On Lie algebras in braided categories

Volume 40 / 1997

Bodo Pareigis Banach Center Publications 40 (1997), 139-158 DOI: 10.4064/-40-1-139-158


The category of group-graded modules over an abelian group $G$ is a monoidal category. For any bicharacter of $G$ this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have $n$-ary multiplications between various graded components. They possess universal enveloping algebras that are Hopf algebras in the given category. Their biproducts with the group ring are noncommutative noncocommutative Hopf algebras some of them known in the literature. Conversely the primitive elements of a Hopf algebra in the category form a Lie algebra in the above sense.


  • Bodo Pareigis

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