SPECIAL VINBERG CONES
Identifikátory výsledku
Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18470%2F21%3A50018372" target="_blank" >RIV/62690094:18470/21:50018372 - isvavai.cz</a>
Výsledek na webu
<a href="https://link.springer.com/article/10.1007/s00031-021-09649-w" target="_blank" >https://link.springer.com/article/10.1007/s00031-021-09649-w</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00031-021-09649-w" target="_blank" >10.1007/s00031-021-09649-w</a>
Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
SPECIAL VINBERG CONES
Popis výsledku v původním jazyce
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.
Název v anglickém jazyce
SPECIAL VINBERG CONES
Popis výsledku anglicky
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.
Klasifikace
Druh
J<sub>imp</sub> - Článek v periodiku v databázi Web of Science
CEP obor
—
OECD FORD obor
10101 - Pure mathematics
Návaznosti výsledku
Projekt
<a href="/cs/project/GA18-00496S" target="_blank" >GA18-00496S: Singulární prostory ze speciální holonomie a foliací</a><br>
Návaznosti
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Ostatní
Rok uplatnění
2021
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název periodika
Transformation Groups
ISSN
1083-4362
e-ISSN
—
Svazek periodika
26
Číslo periodika v rámci svazku
2
Stát vydavatele periodika
US - Spojené státy americké
Počet stran výsledku
26
Strana od-do
377-402
Kód UT WoS článku
000637460500001
EID výsledku v databázi Scopus
2-s2.0-85107646905