Throughout, all algebras, coalgebras and modules are taken over a unitary commutative ring \(k\).
In this context, an infinitesimal bialgebra of weight \(\lambda\in k\) is a module \(A\) which is simultaneously an algebra (with multiplication \(\cdot\)) and a coalgebra (with comultiplication \(\Delta\)) such that \[\Delta(ab) = a\cdot \Delta(b) + \Delta(a) \cdot b + \lambda(a\otimes b) \] for all \(a, b\in A\). Note that the authors do not require algebras (resp. coalgebras) to have a unit (resp. a counit): if \(A\) is additionally a unitary algebra, then it is an infinitesimal unitary bialgebra of weight \(\lambda\). (The case when \(A\) is both unitary and counitary, and \(\lambda = 0\), is known to be trivial: see Remark 2.3(2).)
This notion is a generalisation of several other definitions of infinitesimal bialgebras in the literature: references to previous work on the specific cases \(\lambda = 0\) and \(\lambda = -1\) are given in the Introduction, and Remark 2.3(3) explains how to reduce from the case \(\lambda\in k^\times\) to the case \(\lambda = -1\).
Any unitary algebra \(A\) can be made into an infinitesimal unitary bialgebra of weight 0 in a trivial way by setting \(\Delta = 0\). In Example 2.4, the authors list three other known examples with nontrivial coproducts: two of weight 0 (the polynomial algebra \(k[x_1, x_2, \dots]\), and the path algebra \(kQ\) of a quiver \(Q\)) and one of weight -1 (the tensor algebra \(T(V)\) for a \(k\)-module \(V\)).
The main result of §2.2 is Theorem 2.11: if \(M^*\) is the free monoid on the set \(M\), \(kM^*\) is its monoid algebra over \(k\), and \(\lambda \in k\) is arbitrary, then the comultiplication \(\Delta_\varepsilon\) defined recursively by \[ \begin{cases} \Delta_\varepsilon(1) = -\lambda(1\otimes 1),\\ \Delta_\varepsilon(a_1) = 1\otimes 1,\\ \Delta_\varepsilon(a_1 w) = a_1\cdot \Delta_\varepsilon(w) + \Delta_\varepsilon(a_1) \cdot w + \lambda a_1\otimes w, \end{cases}\] (for all \(a_1\in M\) and \(w\in M^*\)) defines the structure of a weighted infinitesimal unitary bialgebra of weight \(\lambda\) on \(kM^*\). This allows the authors to deduce in Theorem 2.19 that this \(kM^*\) has the structure of a pre-Lie algebra, and hence also of a Lie algebra.
An explicit calculation of \(\Delta_\varepsilon(x^n)\) in the simplest nontrivial case, when \(M = \{x\}\) and \(kM^* = k[x]\), is recorded as Example 2.4(5).
In Theorem 3.12, it is shown that (in the special case when \(\lambda = 0\) and \(k\) is a field of characteristic \(0\)) \(kM^*\) may be given a bijective antipode, making it into an infinitesimal unitary Hopf algebra (of weight 0) in the sense of Aguiar, but with a coproduct that appears to be new.