On Lie algebras in braided categories

Bodo Pareigis

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 139-158
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Pareigis, Bodo. "On Lie algebras in braided categories." Banach Center Publications 40.1 (1997): 139-158. <http://eudml.org/doc/252225>.

@article{Pareigis1997,
abstract = {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.},
author = {Pareigis, Bodo},
journal = {Banach Center Publications},
keywords = {braided category; universal enveloping algebra; braided Hopf algebra; graded Lie algebra; categories of graded vector spaces; braided monoidal categories; Lie algebras; color algebras; universal enveloping algebras; biproducts; noncommutative noncocommutative Hopf algebras},
language = {eng},
number = {1},
pages = {139-158},
title = {On Lie algebras in braided categories},
url = {http://eudml.org/doc/252225},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Pareigis, Bodo
TI - On Lie algebras in braided categories
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 139
EP - 158
AB - 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.
LA - eng
KW - braided category; universal enveloping algebra; braided Hopf algebra; graded Lie algebra; categories of graded vector spaces; braided monoidal categories; Lie algebras; color algebras; universal enveloping algebras; biproducts; noncommutative noncocommutative Hopf algebras
UR - http://eudml.org/doc/252225
ER -

References

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  1. [A] George E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications. Vol. 2, Addison-Wesley, 1976. 
  2. [FM] Davida Fischman and Susan Montgomery, A Schur Double Centralizer Theorem for Cotriangular Hopf Algebras and Generalized Lie Algebras, J. Algebra 168 (1994), 594-614. Zbl0818.16031
  3. [M94a] Shahn Majid, Crossed Products by Braided Groups and Bosonization, J. Algebra 163 (1994), 165-190. Zbl0807.16036
  4. [M94b] Shahn Majid, Algebra and Hopf Algebras in Braided Categories, in: Advances in Hopf Algebras. LN pure and applied mathematics 158 (1994) 55-105. Zbl0812.18004
  5. [R] David Radford, The Structure of Hopf Algebras with a Projection, J. Algebra 92 (1985), 322-347. Zbl0549.16003
  6. [T] Earl J. Taft, The Order of the Antipode of Finite-dimensional Hopf algebras. Proc. Nat. Acad. Sci. USA 68 (1971), 2631-2633. Zbl0222.16012

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