SPECIAL VINBERG CONES

Abstract

The paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of "positive definite matrices" in the Vinberg commutative algebra H-n, of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n x n matrices A = (a(ij)), where a(ij) is an element of R, the entry a(ij) for i < j belongs to some Euclidean vector space (V-ij, g) and a(ij) = a(ij)* = g(a(ij), .) is an element of V-ij* belongs to the dual space V-ij*. The multiplication of T-Hermitian matrices is defined by a system of "isometric" bilinear maps V-ij x V-jk -> V-ij, i < j < k, such that vertical bar a(i)(j) . a(jk)vertical bar, = vertical bar a(ij)vertical bar, . vertical bar a(jk)vertical bar, a(lm) is an element of V-lm. For n = 2, the Hermitian T-algebra H-2 = 9 H-2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space R-1,R-1 circle plus V. A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra 9 6(V, S) associated to a Clifford Cl(V)-module S together with an "admissible" Euclidean metric g(s). We generalize the construction of rank 2 Vinberg algebras H-2 (V) and special Vinberg algebras H-3 (V, S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S-0 circle plus S-1 is a Z(2)-graded Clifford Cl(V)-module with an admissible pseudoEuclidean metric. The associated cone V is a homogeneous, but not convex cone in H-m, m = 2, 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez' quantum-mechanical interpretation of the Vinberg cone V-2 subset of H-2(V) to the special rank 3 case.