Generators and relations for Lie superalgebras of Cartan type

Abstract

We give an analog of a Chevalley-Serre presentation for the Lie superalgebras W pnq and Spnq of Cartan type. These are part of a wider class of Lie superalgebras, the so-called tensor hierarchy algebras, denoted W pgq and Spgq, where g denotes the Kac-Moody algebra Ar, Dr or Er. Then W pAn´1q and SpAn´1q are the Lie superalgebras W pnq and Spnq. The algebras W pgq and Spgq are constructed from the Dynkin diagram of the Borcherds-Kac-Moody superalgebras Bpgq obtained by adding a single grey node (representing an odd null root) to the Dynkin diagram of g. We redefine the algebras W pArq and SpArq in terms of Chevalley generators and defining relations. We prove that all relations follow from the defining ones at level ě´2. The analogous definitions of the algebras in the D-and E-series are given. In the latter case the full set of defining relations is conjectured.

Tensor hierarchy algebras and extended geometry. Part II. Gauge structure and dynamics

“…It has been shown to have close relations to certain classes of Lie superalgebras [2][3][4][5]. The purpose of the present paper is to demonstrate how indeed the recently invented tensor hierarchy algebras (THA's) [6,7] are to be seen as the algebraic structure responsible for and underlying extended geometry, in its most general setting. Besides unifying double geometry [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] and exceptional geometry , one of the advantages of the framework of extended geometry is that it opens a window to situations with infinite-dimensional structure groups [42][43][44].…”

Generators and relations for (generalised) Cartan type superalgebras

“…We then define W pgq as W pgq " Ă W pgq{J, where J is the maximal ideal of Ă W pgq intersecting the local part trivially. The following theorem summarises the main results of [1] (where the proof can be found).…”

“…This talk, given by JP at "The 32nd International Colloquium on Group Theoretical Methods in Physics (Group32)" in Prague, [9][10][11][12][13] July, 2018, is based on [1], where more details and references can be found.…”

L ∞ algebras for extended geometry

“…[34][35][36]48]. Then one needs a tensor hierarchy algebra [49,50], a generalisation of the Cartan-type superalgebras W (n) and S(n) in Kac's classification [51]. Tensor hierarchy algebras are non-contragredient superalgebras, and therefore a priori not defined by standard Chevalley-Serre relations from a Dynkin diagram.…”

“…In ref. [50] we presented a set of generators and relations for the (finite-dimensional) tensor hierarchy algebras W (r + 1) = W (A r ) and S(r + 1) = S(A r ), based on the same Dynkin diagram as that of B(A r ). The straightforward generalisation of these relations seems to provide a good definition of W (g) and S(g) in general (see the talk by JP at the present meeting [52]).…”