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 × n matrices A = (a_ij), where a_ii ∈ ℝ, the entry a_ij for i < j belongs to some Euclidean vector space (V_ij ; 𝔤) and \( {a}_{ji}={a}_{ij}^{\ast }=\mathfrak{g}\left({a}_{ij},\cdot \right)\in {V}_{ij}^{\ast } \) belongs to the dual space \( {V}_{ij}^{\ast }. \) The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps V_ij × V_jk → V_ij ; i < j < k, such that |a_ij ⋅ a_jk| = |a_ij| ⋅ |a_ik|, a_lm ∈ V_lm. For n = 2, the Hermitian T-algebra ℋ_n = ℋ₂ (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^1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ₃(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 ℋ₂(V ) and special Vinberg algebras ℋ₃(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S₀ ⊕ S₁ is a ℤ₂-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean 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 𝒱₂ ⊂ ℋ₂(V ) to the special rank 3 case.

本文致力于均匀凸锥的Vinberg理论的推广。这样的锥体被描述为一组在Vinberg交换代数ℋ“正定矩阵”的_Ñ埃尔米特T-矩阵。这些代数是欧几里德Jordan代数的一般化和包括Ñ × Ñ矩阵甲=（一个_IJ），其中一个_二∈ℝ，条目一个_IJ为我< Ĵ属于一些向量空间（V _IJ ;𝔤）和\（{a} _ {ji} = {a} _ {ij} ^ {\ ast} = \ mathfrak {g} \ left（{a} _ {ij}，\ cdot \ right）\ in {V} _ {ij} ^ {\ ast} \）属于对偶空间\（{V} _ {ij} ^ {\ ast}。\） T-Hermitian矩阵的乘法由“等距”双线性映射V _ij × V _jk → V _ij的系统定义；i <j < k，使得| 一_IJ  ⋅ 一个_JK | = | 一_IJ | ⋅| 一个_IK |， 一个_LM  ∈  V_流明。对于n = 2，埃尔米特T-代数ℋ _Ñ = ℋ ₂（V）由向量空间决定V和同构于一个称为自旋因子代数和相关联的均质凸锥欧几里德若代数在闵可夫斯基向量空间ℝ类时未来定向向量的洛仑兹锥体^1,1 ⊕V。一个特殊的Vinberg埃尔米特T-代数是秩3矩阵代数ℋ ₃（V ;小号）关联到克利福氯（V） -模小号连同“受理”欧几里德度量𝔤_小号。

我们泛指秩2的结构Vinberg代数ℋ ₂（V）和特殊Vinberg代数ℋ ₃（V ;小号）到伪欧几里德情况下，当V是一个伪向量空间和小号=小号₀ ⊕小号₁是一个ℤ ₂ -graded克利福氯（V） -模与可容许伪欧几里德度量。相关联的锥体𝒱是均质的，但在不ℋ凸锥_米; 米= 2; 3.我们计算该圆锥体的Koszul-Vinberg特征函数，并写下相关的三次多项式。我们延长Vinberg锥体𝒱的贝兹”量子力学解释₂ ⊂ℋ ₂（V）对专用秩3的情况。