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In this paper we construct "structural" pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand,…
A metaphor of Loday describes Lie, associative, and commutative associative algebras as ``the three graces'' of the operad theory. In this article, we study the three graces in the category of $\mathfrak{sl}_2$-modules that are sums of…
We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra \pi_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology…
We argue that one can relax the requirements of the non-associative three-algebras recently used in constructing D=3, N=8 superconformal field theories, and introduce the notion of ``relaxed three-algebras''. We present a specific…
For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3-manifolds, including the ``piecewise geometric'' ones in the sense of…
We study real and complex Manin triples for a complex reductive Lie algebra, $\g$. The first part includes, and extends to complex Manin triples, our earlier work [De]. First, we generalize results of E. Karolinsky, on the classification of…
Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with…
We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles…
We regard the classification of rational homotopy types as a problem in algebraic deformation theory: any space with given cohomology is a perturbation, or deformation, of the "formal" space with that cohomology. The classifying space is…
Lie theory is, beyond any doubt, an absolutely essential part of differential geometry. It is therefore necessary to seek its generalization to $\mathbb{Z}$-graded geometry. In particular, it is vital to construct non-trivial and explicit…
Higher structures - infinity algebras and other objects up to homotopy, categorified algebras, `oidified' concepts, operads, higher categories, higher Lie theory, higher gauge theory... - are currently intensively investigated in…
Homomorphic expansions are combinatorial invariants of knotted objects, which are universal in the sense that all finite-type (Vassiliev) invariants factor through them. Homomorphic expansions are also important as bridging objects between…
The three-manifold topological invariants $\hat Z$ capture the half-index of the three-dimensional theory with ${\mathcal{N}}=2$ supersymmetry obtained by compactifying the M5 brane theory on the closed three-manifold. In 2019, surprising…
Clifford geometric algebras of multivectors are introduced which exhibit a bilinear form which is not necessarily symmetric. Looking at a subset of bi-vectors in CL(K^{2n},B), we proof that theses elements generate the Hecke algebra…
The notion of a Hom-Leibniz bialgebra is introduced and it is shown that matched pairs of Hom-Leibniz algebras, Manin triples of Hom-Leibniz algebras and Hom-Leibniz bialgebras are equivalent in a certain sense. The notion of Hom-Leibniz…
To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group B_n. Using Hansen's polynomial covering space theory, we give a new interpretation of…
We give an overview of some of the main results in geometric representation theory that have been proved by means of the Steinberg variety. Steinberg's insight was to use such a variety of triples in order to prove a conjectured formula by…
We study and classify Lie algebras, homogeneous spacetimes and coadjoint orbits ("particles") of Lie groups generated by spatial rotations, temporal and spatial translations and an additional scalar generator. As a first step we classify…
It is easy to find algebras $\mathbb{T}\in\mathcal{C}$ in a finite tensor category $\mathcal{C}$ that naturally come with a lift to a braided commutative algebra $\mathsf{T}\in Z(\mathcal{C})$ in the Drinfeld center of $\mathcal{C}$. In…
We define the notion of hom-Batalin-Vilkovisky algebras and strong differential hom-Gerstenhaber algebras as a special class of hom-Gerstenhaber algebras and provide canonical examples associated to some well-known hom-structures.…