLE]

induces an exact sequence (see [Wat79, §18.1]) in cohomology:

[TABLE]

Since $H^{1}(\mathbb{F},\mathbf{SL}_{2})$ is trivial, this gives an exact sequence

[TABLE]

but, in principle, this does not prove that $\pi_{*}$ is a bijection. It just says that the only element in H^{1}\bigl{(}\mathbb{F},\bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2}\bigr{)} that is sent by $\pi_{*}$ to the trivial element in $H^{1}(\mathbb{F},\mathsf{C}_{2})$ is the trivial element.

Through the isomorphisms in the previous section, it is enough to prove that $\pi$ induces a bijection of pointed sets \pi_{*}:H^{1}\bigl{(}\mathbb{F},\operatorname{\mathbf{Aut}}\bigl{(}\mathsf{J}(W)\times\mathsf{J}(W)\bigr{)}\bigr{)}\rightarrow H^{1}(\mathbb{F},\mathsf{C}_{2}).

Write $\mathcal{J}=\mathsf{J}(W)\times\mathsf{J}(W)$ and identify \bigl{(}\mathbf{SL}(W)\times\mathbf{SL}(W)\bigr{)}\rtimes\mathsf{C}_{2} with $\operatorname{\mathbf{Aut}}(\mathcal{J})$ by means of $\Psi^{2}$ in Proposition 1.2. Then $\mathcal{J}_{\bar{0}}=\mathbb{F}\times\mathbb{F}$, and $\pi$ corresponds to the restriction map $\operatorname{\mathbf{Aut}}(\mathcal{J})\rightarrow\operatorname{\mathbf{Aut}}(\mathcal{J}_{\bar{0}})$, and the induced map in cohomology $\pi_{*}:H^{1}(\mathbb{F},\operatorname{\mathbf{Aut}}(\mathcal{J}))\rightarrow H^{1}(\mathbb{F},\operatorname{\mathbf{Aut}}(\mathcal{J}_{\bar{0}}))$ corresponds to the map that takes the isomorphism class of a twisted form $\mathcal{H}$ of $\mathcal{J}$ to the isomorphism class of $\mathcal{H}_{\bar{0}}$. If $\mathcal{H}_{\bar{0}}$ is trivial ($\mathcal{H}_{\bar{0}}\simeq\mathbb{F}\times\mathbb{F}$) then, by the above, $\mathcal{H}$ is isomorphic to $\mathcal{J}$. Otherwise, $\mathcal{H}_{\bar{0}}$ is a quadratic separable field extension of $\mathbb{F}$, so that $\mathcal{H}_{\bar{1}}$ is a two-dimensional vector space over $\mathcal{H}_{\bar{0}}$, and the multiplication $\mathcal{H}_{\bar{1}}\times\mathcal{H}_{\bar{1}}\rightarrow\mathcal{H}_{\bar{0}}$ is $\mathcal{H}_{\bar{0}}$-bilinear and anticommutative, and hence uniquely determined. It turns out that $\mathcal{H}$ is isomorphic to $\mathsf{J}(W)\otimes_{\mathbb{F}}\mathcal{H}_{\bar{0}}$. This proves the injectivity of $\pi_{*}$. ∎

Corollary 2.2**.**

Any twisted form of $\mathsf{K}_{10}$ splits after a quadratic separable field extension, and for any such field extension $\mathbb{K}$ of $\mathbb{F}$, there is a unique (up to isomorphism) nontrivial twisted form of $\mathsf{K}_{10}$ which splits over $\mathbb{K}$.

If $\mathbb{K}$ is a quadratic separable field extension of $\mathbb{F}$, the unique nontrivial twisted form of $\mathsf{K}_{10}$ that splits over $\mathbb{K}$ corresponds to the nontrivial element in $H^{1}(\mathbb{K}/\mathbb{F},\mathsf{C}_{2})$, and hence if the Galois group $\textrm{Gal}(\mathbb{K}/\mathbb{F})$ is generated by $\sigma$ ($\sigma^{2}=\mathsf{id}$), this twisted form is given by

[TABLE]

where $\tau$ is given in (1.1).

