Research ArticlesNo Access

Representations of $E_{7}$ $E_{7}$ , equivalent combinatorics and algebraic varieties

Xiaoping Xu

Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, P. R. China

School of Mathematics, University of Chinese Academy of Sciences, Beijing 100049, P. R. China

E-mail Address: xiaoping@math.ac.cn

Search for more papers by this author

https://doi.org/10.1142/S0219498818500457Cited by:1 (Source: Crossref)

Abstract

In our earlier work on a new functor from $E_{6}$ $E_{6}$ -Mod to $E_{7}$ $E_{7}$ -Mod, we found a one-parameter ( $c$ $c$ ) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type $E_{7}$ $E_{7}$ in $27$ $27$ variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector $\vec{a} \in ℂ^{27} \ {\vec{0}}$ , we prove that the space forms an irreducible $E_{7}$ -module for any $c \in ℂ$ if $\vec{a}$ is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of $E_{6}$ over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

Communicated by I. M. Musson

Keywords:

AMSC: 17B10, 17B25, 22E46

References

1. G. Benkart, D. Britten and F. W. Lemire, Modules with bounded multiplicities for simple Lie algebras, Math. Z. 225 (1997) 333–353. Crossref, Web of Science, Google Scholar
2. D. Britten, V. Futorny and F. W. Lemire, Simple $A_{2}$ -modules with a finite-dimensional weight space, Comm. Algebra 23 (1995) 467–510. Crossref, Web of Science, Google Scholar
3. D. Britten, J. Hooper and F. W. Lemire, Simple $C_{n}$ -modules with multiplicities 1 and applications, Canad. J. Phys. 72 (1994) 326–335. Crossref, Web of Science, Google Scholar
4. D. Britten and F. W. Lemire, A classification of simple Lie modules having 1-dimensional weight space, Trans. Amer. Math. Soc. 299 (1987) 683–697. Crossref, Web of Science, Google Scholar
5. D. Britten and F. W. Lemire, On modules of bounded multiplicities for symplectic algebras, Trans. Amer. Math. Soc. 351 (1999) 3413–3431. Crossref, Web of Science, Google Scholar
6. M. Davidson, T. Enright and R. Stanke, Differential Operators and Highest Weight Representations, Memoirs of American Mathematical Society, Vol. 94, No. 455 (American Mathematical Society, Providence, RI, 1991). Google Scholar
7. F. M. Fernández and E. A. Castro, Algebraic Methods in Quantum Chemistry and Physics (CRC Press, 1996). Google Scholar
8. S. L. Fernando, Lie algebra modules with finite-dimensional weight spaces, I, Trans. Amer. Math. Soc. 322 (1990) 757–781. Web of Science, Google Scholar
9. L. Frappat, A. Sciarrino and P. Sorba, Dictionary on Lie Algebras and Superalgebras (Academic Press, 2000). Google Scholar
10. V. Futorny, The weight representations of semisimple finite-dimensional Lie algebras, Ph.D. thesis, Kiev University (1987). Google Scholar
11. H. Georgi, Lie Algebras in Particle Physics, 2nd edn. (Perseus Books Group, 1999). Google Scholar
12. R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, in The Schur Lectures (1992) (Tel Aviv), Israel Mathematical Conference Proceedings, Vol. 8 (Bar-Ilan University, Ramat Gan, 1995), pp. 1–182. Google Scholar
13. V. Kac, Infinite-dimensional Lie Algebras (Birkhäser, Boston, 1982). Google Scholar
14. F. S. Levin, An Introduction to Quantum Theory (Cambridge University Press, 2002). Google Scholar
15. W. Ludwig and C. Falter, Symmetries in Physics, 2nd edn. (Springer-Verlag, Berlin, 1996). Crossref, Google Scholar
16. O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000) 537–592. Crossref, Web of Science, Google Scholar
17. G. Shen, Graded modules of graded Lie algebras of Cartan type (I) — mixed product of modules, Sci. China A 29 (1986) 570–581. Google Scholar
18. X. Xu, Kac–Moody Algebras and their Representations (China Science Press, Beijing, 2007). Google Scholar
19. X. Xu, A new functor from $E_{6}$ -Mod to $E_{7}$ -Mod, Comm. Algebra 43(9) (2015) 3589–3636. Crossref, Web of Science, Google Scholar
20. X. Xu, A new functor from $D_{5}$ -Mod to $E_{6}$ -Mod, Acta Math. Sin. (Engl. Ser.) 32(8) (2016) 867–892. Crossref, Web of Science, Google Scholar
21. X. Xu, Conformal oscillator representations of orthogonal Lie algebras, Sci. China: Math. (Engl. Ser.) 59(1) (2016) 37–48. Crossref, Web of Science, Google Scholar
22. X. Xu and Y. Zhao, Extensions of the conformal representations for orthogonal Lie algebras, J. Algebra 377 (2013) 97–124. Crossref, Web of Science, Google Scholar