The Tits–Kantor–Koecher Construction and Birepresentations of the Jordan Superpair SH(1, n)

Abstract

We introduce the concept of bimodule over a Jordan superpair and the Tits– Kantor–Koecher construction for bimodules. Using the construction we obtain the classification of irreducible bimodules over the Jordan superpair SH(1, n). We also prove semisimplicity for a class of finite dimensional SH(1, n)-bimodules for n ≥ 3.

Key Words:

• Jordan superpair
• Graded Lie superalgebra
• Bimodule
• Birepresentation
• Tits–Kantor–Koecher construction

Mathematics Subject Classification:

• 17C70 (17B70)

Acknowledgments

We would like to expresses our gratitude to Alexander S. Stern for the attention to the work and helpful remarks. We thank I. Dimitrov and G. Seligman for helpful discussions. We also thank the referee for many helpful comments.

Notes

^#Communicated by E. Zelmanov.