Hence, with $\mathbb{K}=\mathbb{F}(\alpha)$, $\alpha^{2}\in\mathbb{F}$, $\sigma(\alpha)=-\alpha$, so if $\{u,v\}$ is a symplectic basis of $W=(\mathsf{K}_{3})_{\bar{1}}$,

[TABLE]

Remark 2.3**.**

• •

Both $\mathsf{K}_{3}$ and $\mathsf{J}(W)$ are rigid: any twisted form of any of these superalgebras is isomorphic to it. This is because $\operatorname{\mathbf{Aut}}(\mathsf{K}_{3})\simeq\operatorname{\mathbf{Aut}}(\mathsf{J}(W))\simeq\mathbf{SL}(W)\simeq\mathbf{SL}_{2}$.

• •

The proof of Theorem 2.1 shows that the twisted forms of $\mathsf{K}_{3}\times\mathsf{K}_{3}$ (respectively $\mathsf{J}(W)\times\mathsf{J}(W)$) are, up to isomorphism, the superalgebras $\mathsf{K}_{3}\otimes_{\mathbb{F}}\mathbb{K}$ (resp. $\mathsf{J}(W)\otimes_{\mathbb{F}}\mathbb{K}$), where $\mathbb{K}$ is quadratic étale algebra over $\mathbb{F}$, and any two such forms are isomorphic if and only if so are the corresponding étale algebras.

3. Gradings

Given an abelian group $G$, a $G$-grading on a superalgebra $\mathcal{A}=\mathcal{A}_{\bar{0}}\oplus\mathcal{A}_{\bar{1}}$ is a decomposition into a direct sum of subspaces: $\mathcal{A}=\bigoplus_{g\in G}\mathcal{A}_{g}$, such that $\mathcal{A}_{g}\mathcal{A}_{h}\subseteq\mathcal{A}_{gh}$ for any $g,h\in G$, and each homogeneous component is a subspace in the ‘super’ sense of $\mathcal{A}$: $\mathcal{A}_{g}=(\mathcal{A}_{g}\cap\mathcal{A}_{\bar{0}})\oplus(\mathcal{A}_{g}\cap\mathcal{A}_{\bar{1}})$. We will write $\deg(x)=g$ in case $0\neq x\in\mathcal{A}_{g}$. Two $G$-gradings $\Gamma:\mathcal{A}=\bigoplus_{g\in G}\mathcal{A}_{g}$ and $\Gamma^{\prime}=\bigoplus_{g\in G}\mathcal{A}^{\prime}_{g}$, are isomorphic if there is an automorphism $\varphi\in\mathsf{Aut}(\mathcal{A})$ such that $\varphi(\mathcal{A}_{g})=\mathcal{A}^{\prime}_{g}$ for any $g\in G$.

A grading by $G$ is equivalent to a homomorphism of affine group schemes $G^{D}\rightarrow\operatorname{\mathbf{Aut}}(\mathcal{A})$ (see [EK13]), and this shows that two superalgebras with isomorphic group-schemes of automorphisms have equivalent classifications of $G$-gradings up to isomorphism. Therefore, in order to classify gradings on $\mathsf{K}_{10}$ it is enough to classify gradings on $\mathsf{K}_{3}\times\mathsf{K}_{3}$ (or $\mathsf{J}(W)\times\mathsf{J}(W)$), and this is straightforward.

Actually, fix a symplectic basis $\{u,v\}$ of $W=(\mathsf{K}_{3})_{\bar{1}}$.

Definition 3.1**.**

Given an abelian group $G$, consider the following gradings ($e$ denotes the neutral element of $G$):

• •

For $g_{1},g_{2}\in G$, denote by $\Gamma^{1}(G;g_{1},g_{2})$ the $G$-grading given by:

[TABLE]

• •

For $g,h\in G$ with $h^{2}=e\neq h$, denote by $\Gamma^{2}(G;g,h)$ the $G$-grading given by:

[TABLE]

Proposition 3.2**.**

Any grading on $\mathsf{K}_{3}\times\mathsf{K}_{3}$ by the abelian group $G$ is isomorphic to either $\Gamma^{1}(G;g_{1},g_{2})$ or to $\Gamma^{2}(G;g,h)$ (for some $g_{1},g_{2}$ or $g,h$ in $G$).

