Iwasawa nilpotency degree of non compact symmetric cosets in N-extended Supergravity

TL;DR

This paper investigates the polynomial structure of coset representatives in non-compact symmetric spaces within supergravity, linking algebraic properties to physical scalar and charge representations across different dimensions.

Contribution

It computes the Iwasawa nilpotency degree for all faithful Lie algebra representations in supergravity, connecting algebraic structures to physical scalar and charge couplings.

Findings

Nilpotency degree depends only on space-time dimension in magic supergravity.

Results align with Tits-Satake projections for symmetric spaces.

Provides algebraic insights relevant for supergravity ultraviolet behavior.

Abstract

We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU(2)_P subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian. We consider symmetric scalar manifolds in N-extended supergravity in various space-time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits-Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space-time dimension of the theory. These results should be helpful within a deeper investigation of the corresponding supergravity theory, e.g. in studying ultraviolet properties of maximal supergravity in various dimensions.