Special Vinberg cones of rank 4

Journal of High Energy Physics. 2026. Vol. 2026. No. 1. Article 1.

Alekseevsky D. V., Osipov P.

E.B. Vinberg developed a theory of homogeneous convex cones , which has many applications. He gave a construction of such cones in terms of non-associative rank n matrix T-algebras , that consist of vector-valued n × n matrices X = ||xij||, xij ∈ Vij where Vij are Euclidean vector spaces. The multiplication in a T-algebra is determined by a system of isometric maps Vij × Vjk → Vik, s.t. |vij · vjk| = |vij| · |vjk| that satisfies some axioms. A T-algebra is determined by its associative subalgebra of upper triangular matrices or its niladical , called the Nil-algebra. The connected Lie group of the upper triangular (non-degenerate) matrices acts in the vector space

of Hermitian matrices and the orbit C = G(I) ⊂ Hermn of the identity matrix I is a convex cone with a simply transitive action of G. Conversely, any homogeneous convex cone is obtained by this construction.

Generalizing the notion of rank 3 Clifford T-algebra [1, 2], we define notions of rank n special T-algebra and Clifford Nil-algebra, which define a special Vinberg cone. We associate with a Clifford Nil-algebra

a directed acyclic graph of diameter 1 and show that Clifford Nil-algebras with given graph Γ bijectively correspond to its admissible equipments. This gives an effective method of classification of Clifford Nil-algebras and associated special Vinberg cones. We apply this approach for explicit classification of rank 4 special Vinberg cones.

Keywords: Supergravity Models, Differential and Algebraic Geometry

Publication based on the results of: The Methods of Mirror Symmetry in Birational Geometry, Mathematical Physics and Arithmetic (2025)