Universal enveloping algebras of differential graded Poisson algebras

2014/03/12 by Jiafeng Lü, Lü, Jiafeng, Xingting Wang +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #16E45 #16S10 #17B35 #17B63 #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #Sphingolipid Metabolism and Signaling #math.RA #msc:16E45 #msc:16S10 #msc:17B35 #msc:17B63

paper · pdf · doi:10.48550/arxiv.1403.3130

37 pages, the abstract is rewritten, another construction of the universal enveloping algebra is given and several typos are fixed

openalex publication_date 2014/03/12 · arxiv created 2014/03/25 · arxiv updated 2014/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we introduce the notion of differential graded Poisson algebra and study its universal enveloping algebra. From any differential graded Poisson algebra A, we construct two isomorphic differential graded algebras: A^e and A^E. It is proved that the category of differential graded Poisson modules over A is isomorphic to the category of differential graded modules over A^e, and A^e is the unique universal enveloping algebra of A up to isomorphisms. As applications of the universal property of A^e, we prove that (A^e)^op≅ (A^op)^e and (A⊗_\BbbkB)^e≅ A^e⊗_\BbbkB^e as differential graded algebras. As consequences, we obtain that ``e'' is a monoidal functor and establish links among the universal enveloping algebras of differential graded Poisson algebras, differential graded Lie algebras and associative algebras.

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