# 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.

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## 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…