Abstract：This paper studies p-differential graded Poisson Hopf algebras and proves that the tensor product of p-differential graded Poisson Hopf algebras is also a p-differential graded Poisson Hopf algebra. As an application, we show that the category of p-differential graded Poisson Hopf algebras, denoted by dg-PHA, belongs to symmetric monoidal category.
[1] HUEBSCHMANN J. Poisson cohomology and quantization[J]. Journal Für Die Reine Und Angewandte Mathematik,1990, 408:57-113.
[2] XU X P. Novikov-Poisson algebras[J]. Journal of Algebra,1997, 190(2):253-279.
[3] CASAS J M,DATUASHVILI T. Noncommutative Leibniz-Poisson algebras[J]. Communications in Algebra,2006, 34(7):2507-2530.
[4] MICHELV D B. Double Poisson algebras[J]. Transactions of the American Mathematical Society, 2008, 360(11):5711-5769.
[5] DRINFEL'D V G.Quantum groups[J]. Journal of Soviet Mathematics, 1988, 41(2):898-915.
[6] LYU J F, WANG X T, ZHUANG G B. Universal enveloping algebras of Poisson Hopf algebras[J]. Journal of Algebra, 2015, 426:92-136.
[7] BECK K A, SATHER-WAGSTAFF S. A somewhat gentle introduction to differential graded commutative algebra[C]//Connections Between Algebra, Combinatorics, and Geometry Springer Proceeding in Mathematics and Statistics. New York:Springer, 2014, 76(2):3-99.
[8] FRANKILD A, JØRGENSEN P. Homological identities for differential graded algebras[J]. Journal of Algebra, 2003, 265(1):114-135.
[9] LYU J F, WANG X T, ZHUANG G B. DG Poisson algebra and its universal enveloping algebra[J]. Science China Mathematics, 2016, 59(5):849-860.
[10] CATTANEO A S, FIORENZA D, LONGONI R. Graded Poisson algebras[J]. Encyclopedia of Mathematical Physics, 2006(2):560-567.
[11] NASTASESCU C, TORRECILLAS B. Graded coalgebras[J]. Tsukuba Journal of Mathematics, 1993, 17(2):461-479.
[12] SWEEDLER M E.Hopf Algebras[M]. New York:Benjamin, 1969.
[13] UMBLE R N. The deformation complex for differential graded Hopf algebras[J]. Journal of Pure and Applied Algebra, 1996, 106(2):199-222.
2018, Vol. 45 
