Exceptional Lie algebras, SU(3) and Jordan pairs: part 2. Zorn-type representations
@article{Marrani2011ExceptionalLA, title={Exceptional Lie algebras, SU(3) and Jordan pairs: part 2. Zorn-type representations}, author={Alessio Marrani and Piero Truini}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2011}, volume={47} }
A representation of the exceptional Lie algebras reflecting a simple unifying view, based on realizations in terms of Zorn-type matrices, is presented. The role of the underlying Jordan pair and Jordan algebra content is crucial in the development of the structure. Each algebra contains three Jordan pairs sharing the same Lie algebra of automorphisms and the same external su(3) symmetry. The applications in physics are outlined.
27 Citations
Jordan pairs, E6 and U-duality in five dimensions
- Mathematics
- 2012
By exploiting the Jordan pair structure of U-duality Lie algebras in D = 3 and the relation to the super-Ehlers symmetry in D = 5, we elucidate the massless multiplet structure of the spectrum of a…
Jordan algebraic interpretation of maximal parabolic subalgebras: exceptional Lie algebras
- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2020
With this paper we start a programme aiming at connecting two vast scientific areas: Jordan algebras and representation theory. Within representation theory, we focus on non-compact, real forms of…
Exceptional Periodicity and Magic Star Algebras. II : Gradings and HT-Algebras
- Mathematics
- 2019
We continue the study of Exceptional Periodicity and Magic Star algebras, which provide non-Lie, countably infinite chains of finite dimensional generalizations of exceptional Lie algebras. We…
Exceptional lie algebras at the very foundations of space and time
- MathematicsP-Adic Numbers, Ultrametric Analysis, and Applications
- 2016
While describing the results of our recent work on exceptional Lie and Jordan algebras, so tightly intertwined in their connection with elementary particles, we will try to stimulate a critical…
Exceptional lie algebras at the very foundations of space and time
- Mathematics
- 2015
While describing the results of our recent work on exceptional Lie and Jordan algebras, so tightly intertwined in their connection with elementary particles, we will try to stimulate a critical…
The Magic of Being Exceptional
- MathematicsJournal of Physics: Conference Series
- 2019
Starting from the Jordan algebraic interpretation of the “Magic Star” embedding within the exceptional sequence of simple Lie algebras, we exploit the so-called spin factor embedding of rank-3 Jordan…
Exceptional Periodicity and Magic Star Algebras. I : Foundations
- Mathematics
- 2019
We introduce and start investigating the properties of countably infinite, periodic chains of finite dimensional generalizations of the exceptional Lie algebras: each exceptional Lie algebra (but…
Magic Star and Exceptional Periodicity: an approach to Quantum Gravity
- MathematicsJournal of Physics: Conference Series
- 2019
We present a periodic infinite chain of finite generalisations of the exceptional structures, including the exceptional Lie algebra e8, the exceptional Jordan algebra (and pair) and the octonions. We…
Sextonions, Zorn Matrices, and e_{71/2}
- Mathematics
- 2015
By exploiting suitably constrained Zorn matrices, we present a new construction of the algebra of sextonions (over the algebraically closed field C). This allows for an explicit construction, in…
Constructing numbers in quantum gravity: infinions
- PhysicsJournal of Physics: Conference Series
- 2019
Based on the Cayley-Dickson process, a sequence of multidimensional structured natural numbers (infinions) creates a path from quantum information to quantum gravity. Octonionic structure,…
References
SHOWING 1-10 OF 157 REFERENCES
Three graded exceptional algebras and symmetric spaces
- Mathematics
- 1986
The exceptional algebras of typeE7 are studied from the point of view of their three graded structure. The connection between three-grading and the Jordan Pair structure of such Lie algebras is…
The exceptional Lie algebra E7(−25): multiplets and invariant differential operators
- Mathematics
- 2009
In the present paper, we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional algebra E7(−25). Our choice of this…
Jordan Pairs and Hopf Algebras
- Mathematics
- 2000
Abstract A (quadratic) Jordan pair is constructed from a Z -graded Hopf algebra having divided power sequences over all primitive elements and with three terms in the Z -grading of the primitive…
Jordan pairs, E6 and U-duality in five dimensions
- Mathematics
- 2012
By exploiting the Jordan pair structure of U-duality Lie algebras in D = 3 and the relation to the super-Ehlers symmetry in D = 5, we elucidate the massless multiplet structure of the spectrum of a…
Conformal and Quasiconformal Realizations¶of Exceptional Lie Groups
- Mathematics
- 2000
Abstract: We present a nonlinear realization of E8(8) on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined “light cone” in ℝ57. This…
Invariant differential operators for non-compact Lie groups: the SO* (12) case
- Mathematics
- 2008
In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so* (12). We give the main multiplets of…
An E6⊗U(1) invariant quantum mechanics for a Jordan pair
- Mathematics
- 1982
Quantum mechanical spaces associated with geometries based on exceptional groups are of interest as models for internal (quark) symmetries. Using the concept of a Jordan pair, two copies of complex…
Invariant Differential Operators for Non-compact Lie Groups: The Sp(n, IR) Case
- Mathematics
- 2012
In the present paper we continue the project of systematic explicit construction of invariant differential operators. On the example of the non-compact exceptional group $E_{6(-14)}$ we give the…
Mapping the geometry of the F4 group
- Mathematics
- 2007
In this paper, we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the…