Braided Enveloping Algebras Associated to Quantum Parabolic Subalgebras

Abstract

Associated to each subset J of the nodes I of a Dynkin diagram is a triangular decomposition of the corresponding Lie algebra 𝔤 into three subalgebras (generated by e _j , f _j for j ∈ J and h _i for i ∈ I), (generated by f _d , d ∈ D = I∖J) and its dual .

We demonstrate a quantum counterpart, generalising work of Majid and Rosso, by exhibiting analogous triangular decompositions of U _q (𝔤) and identifying a graded braided Hopf algebra that quantizes . This algebra has many similar properties to , in many cases being a Nichols algebra and therefore completely determined by its associated braiding.

Key Words:

• Braided Hopf algebra
• Nichols algebra
• Quantized enveloping algebra

2000 Mathematics Subject Classification:

• Primary: 17B37, 20G42

ACKNOWLEDGMENTS

I am very grateful to Shahn Majid for the initial suggestions that led to the work here and for much help and encouragement during its completion. I also gratefully acknowledge the assistance provided by Jonathan Dixon, particularly relating to the representation-theoretic aspects of this work. I would also like to thank Stefan Kolb for several helpful conversations.

The majority of the work in this article has appeared in the author's Ph.D. thesis [11], completed at Queen Mary, University of London under the supervision of Prof. Majid and funded by an EPSRC Doctoral Training Account. The remainder has been completed during the author's research fellowship at Keble College, Oxford. I would also like to thank the Mathematical Institute at Oxford for its provision of facilities.

I would also like to thank the referee for correcting an error in a previous version and for several other helpful comments.

Notes

Communicated by T. Lenagan.