Automorphisms for Skew PBW Extensions and Skew Quantum Polynomial Rings

Abstract

In this work, I study the automorphisms of skew PBW extensions and skew quantum polynomials. I use Artamonov's works as reference for getting the principal results about automorphisms for generic skew PBW extensions and generic skew quantum polynomials. In general, if I have an endomorphism on a generic skew PBW extension and there are some x _i , x _j , x _u such that the endomorphism is not zero on these elements and the principal coefficients are invertible, then endomorphisms act over x _i as a _i x _i for some a _i in the ring of coefficients. Of course, this is valid for quantum polynomial rings, with r = 0, as such Artamonov shows in his work. We use this result for giving some more general results for skew PBW extensions, using filtred-graded techniques. Finally, I use localization to characterize some class the endomorphisms and automorphisms for skew PBW extensions and skew quantum polynomials over Ore domains.

Key Words:

• Automorphisms
• Endomorphisms
• Filtration
• Graded
• Ore domain
• Quantum polynomials
• Skew PBW extensions
• Skew quantum polynomials

2010 Mathematics Subject Classification:

• 16S32
• 16S36
• 16W20

ACKNOWLEDGMENTS

I thank Professor Oswaldo Lezama at the National University of Colombia for his advice and suggestions.

Notes

Communicated by V. A. Artamonov.