Special Vinberg cones of rank 4

Abstract

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 $\cal{T}$, that consist of vector-valued $n \times n$ matrices $X = ||x_{ij}||, \, x_{ij} \in V_{ij} $ where $V_{ij}$ are Euclidean vector spaces. The multiplication in a T-algebra is determined by a system of isometric maps $V_{ij} \times V_{jk} \to V_{ik}$, s.t. $|v_{ij}\cdot v_{jk}| = |v_{ij}|\cdot |v_{jk}|$ that satisfies some axioms. A T-algebra is determined by its associative subalgebra of upper triangular matrices or its niladical $\mathcal{N}$, called the Nil-algebra. The connected Lie group $G$ of the upper triangular (non-degenerate) matrices acts in the vector space $Herm_n \subset\cal{T}$ of Hermitian matrices and the orbit $C = G(I)\subset Herm_n$ 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, 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 $\Gamma$ 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 in terms of admissible equipment.