Centroids of Quadratic Jordan Superalgebras

Abstract

The centroid of a Jordan superalgebra consists of the natural “superscalar multiplications” on the superalgebra. A philosophical question is whether the natural concept of “scalar” in the category of superalgebras should be that of superscalars or ordinary scalars. Basic examples of Jordan superalgebras are the simple Jordan superalgebras with semisimple even part, which were classified over an algebraically closed field of characteristic ≠ 2 by Racine and Zelmanov. Here, we determine the centroids of the analog of these superalgebras over general rings of scalars and show that they have no odd centroid, suggesting that ordinary scalars are the proper concept.

Key Words:

• Centroid
• Jordan superalgebra
• Superscalar

2000 Mathematics Subject Classification:

• Primary 17
• Secondary 16

ACKNOWLEDGMENTS

The author would like to thank the referee and Kevin McCrimmon for their careful consideration of this work and their valuable suggestions.

Notes

¹Thanks to Ottmar Loos for suggesting the full-valued case.

Communicated by I. P. Shestakov.