Dual Lie elements and a derivation for the cofree coassociative coalgebra

Griffing, Gary

doi:10.1090/S0002-9939-1995-1273493-6

Dual Lie elements and a derivation for the cofree coassociative coalgebra
HTML articles powered by AMS MathViewer

by Gary Griffing PDF

Proc. Amer. Math. Soc. 123 (1995), 3269-3277 Request permission

Abstract:

We construct a derivation D in the Hopf algebra TcV, the cofree coassociative coalgebra on a vector space V. We then define the subspace of TcV consisting of dual Lie elements, which is analogous to the subspace of the Hopf algebra TV, the free associative algebra on V, consisting of Lie elements. Thereafter, we formulate a dual Dynkin-Specht-Wever theorem. Using our map D, we then give very short proofs of both the dual Dynkin-Specht-Wever and dual Friedrichs’ theorems, each of which characterizes the space of dual Lie elements in TcV at characteristic 0.

References

M. André, Hopf algebras with divided powers, J. Algebra 18 (1971), 19–50. MR 277590, DOI 10.1016/0021-8693(71)90126-8
Richard E. Block and Pierre Leroux, Generalized dual coalgebras of algebras, with applications to cofree coalgebras, J. Pure Appl. Algebra 36 (1985), no. 1, 15–21. MR 782637, DOI 10.1016/0022-4049(85)90060-X
Richard E. Block, Commutative Hopf algebras, Lie coalgebras, and divided powers, J. Algebra 96 (1985), no. 1, 275–306. MR 808852, DOI 10.1016/0021-8693(85)90050-X
Gary Griffing, The dual Friedrichs’ theorem, J. Algebra 132 (1990), no. 2, 446–460. MR 1061490, DOI 10.1016/0021-8693(90)90140-J
Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927
Walter Michaelis, Lie coalgebras, Adv. in Math. 38 (1980), no. 1, 1–54. MR 594993, DOI 10.1016/0001-8708(80)90056-0
Walter Michaelis, The dual Poincaré-Birkhoff-Witt theorem, Adv. in Math. 57 (1985), no. 2, 93–162. MR 803009, DOI 10.1016/0001-8708(85)90051-9
Walter Michaelis, The primitives of the continuous linear dual of a Hopf algebra as the dual Lie algebra of a Lie coalgebra, Lie algebra and related topics (Madison, WI, 1988) Contemp. Math., vol. 110, Amer. Math. Soc., Providence, RI, 1990, pp. 125–176. MR 1079105, DOI 10.1090/conm/110/1079105
Rimhak Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. (2) 68 (1958), 210–220. MR 100011, DOI 10.2307/1970243
Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
David Wigner, An identity in the free Lie algebra, Proc. Amer. Math. Soc. 106 (1989), no. 3, 639–640. MR 969322, DOI 10.1090/S0002-9939-1989-0969322-5