Special Vinberg cones of rank 4

A bstract E.B. Vinberg developed a theory of homogeneous convex cones $$C\subset V={\mathbb{R}}^{n}$$ , which has many applications. He gave a construction of such cones in terms of non-associative rank n matrix T-algebras $$\mathcal{T}$$ , that consist of vector-valued n × n matrices X = || x ij ||, x ij ∈ V ij where V ij are Euclidean vector spaces. The multiplication in a T-algebra is determined by a system of isometric maps V ij × V jk → V ik , s.t. | v ij · v jk | = | v ij | · | v jk | that satisfies some axioms. A T-algebra is determined by its associative subalgebra of upper triangular matrices $$\mathcal{G}$$ or its niladical $$\mathcal{N}$$ , called the Nil-algebra. The connected Lie group $$G\subset \mathcal{G}$$ of the upper triangular (non-degenerate) matrices acts in the vector space $$Her{m}_{n}\subset \mathcal{T}$$ of Hermitian matrices and the orbit C = G ( I ) ⊂ Herm n 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 $$\mathcal{N}$$ a directed acyclic graph $$\Gamma =\Gamma (\mathcal{N})$$ 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.