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Infinitesimal bialgebras, pre-Lie and dendriform algebras. (English) Zbl 1059.16027
Bergen, Jeffrey (ed.) et al., Hopf algebras. Proceedings from the international conference, DePaul University, Chicago, IL, USA held during the 2001–2002 academic year. New York, NY: Marcel Dekker (ISBN 0-8247-5566-9/pbk). Lect. Notes Pure Appl. Math. 237, 1-33 (2004).
Let \((A,\cdot,\Delta)\) be an infinitesimal bialgebra, i.e., \(A\) is an algebra with product \(\cdot\), a coalgebra with coproduct \(\Delta\) which is also a derivation, \(\Delta(a\cdot b)=a\cdot b_1\otimes b_2+a_1\otimes a_2\cdot b\). Above, Sweedler’s notation for the coproduct is being used, where \(\Delta(a)=a_1\otimes a_2\), and a summation is implicit. No units or counits are assumed. See S. A. Joni and G.-C. Rota [Stud. Appl. Math 61, 93–139 (1979; Zbl 0471.05020)] for more details about infinitesimal bialgebras.
The paper under review introduces the notion of an infinitesimal Hopf bimodule over \(A\) as a vector space \(M\) with operations \[ \lambda\colon A\otimes M\to M,\quad\Lambda\colon M\to A\otimes M,\quad\xi\colon M\otimes A\to M,\quad\Xi\colon M\to M\otimes A \] that satisfy certain compatibility requirements. Proven is that if \(M\) is an infinitesimal Hopf bimodule over \(A\), then \(M\) is also a bimodule over the Drinfeld double, \(D(A)\). Additionally, given an infinitesimal bialgebra \(A\) as above, then the operation \(a\circ b=b_1ab_2\) affords \((A,\circ)\) the structure of a pre-Lie algebra. Also proven is that given a quasitriangular infinitesimal bialgebra \((A,r)\), \(r=\sum u_i\otimes v_i\), then \((A,\succ,\prec)\) becomes a dendriform algebra, where \[ x\succ y=\sum_iu_ixv_iy,\quad x\prec y=\sum_ixy_iyv_i. \] See [J.-L. Loday, Lect. Notes Math. 1763, 7–66 (2001; Zbl 0999.17002)] for a discussion of dendriform algebras. Moreover, if \(A\) is an infinitesimal bialgebra, then there is a dendriform algebra structure on \(\text{End}(A)\oplus A\oplus A^*\). Then it is shown that an infinitesimal Hopf bimodule over \(A\) is a bimodule over the pre-Lie algebra \((A,\circ)\) via \[ a\circ m=m_{-1}am_0+m_0am_1,\quad m\circ a=a_1ma_2. \] This is followed by a construction of an infinitesimal Hopf bimodule from a bimodule over a quasitriangular infinitesimal bialgebra, and these are related to brace algebras. In particular, every infinitesimal bialgebra is a brace algebra, and an outline of the proof is given. Finally, infinitesimal bialgebras are shown to be comonoidal objects in a monoidal category.
For the entire collection see [Zbl 1030.00029].

MSC:
16T10 Bialgebras
16T05 Hopf algebras and their applications
17B62 Lie bialgebras; Lie coalgebras
17A30 Nonassociative algebras satisfying other identities
18D50 Operads (MSC2010)
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