Moreover, no grading of the first type ($\Gamma^{1}(G;g_{1},g_{2})$) is isomorphic to a grading of the second type ($\Gamma^{2}(G;g,h)$), and

• •

$\Gamma^{1}(G,g_{1},g_{2})$* is isomorphic to $\Gamma^{1}(G;g_{1}^{\prime},g_{2}^{\prime})$ if and only if the sets*

$\{g_{1},g_{1}^{-1},g_{2},g_{2}^{-1}\}$* and $\{g_{1}^{\prime},(g_{1}^{\prime})^{-1},g_{2}^{\prime},(g_{2}^{\prime})^{-1}\}$ coincide.*

• •

$\Gamma^{2}(G;g,h)$* is isomorphic to $\Gamma^{2}(G;g^{\prime},h^{\prime})$ if and only if $h^{\prime}=h$ and $g^{\prime}\in\{g,gh,g^{-1},g^{-1}h\}$.*

Proof.

Any $G$-grading on $\mathcal{J}:=\mathsf{K}_{3}\times\mathsf{K}_{3}$ induces a $G$-grading on $\mathcal{J}_{\bar{0}}$, which is isomorphic to $\mathbb{F}\times\mathbb{F}$, and hence we are left with two cases:

1. (1)

The grading on $\mathcal{J}_{\bar{0}}$ is trivial, i.e., $\mathcal{J}_{\bar{0}}$ is contained in the homogeneous component $\mathcal{J}_{e}$. Then, with $W=(\mathsf{K}_{3})_{\bar{1}}$, both $W\times 0=(a,0)\mathcal{J}_{\bar{1}}$ and $0\times W=(0,a)\mathcal{J}_{\bar{1}}$ are graded subspaces of $\mathcal{J}_{\bar{1}}$. Hence we can take bases $\{u_{i},v_{i}\}$ of $W$, $i=1,2$, such that $\{(u_{1},0),(v_{1},0),(0,u_{2}),(0,v_{2})\}$ is a basis of $\mathcal{J}_{\bar{1}}$ consisting of homogeneous elements. We can adjust $v_{i}$, $i=1,2$, so that $(u_{i}\mid v_{i})=1$. If $\deg(u_{i},0)=g_{i}$, $i=1,2$, the grading is isomorphic to $\Gamma^{1}(G;g_{1},g_{2})$. 2. (2)

The grading on $\mathcal{J}_{\bar{0}}$ is not trivial. Then there is an element $h\in G$ of order $2$ such that $\deg(a,a)=e$ and $\deg(a,-a)=h$. (Note that $(a,a)$ is the unity element of $\mathcal{J}_{\bar{0}}\,(\simeq\mathbb{F}\times\mathbb{F})$, so it is always homogeneous of degree $e$.) As $\mathcal{J}_{\bar{0}}=(\mathcal{J}_{\bar{1}})^{2}$, there are homogeneous elements $(u_{1},u_{2}),(v_{1},v_{2})\in\mathcal{J}_{\bar{1}}$ such that $(u_{1},u_{2})(v_{1},v_{2})=(a,a)$. If $g=\deg(u_{1},u_{2})$, this grading is isomorphic to $\Gamma^{2}(G;g,h)$.

The conditions for isomorphism are clear. ∎

Any grading $\Gamma^{1}(G;g_{1},g_{2})$ is a coarsening of the grading \Gamma^{1}\bigl{(}\mathbb{Z}^{2};(1,0),(0,1)\bigr{)}, while any grading $\Gamma^{2}(G;g,h)$ is a coarsening of the grading \Gamma^{2}\bigl{(}\mathbb{Z}\times\mathbb{Z}_{2};(1,\bar{0}),(0,\bar{1})\bigr{)}, where $\mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z}$. As an immediate consequence, we obtain the next result.

