Tensor hierarchy algebras and extended geometry. Part I. Construction of the algebra
Martin Cederwall, Jakob Palmkvist
Abstract
A bstract Tensor hierarchy algebras constitute a class of non-contragredient Lie superalgebras, whose finite-dimensional members are the “Cartan-type” Lie superalgebras in Kac’s classification. They have applications in mathematical physics, especially in extended geometry and gauged supergravity. We further develop the recently proposed definition of tensor hierarchy algebras in terms of generators and relations encoded in a Dynkin diagram (which coincides with the diagram for a related Borcherds superalgebra). We apply it to cases where a grey node is added to the Dynkin diagram of a rank r + 1 Kac-Moody algebra $$ \mathfrak{g} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> + , which in turn is an extension of a rank r finite-dimensional semisimple simply laced Lie algebra $$ \mathfrak{g} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> . The algebras are specified by $$ \mathfrak{g} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> together with a dominant integral weight λ . As a by-product, a remarkable identity involving representation matrices for arbitrary integral highest weight representations of $$ \mathfrak{g} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> is proven. An accompanying paper [1] describes the application of tensor hierarchy algebras to the gauge structure and dynamics in models of extended geometry.