Octonions and subalgebras of the exceptional algebras

1989 ◽  
Vol 30 (3) ◽  
pp. 585-593 ◽  
Author(s):  
F. Buccella ◽  
A. Della Selva ◽  
A. Sciarrino
Keyword(s):  
Exceptional Algebras

SPECTRAL DECOMPOSITION OF R-MATRICES FOR EXCEPTIONAL LIE ALGEBRAS

1991 ◽  
Vol 06 (10) ◽  
pp. 923-927 ◽  
Author(s):  
S.M. SERGEEV
Keyword(s):  
Lie Algebras ◽  
Spectral Decomposition ◽  
Spectral Decompositions ◽  
Exceptional Lie Algebras ◽  
Exceptional Algebras ◽  
Vector Representations

In this paper spectral decompositions of R-matrices for vector representations of exceptional algebras are found.

Three graded exceptional algebras and symmetric spaces

1986 ◽  
Vol 33 (1) ◽  
pp. 47-65 ◽  
Author(s):  
P. Truini ◽  
G. Olivieri ◽  
L. C. Biedenharn
Keyword(s):  
Symmetric Spaces ◽  
Exceptional Algebras

q-fermionic operators and quantum exceptional algebras

1991 ◽  
Vol 24 (4) ◽  
pp. L179-L183 ◽  
Author(s):  
L Frappat ◽  
P Sorba ◽  
A Sciarrino
Keyword(s):  
Fermionic Operators ◽  
Exceptional Algebras

scholarly journals GROUP THEORY FACTORS FOR FEYNMAN DIAGRAMS

1999 ◽  
Vol 14 (01) ◽  
pp. 41-96 ◽  
Author(s):  
T. VAN RITBERGEN ◽  
A. N. SCHELLEKENS ◽  
J. A. M. VERMASEREN
Keyword(s):  
Group Theory ◽  
Feynman Diagrams ◽  
Computer Algorithms ◽  
Coxeter Number ◽  
Exceptional Algebras ◽  
Dual Coxeter Number ◽  
Invariant Tensors

We present algorithms for the group independent reduction of group theory factors of Feynman diagrams. We also give formulas and values for a large number of group invariants in which the group theory factors are expressed. This includes formulas for various contractions of symmetric invariant tensors, formulas and algorithms for the computation of characters and generalized Dynkin indices and trace identities. Tables of all Dynkin indices for all exceptional algebras are presented, as well as all trace identities to order equal to the dual Coxeter number. Further results are available through efficient computer algorithms.

scholarly journals Spectral theories and topological strings on del Pezzo geometries

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Sanefumi Moriyama
Keyword(s):  
Topological Strings ◽  
Chemical Potential ◽  
Group Structure ◽  
Partition Functions ◽  
Spectral Operator ◽  
Exceptional Algebras ◽  
Del Pezzo ◽  
Classical Phase Space ◽  
Group Symmetries ◽  
Mirror Map

Abstract Motivated by understanding M2-branes, we propose to reformulate partition functions of M2-branes by quantum curves. Especially, we focus on the backgrounds of del Pezzo geometries, which enjoy Weyl group symmetries of exceptional algebras. We construct quantum curves explicitly and turn to the analysis of classical phase space areas and quantum mirror maps. We find that the group structure helps in clarifying previous subtleties, such as the shift of the chemical potential in the area and the identification of the overall factor of the spectral operator in the mirror map. We list the multiplicities characterizing the quantum mirror maps and find that the decoupling relation known for the BPS indices works for the mirror maps. As a result, with the group structure we can present explicitly the statement for the correspondence between spectral theories and topological strings on del Pezzo geometries.

Physical theories in hypercomplex geometric description

2014 ◽  
Vol 11 (06) ◽  
pp. 1450062 ◽  
Author(s):  
Alexander P. Yefremov
Keyword(s):  
Relativity Theory ◽  
Direct Products ◽  
Yang Mills ◽  
Equations Of Electrodynamics ◽  
Quaternion Space ◽  
Inertial Frames ◽  
Exceptional Algebras ◽  
Physical Laws ◽  
Jacobi Equations ◽  
Mills Field

Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

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