Corollary 3.3**.**

Up to equivalence, there are exactly two fine gradings on $\mathsf{K}_{3}\times\mathsf{K}_{3}$: \Gamma^{1}\bigl{(}\mathbb{Z}^{2};(1,0),(0,1)\bigr{)} and \Gamma^{2}\bigl{(}\mathbb{Z}\times\mathbb{Z}_{2};(1,\bar{0}),(0,\bar{1})\bigr{)}.

Proposition 3.2 and Corollary 3.3 have completely similar counterparts for $\mathsf{J}(W)\times\mathsf{J}(W)$.

In order to transfer these results to Kac’s superalgebra $\mathsf{K}_{10}$, take into account that $\mathsf{K}_{10}$ is generated by its odd part, as \bigl{(}(\mathsf{K}_{10})_{\bar{1}}\bigr{)}^{2}=(\mathsf{K}_{10})_{\bar{0}}, so any grading is determined by its restriction to the odd part, and use the commutativity of the diagram

[TABLE]

where the vertical arrows are given by the restrictions to the odd parts, and the bottom isomorphism is given by the natural identification $W\times W\rightarrow(W\otimes a)\oplus(a\otimes W)$, $(v,w)\mapsto v\otimes a+a\otimes w$.

Thus Definition 3.1 transfers to $\mathsf{K}_{10}$ as follows:

Definition 3.4**.**

Given an abelian group $G$, consider the following $G$-gradings on $\mathsf{K}_{10}$:

• •

For $g_{1},g_{2}\in G$, denote by $\Gamma^{1}_{\mathsf{K}_{10}}(G;g_{1},g_{2})$ the $G$-grading determined by:

[TABLE]

• •

For $g,h\in G$ with $h^{2}=e\neq h$, denote by $\Gamma^{2}_{\mathsf{K}_{10}}(G;g,h)$ the $G$-grading determined by:

[TABLE]

And Proposition 3.2 and Corollary 3.3 are now transferred easily to $\mathsf{K}_{10}$:

Theorem 3.5**.**

Any grading on $\mathsf{K}_{10}$ by the abelian group $G$ is isomorphic to either $\Gamma^{1}_{\mathsf{K}_{10}}(G;g_{1},g_{2})$ or to $\Gamma^{2}_{\mathsf{K}_{10}}(G;g,h)$ (for some $g_{1},g_{2}$ or $g,h$ in $G$).

No grading of the first type is isomorphic to a grading of the second type, and

• •

$\Gamma^{1}_{\mathsf{K}_{10}}(G;g_{1},g_{2})$* is isomorphic to $\Gamma^{1}_{\mathsf{K}_{10}}(G;g_{1}^{\prime},g_{2}^{\prime})$ if and only if the sets*

$\{g_{1},g_{1}^{-1},g_{2},g_{2}^{-1}\}$* and $\{g_{1}^{\prime},(g_{1}^{\prime})^{-1},g_{2}^{\prime},(g_{2}^{\prime})^{-1}\}$ coincide.*

• •

$\Gamma^{2}_{\mathsf{K}_{10}}(G;g,h)$* is isomorphic to $\Gamma^{2}_{\mathsf{K}_{10}}(G;g^{\prime},h^{\prime})$ if and only if $h^{\prime}=h$ and $g^{\prime}\in\{g,gh,g^{-1},g^{-1}h\}$.*

Moreover, there are exactly two fine gradings on $\mathsf{K}_{10}$ up to equivalence, namely \Gamma^{1}_{\mathsf{K}_{10}}\bigl{(}\mathbb{Z}^{2};(1,0),(0,1)\bigr{)} and \Gamma^{2}_{\mathsf{K}_{10}}\bigl{(}\mathbb{Z}\times\mathbb{Z}_{2};(1,\bar{0}),(0,\bar{1})\bigr{)}.

Remark 3.6**.**

Any grading on $\mathsf{K}_{10}$ by an abelian group $G$ extends naturally to a grading by either $\mathbb{Z}\times G$ or $\mathbb{Z}_{2}^{2}\times G$ on the exceptional simple Lie superalgebra $F(4)$, which is obtained from $\mathsf{K}_{10}$ using the well-known Tits-Kantor-Koecher construction. However, not all gradings on $F(4)$ are obtained in this way.