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A Short Note on the Conjugation in the Leibniz-Hopf Algebra
Neşet Deniz Turgay
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DOI:10.17265/2159-5291/2014.01.006
Independent Scholar, 126/7 Sok. No:10/1, Daire 3, Evka 3, Bornova- Izmir 35050, Turkey
The Leibniz-Hopf algebra is the free associativealgebra with one generator in each positive degree and coproduct is given by the Cartan formula. It has been also known as the ‘ring of noncommutative symmetric functions’ [1], and to be isomorphic to the Solomon Descent algebra [12]. This Hopf algebra has links with algebra,topology and combinatorics. In this article we consider another approach of proof for the antipode formula in the Leibniz-Hopf algebra by using some properties of words in [2].
Hopf algebra, Leibniz-Hopf algebra, antipode, Steenrod algebra.
[2] N.D.Turgay, Some notes on the dual Leibniz-Hopf algebra, Journal of Mathematics and System Science 3 (2013) 573-576.
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[4] N.E. Steenrod, Cohomology Operations. Ann. Math. Studies, Vol. 50. Princeton: Princeton Univ. Press.1962.
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[10] M.D. Crossley, Some Hopf Algerabs of Words, Glasg. Math. J. 48 (2006) 575-582.
[11] R. Ehrenborg, On posets and Hopf algebras, Adv. Math. 119 (1996) 1-25.
[12] C. Malvenuto, C. Reutenauer, Duality between Quasi-Symmetric Functions and the Solomon Descent Algebra, J. Algebra 177 (1995) 967-982.
[13] N.D. Turgay, Conjugation invariants in Word Hopf algebras, Ph.D. Thesis, Swansea University, Swansea, 2012.
[14] M.E. Sweedler Hopf algebras, Benjamin, 1969.