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 = ||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 \(\mathcal{G}\) 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) ⊂ 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 \(\mathcal{N}\) 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.
