BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Marco Farinati (University of Buenos Aires\, Argentina) DTSTART:20250602T150000Z DTEND:20250602T160000Z DTSTAMP:20260423T035753Z UID:ENAAS/125 DESCRIPTION:Title: The Tits construction for SHORT $\\mathfrak{sl}_2$-super-structures.\n by Marco Farinati (University of Buenos Aires\, Argentina) as part of Euro pean Non-Associative Algebra Seminar\n\n\nAbstract\nWhen 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 "alg ebra" with an operation $m:A\\otimes A\\to A$\, one may decompose $A=\\opl us_n V_n\\otimes M_n$\, where $V_n$ is the irreducible $\\mathfrak{sl}_2$- module of highest weight $n$ (and dimension $n+1$) and $M_n$ is just a "mu ltiplicity" vector space. If the operation $m$ is $\\mathfrak{sl}_2$-linea r\, then\, a priori\, several restrictions for the operation $m$ can be d educed\, and algebraic identities (e.g. associativity\, Jacobi identity\, symmetry\, antisymmetry\,...) of $A$ can be translated into operations and identities on the $M_n$'s.\n\nIn case the only isotypical components that appear are the trivial ($V_0$) and the adjoint ($V_2$)\, then the $\\math frak{sl}_2$ structure is called VERY SHORT. In case the only isotypical co mponents that appear are the trivial ($V_0$)\, the adjoint ($V_2$)\, and t he defining 2-dimensional representation ($V=\\mathcal{C}^2=V_1$) then the $\\mathfrak{sl}_2$ structure is called SHORT.\n\nThe case of VERY SHORT $ \\mathfrak{sl}_2$ Lie algebras is a classical object studied by Tits and l eads to Jordan algebras. There is also a kind of reciprocal knowledge like TKK-construction (TKK from Tits-Kantor-Koecher): given a Jordan algebra o ne can assign to it a natural (but not functorial) Lie algebra. The functo riality problem was solved by Alison and Gao\, we call it the TAG construc tion.\n\nWhen the natural representation $V$ also appears\, Elduque et al. made the "translation" from Lie axioms into an object called "Jordan trip le".\n\nIf the algebra is a SUPER Lie algebra\, but VERY SHORT\, then both TKK and TAG constructions were generalized to the super case by Barbier a nd Shang.\n\nIn this talk\, I will show that TKK and TAG constructions can be extended to the SHORT super case. That is\, one can make a constructio n 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 jus t a Jordan algebra\, but super\, we generalize Shang's work for super Jord an algebras. On the way of doing that\, adapting to the short case an intr insic description of Shang of the Jordan algebra associated to a very shor t Lie algebra\, we discover two different possible ternary Jordan structur e on the "Jordan data" associated to a short Lie algebra: one was consider ed previously by Elduque et al (in the non super case) and the other can b e described in a simpler way using the Lie-intrinsic description\, and it happens that this second one is more suitable for the functorial generaliz ation of TKK and TAG construction for short (and super) construction.\n LOCATION:https://researchseminars.org/talk/ENAAS/125/ END:VEVENT END:VCALENDAR