Dual Alternative Algebras in Characteristic Three

Abstract

We prove that the dimension of the arity n component of the operad of dual alternative algebras over a field of characteristic three is equal to 2 ⁿ  − n, and describe the structure of the corresponding S _n -module.

Key Words:

• Dual alternative algebras
• Modular representations
• Operads
• Varieties of algebras

2000 Mathematics Subject Classification:

• Primary 18D50
• Secondary 16S15
• 20C20

ACKNOWLEDGMENTS

Thanks are due to Ivan Yudin whose questions led us to discovering a gap in the original proof of the main theorem, and made us present the argument in a hopefully more conceptual way, and to the referee who pointed out that the conjecture of Dzhumadil'daev and Zusmanovich was actually proved in [12] before being formulated in [4]. Most of this project was completed while the author was supported by Grant GeoAlgPhys 2011–2013 awarded by the University of Luxembourg.

Notes

Communicated by I. Shestakov.