Monads and distributive laws for Rota-Baxter and differential algebras

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• 作者：Shilong Zhang^a ; ^{2663067567@qq.com" class="auth_mail" title="E-mail the corresponding author} ; Li Guo^b ; ^c ; ^{liguo@newark.rutgers.edu" class="auth_mail" title="E-mail the corresponding author} ; William Keigher^c ; ^{keigher@newark.rutgers.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 18C15 ; 13Nxx ; 16W99 ; secondary ; 18C20 ; 12H05
• 刊名：Advances in Applied Mathematics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：72
• 期：Complete
• 页码：139-165
• 全文大小：500 K

文摘

In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota–Baxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws which can be tracked back to the distributive law of multiplication over addition. The monad giving Rota–Baxter algebras and the comonad giving differential algebras are constructed. Then we obtain monads and comonads giving the composite structures of differential and Rota–Baxter algebras. As a consequence, a mixed distributive law of the monad giving Rota–Baxter algebras over the comonad giving differential algebras is established.