On the irreducible representations of the Jordan triple system of $p \times q$ matrices

Abstract

Let $\mathcal{J}_{\field}$ be the Jordan triple system of all $p \times q$ ($p\neq q$; $p,q >1)$ rectangular matrices over a field $\field$ of characteristic 0 with the triple product $\{x,y,z\}= x y^t z+ z y^t x $, where $y^t$ is the transpose of $y$. We study the universal associative envelope $\mathcal{U}(\mathcal{J}_{\field})$ of $\mathcal{J}_{\field}$ and show that $\mathcal{U}(\mathcal{J}_{\field}) \cong M_{p+q \times p+q}(\field)$, where $M_{p+q\times p+q} (\field)$ is the ordinary associative algebra of all $(p+q) \times (p+q)$ matrices over $\field$. It follows that there exists only one nontrivial irreducible representation of $\mathcal{J}_{\field}$. The center of $\mathcal{U}(\mathcal{J}_{\field})$ is deduced.

Keywords

• Jordan triple system
• rectangular matrix
• universal associative envelope
• representation theory

References

1. C. Chu, Jordan triples and Riemannian symmetric spaces, Adv. Math., 219 (2008), 2029-2057.
2. E. Corrigan and T. Hollowood, String construction of a commutative nonassociative algebra related to the exceptional Jordan algebra, Phys. Lett., 203 (1988), 47-51.
3. H. Elgendy, On the universal envelope of a Jordan triple system of $n \times n$ matrices, J. Algebra Appl., 21(6) (2022), 2250126 (19 pp).
4. H. Elgendy, Representations of special Jordan triple systems of all symmetric and hermitian $n$ by $n$ matrices, Linear Multilinear Algebra, DOI: 10.1080/03081087.2021.1970097, in press.
5. D. Fairlie and C. Manogue, Lorentz invariance and the composite string, Phys. Rev., 34 (1986), 1832-1834.
6. J. Faulkner and J. Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc., 9 (1977), 1-35.
7. R. Foot and G. Joshi, String theories and the Jordan algebras, Phys. Lett., 199 (1987), 203-208.
8. P. Goddard, W. Nahm, D. Olive, H. Ruegg and A. Schwimmer, Fermions and octonions, Comm. Math. Phys., 112 (1987), 385-408.

9. M. Günaydin and F. Gürsey, Quark structure and octonions, J. Mathematical Phys., 14 (1973), 1651-1667.
10. M. Günaydin, G. Sierra and P. Townsend, The geometry of $N=2$ Maxwell-Einstein supergravity and Jordan algebras, Nuclear Phys., 242 (1984), 244-268.
11. F. Gürsey, Super Poincar$\acute{e}$ groups and division algebras, Modern Phys. Lett., 2 (1987), 967-976.
12. N. Jacobson, Lie and Jordan triple systems, Amer. J. Math., 71 (1949), 149-170.
13. W. Kaup and D. Zaitsev, On symmetric Cauchy-Riemann manifolds, Adv. Math., 149 (2000), 145-181.
14. M. Koecher, Imbedding of Jordan algebras into Lie algebras I, Amer. J. Math., 89 (1967), 787-816.
15. M. Koecher, An Elementary Approach to Bounded Symmetric Domains, Lecture Notes, Rice University, Houston, 1969.
16. K. McCrimmon, Jordan algebras and their applications, Bull. Amer. Math. Soc., 84 (1978), 612-627.
17. K. McCrimmon, A Taste of Jordan Algebras, Universitext, Springer-Verlag, 2004.
18. K. Meyberg, Lectures on Algebras and Triple Systems, Lecture Notes, University of Virginia, Charlottesville, 1972.
19. E. Neher, Jordan Triple Systems by the Grid Approach, Lecture Notes in Mathematics, Vol. 1280, Springer-Verlag, Berlin, 1987.
20. S. Okubo, Triple products and Yang-Baxter equation. II. Orthogonal and symplectic ternary systems, J. Math. Phys., 34 (1993), 3292-3315.
21. G. Sierra, An application of the theories of Jordan algebras and Freudenthal triple systems to particles and strings, Classical Quantum Gravity, 4 (1987), 227-236.
22. H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math., 108 (1986), 1-25.