The paper under review studies in detail and further develops one of the topics considered in the program paper from the same volume of the Springer Lecture Notes in Mathematics [J.-L. Loday, ibid. 1763, 7-66 (2001; Zbl 0999.17002)].
Loday introduced for dialgebras a natural homology \(HY_{\ast}\) with trivial coefficients. The main purpose of the present paper is to define the dialgebra (co)homology with coefficients, which, in the case of constant coefficients, recovers the considerations of Loday. First, the notion of coefficients is made explicit in this context. Then the author constructs the theory and performs a few important computations. As in the case of constant coefficients, again one finds some nice combinatorial relations with planar binary trees and operations on them. A feature of the theory \(HY\) developed by the author is that the categories of coefficients for homology and cohomology are different.
In the first two sections the author considers dialgebras, their cohomology with coefficients in representations and abelian extensions of dialgebras. Section 3 studies universal enveloping algebras of dialgebras and the analogue of the Poincaré-Birkhoff-Witt theorem. Section 4 is devoted to dialgebra homology and corepresentations. Section 5 studies dialgebra (co)homology as a derived functor. Section 6 considers the homology of bar-unital dialgebras, where the dialgebra is called bar-unital if there exists a bar-unit 1 such that \(1\vdash a=a\dashv 1=a\) for all elements \(a\) of the dialgebra. Finally, the author studies in Section 7 the \(HY\)-unital dialgebras with the characteristic property \(HY_n=0\) for all \(n\geq 1\). (The bar-unital algebras are always \(HY\)-unital.)
For the third and fourth paper in this series see Zbl 0999.17004 and Zbl 0999.17005.
For the entire collection see [Zbl 0970.00010].