Related papers: Integrability from categorification and the 2-Kac-…
In this paper, we give the necessary and sufficient conditions of the integrability of relative Rota-Baxter Lie algebras via double Lie groups, matched pairs of Lie groups and factorization of diffeomorphisms respectively. We use the…
We discuss a new class of strong homotopy algebras constructed via inner deformations. Such deformations have a number of remarkable properties. In the simplest case, every one-parameter family of associative algebras leads to an…
It is well known that the validity of the so called Lenard-Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the…
In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie…
Using a Poisson bracket representation, in 3D, of the Lie algebra $\mathfrak{sl}(2)$, we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras…
Homotopy algebraic methods have become increasingly influential in studying field theories. We consider semi-holomorphic Chern-Simons theory and its relation with the principal chiral model. In particular, we establish an explicit…
In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we…
We describe a simple procedure for constructing a Lax pair for suitable 2-dimensional $\sigma$-models appearing in Poisson-Lie T-duality
The paper naturally continues series of works on identical relations of group rings, enveloping algebras, and other related algebraic structures. Let $L$ be a Lie algebra over a field of characteristic $p>0$. Consider its symmetric algebra…
Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which…
Let $A=F[x,y]$ be the polynomial algebra on two variables $x,y$ over an algebraically closed field $F$ of characteristic zero. Under the Poisson bracket, $A$ is equipped with a natural Lie algebra structure. It is proven that the maximal…
We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all…
Given a homotopy Lie algebra (i.e. an $L_\infty$-algebra) $\mathfrak{g}$, we show concretely how the Lada-Markl $\mathfrak{g}$-modules (i.e. representations) assemble into a symmetric monoidal dg-category. Considering the homotopy…
We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is…
We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these…
Let $A$ be a Cartan matrix and $G(A)$ be the Kac-Moody group associated to Cartan matrix $A$. In this paper, we discuss the computation of the rank $i_k$ of homotopy group $\pi_k(G(A))$. For a large class of Kac-Moody groups, we construct…
The notion of Poisson dialgebras was introduced by Loday. In this article, we propose a new definition with some modifications that is supported by several canonical examples coming from Poisson algebra modules, averaging operators on…
In this paper we introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (i.e., Poisson group(oid)s) and multiplicative closed 2-forms (e.g., symplectic…
In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split…
It is pointed out that affine Lie algebras appear to be the natural mathematical structure underlying the notion of integrability for two-dimensional systems. Their role in the construction and classification of 2D integrable systems is…