Related papers: The left-greedy Lie algebra basis and star graphs
In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Groebner-Shirshov bases of free Rota-Baxter algebra, $\lambda$-differential algebra and…
The set-theoretic large cardinal axiom known as I3 posits the existence of a non-trivial rank-to-rank embedding from an initial segment of the universe of sets into itself. Laver showed that the algebra generated by a single such embedding…
In this paper, we study the concept of free pre-Lie algebra generated by a (non-empty) set. We review the construction of A. Agrachev and R. Gamkrelidze of monomial bases in free pre-Lie algebras. We describe the matrix of the monomial…
We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from…
This paper aims to construct two graded Lie algebras associated with a nonsymmetric operad with multiplication. Maurer-Cartan elements of these graded Lie algebras correspond respectively to Nijenhuis elements and Rota-Baxter elements for…
We construct HNN-extensions of Lie di-algebras in the variety of di-algebras and provide a presentation for the replicated HNN-extension of a Lie di-algebras. Then, by applying the method of Gr\"obner-Shirshov bases for replicated algebras,…
We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-$\Sigma$ operad, preoperad) of rooted planar trees. We determine whether…
We introduce braided Lie bialgebras as the infinitesimal version of braided groups. They are Lie algebras and Lie coalgebras with the coboundary of the Lie cobracket an infinitesimal braiding. We provide theorems of transmutation, Lie…
We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is…
Let $\mathfrak g$ be a Borcherds-Kac-Moody Lie superalgebra (BKM superalgebra in short) with the associated graph $G$. Any such $\mathfrak g$ is constructed from a free Lie superalgebra by introducing three different sets of relations on…
Quite much recent studies has been attracted to the operated algebra since it unifies various notions such as the differential algebra and the Rota-Baxter algebra. An $\Omega$-operated algebra is a an (associative) algebra equipped with a…
We consider a method popular in the literature of associating a two-step nilpotent Lie algebra with a finite simple graph. We prove that the two-step nilpotent Lie algebras associated with two graphs are Lie isomorphic if and only if the…
We give some properties of cosymplectic Lie algebras, we show, in particular, that they support a left symmetric product. We also give some constructions of cosymplectic Lie algebras, as well as a classification in three and…
We describe a canonical decomposition of the cohomology of the Dani-Mainkar metabelian Lie algebras associated with graphs. As applications, we obtain explicit formulas for the third cohomology of any Dani-Mainkar Lie algebra and for the…
Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras $\Psi: \H_{2\ast} (L M) \to {\o}(\Mc)$, where $\H_{2\ast} (LM)$ is the even dimensional part of…
Consider the class RBLie of Lie algebras equipped with a Rota---Baxter operator. Then the forgetful functor RBLie --> Lie has a left adjoint one denoted by $U_{RB}(\cdot)$. We prove an "operator" analogue of the Poincare---Birkhoff---Witt…
Many interesting examples of operator algebras, both self-adjoint and non-self-adjoint, can be constructed from directed graphs. In this survey, we overview the construction of $C^*$-algebras from directed graphs and from two…
In this paper, we consider the construction of linear lexicodes over finite chain rings by using a $B$-ordering over these rings and a selection criterion. % and a greedy Algorithm. As examples we give lexicodes over $\mathbb{Z}_4$ and…
We say that a nonselfadjoint operator algebra is partly free if it contains a free semigroup algebra. Motivation for such algebras occurs in the setting of what we call free semigroupoid algebras. These are the weak operator topology closed…
A theorem of Kontsevich relates the homology of certain infinite dimensional Lie algebras to graph homology. We formulate this theorem using the language of reversible operads and mated species. All ideas are explained using a pictorial…