arXiv research

We extend the construction of the Hennings TQFT for ribbon Hopf algebras to the case of ribbon quasi-Hopf algebras as defined by Drinfeld. Calculations proceed in a similar fashion to the ordinary Hopf algebra case, but also require the handling of the non-trivial coassociator in the triple tensor product of the algebr…

New 3D TQFTs derived from non-semisimple categories.

New invariants from symplectic fermions categorify link and manifold structures.

Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z:\mathcal{B}\to \widehat{\mathbb{A}}$, where $\mathcal{B}$ is the category of bottom tangles in handlebodies and $\widehat{\mathbb{A}}$ is the degree-com…

Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the $SL(2,\mathbb{R})$-action is ergodic in co…

New algebraic structure derived from Hopf algebra and Drinfel'd twist.

Study of cluster and skein algebras for surfaces, showing their connection.

Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.

Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.

We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …

New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.

A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…

Study biderivations in complete Leibniz algebras, extending Lie algebra results.

The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.

The paper classifies Lie algebras with special operators.

Symmetric spaces' connections form Lie admissible triple algebras.

The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.

The paper investigates gradings of complex simple Lie algebras, focusing on $|3|$-gradings and their algebraic structures.

New algebra pong algebra computed for knot Floer homology.

Study resolves conjecture linking two algebraic structures on surfaces.

Study on pre-Lie structures for semisimple Lie algebras over C.

Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.

Characterizes G2-structures on Lie algebras with non-trivial center.

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms o…

This paper presents results on the framization of some knot algebras, defined by the authors. We explain the motivations of the concept of framization, coming from the Yokonuma--Hecke algebras, as well as recent results on the framization of the Temperley--Lieb algebra. Finally, we propose framizations for other knot a…

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algeb…

Similarity algebra extends algebraic structures with quantitative bounds.

Extends Loday-Quillen-Tsygan theorem to bornological Lie algebra homology.

New algebra for twice-punctured torus curves.

We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functio…

New braided Frobenius algebras created from specific Hopf algebras.

Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We stud…

The paper examines differential smoothness in specific algebra types.

Generalizes Hecke algebra for double torus, linking to skein algebra.

2-compatible Lie algebras are quadratic deformations of Lie algebras with specific constraints.

A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the categ…

In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…

We construct the Lie algebra of an n-Lie algebra and we also define the notion of cohomology of an n-Lie algebra.

The study describes the free Lie-Yamaguti algebra.

We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T*-extension of a nilpotent algebra admitting an invertiblederivation and also as the double extension of…

We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space $V$. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtai…

Surveying recent work on Kähler metrics and algebraic variety stability.

If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g_i's, we say that g is simply assembled from the g_i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the g_i's, one obtains a Lie algebra assembled in two step…

Study proves certain algebraic structures are symmetric Frobenius algebras.

The paper explores handlebody versions of various diagram algebras.

A new algebraic structure emerges from reductive homogeneous spaces.

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo $H$-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type $B_n$ with $|2|$-grading do not contain non-Heisenberg pseudo $H$-type Li…

The paper studies the center of the Goldman Lie algebra and its properties.