The classification of simple complex Lie superalgebras of polynomial vector fields and their deformations

Abstract

We overview classifications of simple infinite-dimensional complex $\mathbb{Z}$-graded Lie (super)algebras of polynomial growth, and their deformations. A subset of such Lie (super)algebras consist of vectorial Lie (super)algebras whose elements are vector fields with polynomial, or formal power series, or divided power coefficients. A given vectorial Lie (super)algebra with a (Weisfeiler) filtration corresponding to a maximal subalgebra of finite codimension is called W-filtered; the associated graded algebra is called W-graded. Here, we correct several published results: (1) prove our old claim "the superization of É. Cartan's problem (classify primitive Lie algebras) is wild", (2) solve a tame problem: classify simple W-graded and W-filtered vectorial Lie superalgebras, (3) describe the supervariety of deformation parameters for the serial W-graded simple vectorial superalgebras, (4) conjecture that the exceptional simple vectorial superalgebras are rigid. We conjecture usefulness of our method in classification of simple infinite-dimensional vectorial Lie (super)algebras over fields of positive characteristic.

Theorems & Definitions (1)

• proof : Proof f this theorem occupies the main body of this article. Observe that real forms (classified in an interesting paper CaKa1) do not always survive regrading. Though ${\mathfrak{b}}_{\lambda}(2; r)$ and ${\mathfrak{h}}_{\lambda}(2|2; r)$ are isomorphic, we consider them separately because they are deforms of very distinct structures: the superalgebras with the odd and the even bracket, respectively. In Subsection \ref{['SS:2.22']} we show wherefrom the mysterious at this point isomorphism ${\mathfrak{h}}_{\lambda}(2|2)\simeq {\mathfrak{h}}_{-1-\lambda}(2|2)$ comes. The family ${\mathfrak{h}}_{\lambda}(2|2)\simeq {\mathfrak{b}}_{\lambda}(2; 2)$ should be considered as an exceptional one. The following table lists all W-gradings of serial simple vectorial Lie superalgebras, see Theorem \ref{['th7.3']}. One can directly verify that $\footnotesize {\mathfrak{vect}}(1|1)\simeq {\mathfrak{vect}}(1|1; 1);{\mathfrak{svect}}(2|1)\simeq {\mathfrak{le}}(2; 2); \; {\mathfrak{svect}}(2|1; 1)\simeq {\mathfrak{le}}(2){\mathfrak{sm}}(n)\simeq {\mathfrak{b}}_{2/(n-1)}(n);\text{in particular, ${\mathfrak{sm}}(2)\simeq {\mathfrak{b}}_{2}(2)$, and ${\mathfrak{sm}}(3)\simeq {\mathfrak{b}}_{1}(3)$; hence, ${\mathfrak{sm}}(3)$ is not simple};{\mathfrak{s}}{\mathfrak{le}}'(3)\simeq {\mathfrak{s}}{\mathfrak{le}}'(3; 3);{\mathfrak{b}}_{1/2}(2; 2)\simeq {\mathfrak{h}}_{1/2}(2|2)={\mathfrak{h}}(2|2);\quad {\mathfrak{h}}_{\lambda}(2|2)\simeq {\mathfrak{b}}_{\lambda}(2; 2);{\mathfrak{h}}_{\lambda}(2|2; 1)\simeq {\mathfrak{b}}_{\lambda}(2);\text{${\mathfrak{b}}_{1}'(2; {\rm Reg}_{{\mathfrak{b}}})\simeq {\mathfrak{le}}(2)$,}\text{${\mathfrak{le}}(2;{\rm Reg}_{{\mathfrak{b}}})\simeq {\mathfrak{b}}_1'(2)$},\text{${\mathfrak{b}}_{\infty}'(2; {\rm Reg}_{{\mathfrak{b}}}) \simeq {\mathfrak{b}}_{\infty}'(2)$};{\mathfrak{b}}_{a, b}(2; \mathop{\mathrm{Reg}}\nolimits_{{\mathfrak{b}}})\simeq {\mathfrak{b}}_{-b, -a}(2)\simeq {\mathfrak{b}}_{b, a}(2), \ \ \text{see eq.~}\ref{['reg4']};{\mathfrak{b}}_{\lambda}(2; {\rm Reg}_{{\mathfrak{b}}})\simeq {\mathfrak{b}}_{-1-\lambda}(2)\;\text{ and }\; {\mathfrak{h}}_{\lambda}(2|2)\simeq {\mathfrak{h}}_{-1-\lambda}(2|2), \ \ \text{see eq.