李单子和对偶
Lie monads and dualities
摘要：
我们研究了李代数和李余代数之间的对偶以及它们各自的（余）表示。为了研究无限维情形下的对偶性，我们引入了李群和李群的概念，作为加法幺半范畴中YB-李代数和YB-李余代数的特例.我们证明了李代数和李余代数之间的（强）对偶与关联的李单子和李余单子之间的（等）态射密切相关。在两个霍普夫代数之间的对偶的情况下-在竹内的意义上-我们恢复一个李代数和李余代数之间的对偶-在这个说明中定义的意义上-通过计算的原始和不可分解的元素，分别。
英文摘要：
We study dualities between Lie algebras and Lie coalgebras, and their respective (co)representations. To allow a study of dualities in an infinite-dimensional setting, we introduce the notions of Lie monads and Lie comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive monoidal categories. We show that (strong) dualities between Lie algebras and Lie coalgebras are closely related to (iso)morphisms between associated Lie monads and Lie comonads. In the case of a duality between two Hopf algebras -in the sense of Takeuchi- we recover a duality between a Lie algebra and a Lie coalgebra -in the sense defined in this note- by computing the primitive and the indecomposables elements, respectively.
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