Related papers: The adjoint braid arrangement as a combinatorial L…
The results of this manuscript is the collection of my articles that I published during my PhD thesis. We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra $\mathcal O$ and homotopy…
We investigate the combinatorial properties of the functional equation $\phi[h(z)]=h(qz)$ for the conjugation of a formal diffeomorphism $\phi$ of $\mathbb{C}$ to its linear part $z\mapsto qz$. This is done by interpreting the functional…
This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We…
We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie…
For affine special linear superalgebra $\widehat{sl}(m|n, \Pi)$ defined by an arbitrary system of simple roots $\Pi$ we define the affine super Yangian $Y_{\hbar}(\widehat{sl}(m|n, \Pi))$ as Hopf superalgebra which is a quantization of…
We develop versions of the Poincar\'e-Birkhoff-Witt and Cartier-Milnor-Moore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogues of a Lie algebra in the setting of a braided monoidal category, using the…
We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the *-algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential…
The purpose of this article is to analyze several Lie algebras associated to "orbit configuration spaces" obtained from a group G acting freely, and properly discontinuously on the upper 1/2-plane H^2. The Lie algebra obtained from the…
A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra. Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in…
Lie algebras are extended to the affine case using the heap operation, giving them a definition that is not dependent on the unique element 0, such that they still adhere to antisymmetry and Jacobi properties. It is then looked at how…
Derived braids have been used to classify categorical structures based on the braid underlying a braided monoidal category V. With four-strand braids underlying the composition morphisms of tensor products of categories enriched over V,…
In this paper, we define and study (co)homology theories of a compatible associative algebra $A$. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define…
The Standard Model of particle physics provides very accurate predictions of phenomena occurring at the sub-atomic level, but the reason for the choice of symmetry group and the large number of particles considered elementary, is still…
Given a commutative algebra $\mathcal{A}$, we exhibit a canonical structure of post-Lie algebra on the space $\mathcal{A}\otimes {\rm Der}(\mathcal{A})$ where ${\rm Der}(\mathcal{A})$ is the space of derivations on $\mathcal{A}$, in order…
The goal of this paper is to present some results and (more importantly) state a number of conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the…
In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then…
In this paper, first we give the notion of a compatible $3$-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible $3$-Lie algebras. We also obtain the bidifferential graded Lie algebra…
We give conditions on a diffeological group $G$ and a normal subgroup $H$ under which the quotient group $G/H$ differentiates to a Lie algebra for which $\operatorname{Lie}(G/H) \cong \operatorname{Lie}(G)/\operatorname{Lie}(H)$. Our Lie…
This paper investigates the geometric and algebraic interplay between F-manifolds and a newly defined class of structures termed F$_\text{man}$-algebras. We specialize our study to the category of F-Lie groups, characterized by a Lie group…
In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations…