SPECIAL VINBERG CONES
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
SPECIAL VINBERG CONES
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA18-00496S" target="_blank" >GA18-00496S: Singular spaces from special holonomy and foliations</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Transformation Groups
ISSN
1083-4362
e-ISSN
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Volume of the periodical
26
Issue of the periodical within the volume
2
Country of publishing house
US - UNITED STATES
Number of pages
26
Pages from-to
377-402
UT code for WoS article
000637460500001
EID of the result in the Scopus database
2-s2.0-85107646905