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Lie Groups Quantum Mathematics Seminar

Generalized Cartan superalgebras

Location:  Hill 705
Date & time: Friday, 28 October 2016 at 12:00PM -

Jakob Palmkvist, Texas A&M

 

"Generalized Cartan superalgebras"

Time: 12:00 PM
Location: Hill 705
Abstract: In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, A(0,n)=sl(1|n+1) can be constructed by adding a "gray'' node to the Dynkin diagram of A_n=sl(n+1), corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is W(n), the derivation algebra of the differential algebra in n dimensions, which in turn is generated by the differentials dx^m under the wedge product. I will in my talk relate A(0,n) and W(n) to each other, and show that also W(n) can be constructed from the Dynkin diagram of A(0,n). I will then aim to generalize the result and consider the Kac-Moody algebras E_n, each of which can be extended to an infinite-dimensional Borcherds superalgebra, in the same way as A_n can be extended to A(0,n). At least for n ? 8 there is also a different but related infinite-dimensional Lie superalgebra, in the same way as W(n) is related to A(0,n), which has been named tensor hierarchy algebra because of its applications to gauged supergravity. I will explain these applications, as well as connections to so called exceptional geometry.

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