Universal enveloping algebras of Poisson Hopf algebras

2014/02/09 by Jiafeng Lü, Lü, Jiafeng, Xingting Wang +3
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Rings and Algebras (math.RA) #math.RA

paper · pdf · doi:10.48550/arxiv.1402.2007

37 pages. Reference updated

openalex publication_date 2014/02/09 · arxiv created 2014/03/18 · arxiv updated 2014/03/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For a Poisson algebra A, by exploring its relation with Lie-Rinehart algebras, we prove a Poincaré-Birkoff-Witt theorem for its universal enveloping algebra A^e. Some general properties of the universal enveloping algebras of Poisson Hopf algebras are studied. Given a Poisson Hopf algebra B, we give the necessary and sufficient conditions for a Poisson polynomial algebra B[x; α, δ]_p to be a Poisson Hopf algebra. We also prove a structure theorem for B^e when B is a pointed Poisson Hopf algebra. Namely, B^e is isomorphic to B#_σH(B), the crossed product of B and H(B), where H(B) is the quotient Hopf algebra B^e/B^eB⁺.

Citations

• Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two

Related

• Co-Poisson structures on polynomial Hopf algebras
• Universal Enveloping Algebras of Poisson Superalgebras
• Poisson enveloping algebras and the Poincaré-Birkhoff-Witt theorem
• Hopf-Galois algebras and their Poisson structures
• Poisson Hopf algebras associated to quantized enveloping algebras