Special Vinberg cones, invariant admissible cubics and special real manifolds

By Vinberg theory any homogeneous convex cone V may be realised as the cone of positive Hermitian matrices in a T -algebra of generalised matrices. The level hypersurfaces V_q ⊂ V of homogeneous cubic polynomials q with positive definite Hessian (symmetric) form g(q) := − Hess(log(q))|_{T V_q} are the special real manifolds. Such manifolds occur as scalar manifolds of the vector multiplets in N = 2, D = 5 supergravity and, through the r-map, correspond to Kaehler scalar manifolds in N = 2 D = 4 supergravity. We offer a simplified exposition of the Vinberg theory in terms of Nil-algebras (= the subalgebras of upper triangular matrices in Vinberg T -algebras) and we use it to describe all rational functions on a special Vinberg cone that are G_0- or G′- invariant, where G_0 is the unimodular subgroup of the solvable group G acting simply transitively on the cone, and G′ is the unipotent radical of G_0. The results are used to determine G_0- and G′-invariant cubic polynomials q that are admissible (i.e. such that the hypersurface V_q = {q = 1} ∩ V has positive definite Hessian form g(q)) for rank 2 and rank 3 special Vinberg cones. We get in this way examples of continuous families of non-homogeneous special real manifolds of cohomogeneity less than or equal to two.