~}\ref{['unmyst']}\text{; hence,}\text{the fundamental domain is either } \{\lambda\in{\mathbb C}\mid \mathop{\text{\rm Re}}\nolimits \lambda \geq -\frac{1}{2}\} \text{ or $\{\lambda\in{\mathbb C}\mid \mathop{\text{\rm Re}}\nolimits \lambda \leq -\frac{1}{2}\}$};\widetilde{{\mathfrak{s}}{\mathfrak{b}}}_{\mu}(2^{n-1}-1|2^{n-1})\simeq \widetilde{{\mathfrak{s}}{\mathfrak{b}}}_{\nu}(2^{n-1}-1|2^{n-1})\text{ for $n>2$ even and $\mu\nu\neq 0$},\widetilde{{\mathfrak{s}}{\mathfrak{b}}}_{\mu}(2^{n-1}|2^{n-1}-1)\simeq \widetilde{{\mathfrak{s}}{\mathfrak{b}}}_{\nu}(2^{n-1}|2^{n-1}-1)\text{ for $n>1$ odd}.$ Warning. Isomorphic abstract Lie superalgebras may be quite distinct as filtered or graded: e.g., regradings provide us with isomorphisms (ALSh): ${\mathfrak{k}}(1|2;1)\simeq {\mathfrak{vect}}(1|1)\simeq {\mathfrak{m}}(1;1).$ Observe that only one of the above three non-isomorphic graded algebras is W-graded over ${\mathbb C}$; the situation becomes more complicated over ${\mathbb R}$, cf. CaKa1. The standard grading of ${\mathfrak{k}}(1|2)$, though non-Weisfeiler, is often considered in physics papers. Observe that some real forms of ${\mathfrak{k}}(1|2k; k-1)$ are W-graded. For completeness, our table \ref{['1.4']} includes occasional finite-dimensional versions labeled by "FD" instead of the number. All the Lie superalgebras listed in Theorem \ref{['mainth']} are the results of Cartan prolongation, see Subsection $\ref{['2.6']}$ (or the derived algebras thereof), and therefore are determined by the terms ${\mathfrak{g}}_{i}$ with $i\leq 0$ (or $i\leq 1$ in some cases), see Subsection $\ref{['SS:1.3.3']}$. The "shape" of these terms might drastically vary with $n$ and $r$, though they can be united in the "families" \ref{['1.4']} and \ref{['1.10']}. The Lie superalgebras $\widetilde{{\mathfrak{svect}}}(0|n)$ and $\widetilde{{\mathfrak{s}}{\mathfrak{b}}}_{\mu}(2^{n-1}|2^{n-1}-1)$ depend on an odd parameter for $n$ odd. The exceptional regrading ${\rm Reg}_{{\mathfrak{b}}}$ of ${\mathfrak{b}}_{\lambda}(2)$ and the isomorphism ${\mathfrak{h}}_{\lambda}(2|2)\simeq {\mathfrak{b}}_{\lambda}(2; 2)$ determine an exceptional grading ${\rm Reg}_{{\mathfrak{h}}}$ of ${\mathfrak{h}}_{\lambda}(2|2)$. These regradings (${\rm Reg}_{{\mathfrak{b}}}$ and ${\rm Reg}_{{\mathfrak{h}}}$, see eq. \ref{['1.7']} and Subsection \ref{['SS:1.3.1']}) are irrelevant as far as the classification of simple Lie superalgebras is concerned, thanks to the isomorphism. On the other hand, these regradings contribute to the group of automorphisms of ${\mathfrak{h}}_{\lambda}(2|2)\simeq {\mathfrak{b}}_{\lambda}(2; 2)$. 5) Solution of Problem C is given by Theorem \ref{['thdefb']}, Theorem \ref{['ThRig']} on deformations of serial simple vectorial Lie superalgebras, and Conjecture \ref{['CjRig']} on rigidity of the exceptional simple vectorial Lie superalgebras. The following propositions classify the filtered deformations. For the W-graded exceptional simple vectorial Lie superalgebras, the isomorphism \ref{['1.5.5']} does not take place. The spaces $H^2({\mathfrak{g}}_-; {\mathfrak{g}})$ and $H^2({\mathfrak{h}}_-; {\mathfrak{h}})$ are always distinct in these cases: they consist of different ${\mathfrak{g}}_0$- and ${\mathfrak{h}}_0$-modules.