The Tits construction for SHORT $\mathfrak{sl}_2$-super-structures.

Abstract: When we have an action of $\mathfrak{sl}_2$ on a given structure, one may decompose it on its isotypic components. More concretely, if $A$ is a finite dimensional "algebra" with an operation $m:A\otimes A\to A$, one may decompose $A=\oplus_n V_n\otimes M_n$, where $V_n$ is the irreducible $\mathfrak{sl}_2$-module of highest weight $n$ (and dimension $n+1$) and $M_n$ is just a "multiplicity" vector space. If the operation $m$ is $\mathfrak{sl}_2$-linear, then, a priori, several restrictions for the operation $m$ can be deduced, and algebraic identities (e.g. associativity, Jacobi identity, symmetry, antisymmetry,...) of $A$ can be translated into operations and identities on the $M_n$'s.

In case the only isotypical components that appear are the trivial ($V_0$) and the adjoint ($V_2$), then the $\mathfrak{sl}_2$ structure is called VERY SHORT. In case the only isotypical components that appear are the trivial ($V_0$), the adjoint ($V_2$), and the defining 2-dimensional representation ($V=\mathcal{C}^2=V_1$) then the $\mathfrak{sl}_2$ structure is called SHORT.

The case of VERY SHORT $\mathfrak{sl}_2$ Lie algebras is a classical object studied by Tits and leads to Jordan algebras. There is also a kind of reciprocal knowledge like TKK-construction (TKK from Tits-Kantor-Koecher): given a Jordan algebra one can assign to it a natural (but not functorial) Lie algebra. The functoriality problem was solved by Alison and Gao, we call it the TAG construction.

When the natural representation $V$ also appears, Elduque et al. made the "translation" from Lie axioms into an object called "Jordan triple".

If the algebra is a SUPER Lie algebra, but VERY SHORT, then both TKK and TAG constructions were generalized to the super case by Barbier and Shang.

In this talk, I will show that TKK and TAG constructions can be extended to the SHORT super case. That is, one can make a construction beginning from a Jordan super triple (not just a Jordan algebra) and get a Lie super algebra. In case the Jordan triple is a usual one, we get a reciprocal construction to Elduque's one. In case the Jordan triple is just a Jordan algebra, but super, we generalize Shang's work for super Jordan algebras. On the way of doing that, adapting to the short case an intrinsic description of Shang of the Jordan algebra associated to a very short Lie algebra, we discover two different possible ternary Jordan structure on the "Jordan data" associated to a short Lie algebra: one was considered previously by Elduque et al (in the non super case) and the other can be described in a simpler way using the Lie-intrinsic description, and it happens that this second one is more suitable for the functorial generalization of TKK and TAG construction for short (and super) construction.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar