arXiv research

The paper connects Lie bialgebras, Rota-Baxter Lie algebras, and their properties.

The paper integrates Rota-Baxter Lie algebras into Lie group structures and geometries.

Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.

The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.

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

In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. …

Post-groupoids help solve Yang-Baxter equation using quivers.

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…

Monograph explores algebraic structures related to Yang-Baxter equation.

Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.

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

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of a…

The paper describes metrics on left Leibniz algebras, linking them to quadratic Lie algebras.

Every metric symplectic Lie algebra has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets. Finally, we give a scheme to classify the isomorphism classes of metric symplectic Lie algebras and give a complete list …

Every symplectic Lie algebra with degenerate (including non-abelian nilpotent symplectic Lie algebras) has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets. Finally, we give a scheme to classify the isomorphism …

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…

Hurwitz transformations are defined as specific automorphisms of a Cayley-Dickson algebra. These transformations generate quadratic and nonquadratic forms. We investigate here the Hurwitz transformations corresponding to Cayley-Dickson algebras of dimensions 2m = 2, 4 and 8. The Hurwitz transformations which lead to qu…

The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that th…

Following our approach to metric Lie algebras developed in math.DG/0312243 we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a fu…

Find first (0,2) mirror symmetry examples on Hopf surfaces.

A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.

A general scheme for construction of flat pencils of contravariant metrics and Frobenius manifolds as well as related solutions to WDVV associativity equations is formulated. The advantage is taken from the Rota-Baxter identity and some relation being counterpart of the modified Yang-Baxter identity from the classical …

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…

Any classical r-matrix on the Lie algebra of linear operators on a real vector space V gives rise to a quadratic Poisson structure on V which admits a deformation quantization stemming from the construction of V. Drinfel'd. We exhibit in this article an example of quadratic Poisson structure which does not arise this w…

Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration. Sub-Riemannian cubics are cubics which are restricted to move in a horizontal subspace of the tangent space. When the sub-Riemann…

We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of results in equivariant cohomology, including the Chern-Weil theorem for an arbitrary r…

In this paper, the symmetry group of a differential system of n quadratic homogeneous first order ODEs of n variables is studied. For this purpose, we consider the action of both point and contact transformations to signify the corresponding Lie algebras. We also find the independent differential invariants of these ac…

Study of complex structures on Courant algebroids, linking to Poisson structures.

Study on deformations of symmetric spaces using Jordan algebras.

New method preserves MHD equations on sphere without costly matrix exponentials.

The paper constructs compatible Poisson brackets on gl(N).

A new algebra for Frobenius manifolds solves PDEs and constraints.

To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannia…

We establish explicit formulae for canonical factorizations of extended solutions corresponding to harmonic maps of finite uniton number into the exceptional Lie group $G_2$ in terms of the Grassmannian model for the group of based algebraic loops in $G_2$. A description of the ``Frenet frame data" for such harmonic ma…

In this paper, first we give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid TM to a strict Lie 3-algebroid constructed from derivations. Then we introduce the notion of a quadratic Lie 2-algebroid and define its…

The paper studies geometric structures on SL(n,R) induced by the Killing form.

Study on smoothness of special algebra types.

Group quantization applied to finance models.

The paper explores the geometry of algebraic numbers and their roots.

This paper reinterprets and generalizes Hurwitz--Radon numbers using Lie groups and manifolds.

Investigates smoothness of specific algebra structures.

The theory of Vassiliev invariants deals with many modules of diagrams on which the algebra Lambda defined by Pierre Vogel acts. By specifying a quadratic simple Lie superalgebra, one obtains a character on Lambda. We show the coherence of these characters by building a map of graded algebras beetwen Lambda and a quoti…

We show that the Malcev Lie algebra of the fundamental group of a compact $2n+1$-dimensional Sasakian manifold with $n\ge 2$ admits a quadratic presentation by using Morgan's bigradings of minimal models of mixed-Hodge diagrams. By using bigradings of minimal models, we also simplify the proof of the result of Cappelle…

Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatible connection, permit the definition of an adjoint representation, on the fiber as well as on the tangent of the base. We call (positive) quadratic Lie algebroids, Cartan-Lie algebroids with ad-invariant (Riemannian) metrics on their fibers and base $κ$…

In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type $B_n$ and $D_n$. W…

New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.

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

In this letter, first we give a decomposition for any Lie-Poisson structure $π_g$ associated to the modular vector. In particular, $π_g$ splits into two compatible Lie-Poisson structures if $dim{g} \leq 3$. As an application, we classified quadratic deformations of Lie-Poisson structures on $\mathbb R^3$ up to linear d…