Abstract
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} +, which in turn is an extension of a rank r finite-dimensional semisimple simply laced Lie algebra mathfrak{g} . The algebras are specified by mathfrak{g} 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} is proven. An accompanying paper [1] describes the application of tensor hierarchy algebras to the gauge structure and dynamics in models of extended geometry.
Highlights
In ref. [3], we introduced a set of generators and relations for W (n) and S(n), with an antisymmetry between positive and negative levels, by modifying the presentation of the contragredient Lie superalgebra A(0, n − 1) = sl(1|n)
We further develop the recently proposed definition of tensor hierarchy algebras in terms of generators and relations encoded in a Dynkin diagram
We apply it to cases where a grey node is added to the Dynkin diagram of a rank r +1 Kac-Moody algebra g+, which in turn is an extension of a rank r finite-dimensional semisimple laced Lie algebra g
Summary
[8] is that we label the r nodes in the Dynkin diagram of g (or the corresponding simple roots) by an index i that takes the values i = 2, . Since g is a subalgebra at (p, q) = (0, 0), the subspace at any definite pair of integers (p, q) forms an g-module Our notation for these modules is given in table 1. We introduce basis elements EM and F M for the odd subspaces at (p, q) = (1, 0) and (p, q) = (−1, 0), respectively, which form the g-modules R(−λ) and R(λ). At levels p = 0, ±1 (the local part of the Lie superalgebra with respect to this Z-grading) we have the basis elements shown in table 2.
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