Related papers: Rota-Baxter Algebras and Dendriform Algebras
A Rota-Baxter Leibniz algebra is a Leibniz algebra $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual representation of Rota-Baxter…
The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are…
Loday's dendriform algebras and its siblings pre-Lie and zinbiel have received attention over the past two decades. In recent literature, there has been interest in a generalization of these types of algebra in which each individual…
In this paper, we develop the theory of multiple Rota-Baxter modules over multiple Rota-Baxter algebras. We introduce left, right, and bimodule structures and construct free $\Omega$-operated modules with mixable tensor establishing free…
The concept of Rota-Baxter family algebra is a generalization of Rota-Baxter algebra. It appears naturally in the algebraic aspects of renormalizations in quantum field theory. Rota-Baxter family algebras are closely related to dendriform…
In [{\it On the free implicative semilattice extension of a Hilbert algebra}. Mathematical Logic Quarterly 58, 3 (2012), 188--207], Celani and Jansana give an explicit description of the free implicative semilattice extension of a Hilbert…
Tridendriform algebras are a type of associative algebras, introduced independently by F. Chapoton and by J.-L. Loday and the third author, in order to describe operads related to the Stasheff polytopes. The vector space $\st$ spanned by…
In this paper, we introduce relative Rota-Baxter systems on Leibniz algebras and give some characterizations and new constructions. Then we construct a graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter systems. This…
The free algebra adjunction, between the category of algebras of a monad and the underlying category, induces a comonad on the category of algebras. The coalgebras of this comonad are the topic of study in this paper (following earlier…
In this paper, we consider Rota-Baxter operators on involutive associative algebras. We define cohomology for Rota-Baxter operators on involutive algebras that governs the formal deformation of the operator. This cohomology can be seen as…
We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some…
Aromatic Butcher series were successfully introduced for the study and design of numerical integrators that preserve volume while solving differential equations in Euclidean spaces. They are naturally associated to pre-Lie-Rinehart algebras…
We introduce analogues of algebraic groups called algebraic racks, which are pointed rack objects in the category of schemes over a ground field. Addressing a problem of Loday, we construct functors assigning left and right Leibniz algebras…
We establish a bialgebra structure on Rota-Baxter Lie algebras following the Manin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebras are characterized by generalizing matched pairs of Lie algebras and Manin triples…
As Hopf truss analogues of Rota-Baxter Hopf algebras, the notion of Rota-Baxter systems of Hopf algebras is proposed. We study the relatiohship between Rota-Baxter systems of Hopf algebras and Rota-Baxter Hopf algebras, show that there is a…
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that…
We introduce a notion of ternary $F$-manifold algebras which is a generalization of $F$-manifold algebras. We study representation theory of ternary $F$-manifold algebras. In particular, we introduce a notion of dual representation which…
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted…
We exhibit an adjunction between a category of abstract algebras of partial functions that we call difference-restriction algebras and a category of Hausdorff \'etale spaces. Difference-restriction algebras are those algebras isomorphic to…
This paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving…