Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2503.17781 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916860130230272 |
|---|---|
| author | Alekseevsky, D. V. Osipov, P. |
| author_facet | Alekseevsky, D. V. Osipov, P. |
| contents | 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 $Γ=Γ(\mathcal{N})$ 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 in terms of admissible equipment. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Special Vinberg cones of rank 4 Alekseevsky, D. V. Osipov, P. Differential Geometry 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 $Γ=Γ(\mathcal{N})$ 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 in terms of admissible equipment. |
| title | Special Vinberg cones of rank 4 |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2503.17781 |