Drinfeld algebra deformations, homotopy comodules and the associahedra

Authors:
Martin Markl and Steve Shnider

Journal:
Trans. Amer. Math. Soc. **348** (1996), 3505-3547

MSC (1991):
Primary 17B37; Secondary 18G60

DOI:
https://doi.org/10.1090/S0002-9947-96-01489-4

MathSciNet review:
1321583

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Abstract: The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra and thus finish the program which began in [13], [14]. The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of a map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition 3.3 below) on the bar resolution of a general Drinfel'd algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in [12].

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Additional Information

**Martin Markl**

Affiliation:
Mathematical Institute of the Academy, Žitná 25, 115 67 Praha 1, Czech Republic

Email:
mark@earn.cvut.cz

**Steve Shnider**

Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

Email:
shnider@bimacs.cs.biu.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-96-01489-4

Keywords:
Quasi-bialgebra,
formal deformations,
multicomplex,
associativity constraints

Received by editor(s):
October 3, 1994

Additional Notes:
The first author partially supported by the National Research Counsel, USA

The second author partially supported by a grant from the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities

Article copyright:
© Copyright 1996
American Mathematical Society