Article
Mathematics, Applied
Bjarne Kosmeijer, Hessel Posthuma
Summary: This paper defines a morphism from the deformation complex of a Lie groupoid to the Hochschild complex of its convolution algebra, and proves that it maps the class of a geometric deformation to the algebraic class of the induced deformation in Hochschild cohomology. Applied to the adiabatic groupoid, it is shown that the van Est map to deformation cohomology of Lie algebroids is induced by taking the classical limit of a quantization map on the dual of the Lie algebroid.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Physics, Mathematical
Sergei Merkulov, Marko Zivkovic
Summary: We prove that the action of the Grothendieck-Teichmuller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the duality involution exchanging Lie bracket and Lie cobracket. The proofs are based on a new result in the theory of oriented graph complexes (which can be of independent interest) saying that the involution on an oriented graph complex that changes all directions on edges induces the identity map on its cohomology.
LETTERS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Guilai Liu, Chengming Bai
Summary: This paper introduces a bialgebra theory for transposed Poisson algebras and discusses the concepts of anti-pre-Lie bialgebras and anti-pre-Lie Poisson bialgebras. The paper studies the coboundary cases and the related structures.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2023)
Article
Mathematics
Jan Pulmann, Pavol Severa
Summary: This paper describes a method for quantization of Poisson Hopf algebras in Q-linear symmetric monoidal categories. It is shown that the nerve of a Hopf algebra exhibits braided properties, while the nerve of a Poisson Hopf algebra exhibits infinitesimal braiding. The problem is solved using the standard machinery of Drinfeld associators.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Mohamed Ayadi, Dominique Manchon
Summary: In this paper, we study the twisted pre-Lie algebra of connected finite topological spaces and construct its twisted pre-Lie structure on the doubling space. We describe the enveloping algebra of both twisted pre-Lie algebras and prove that the doubling space is a left module on the twisted pre-Lie algebra. We exhibit natural semi-products of both twisted pre-Lie algebras and both enveloping algebras.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Mathematics
Dan Chen, Yan-Feng Luo, Xiao-Song Peng, Yi Zhang
Summary: The paper introduces the concept of weighted infinitesimal unitary bialgebras and their application on free monoid algebras. By exploring the relationship between weighted infinitesimal unitary bialgebras and pre-Lie algebras, a pre-Lie algebra was constructed on free monoid algebras. Finally, it is shown that the weighted infinitesimal unitary bialgebra on free monoid algebras has an infinitesimal unitary Hopf algebra structure as defined by Aguiar.
COLLOQUIUM MATHEMATICUM
(2021)
Article
Mathematics, Applied
M. Ayadi
Summary: In this paper, we review the construction of a twisted pre-Lie algebra structure on the species of finite connected topological spaces. Then, we construct a corresponding non-coassociative permutative (NAP) coproduct on the subspecies of finite connected T0 topological spaces, and prove the freeness of the generated vector space and the duality between the non-associative permutative product and the proposed NAP coproduct. Finally, we establish that the results hold for finite connected topological spaces.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Mathematics, Applied
Wen Teng, Jiulin Jin, Yu Zhang
Summary: In this paper, we generalize known results of nonabelian embedding tensor to the Hom setting. We introduce the concept of Hom-Leibniz-Lie algebra, which is the basic algebraic structure of nonabelian embedded tensors on Hom-Lie algebras and can also be regarded as a nonabelian generalization of Hom-Leibniz algebra. Moreover, we define a cohomology of nonabelian embedding tensors on Hom-Lie algebras with coefficients in a suitable representation. The first cohomology group is used to describe infinitesimal deformations as an application. In addition, Nijenhuis elements are used to describe trivial infinitesimal deformations.
Article
Physics, Mathematical
Honglei Lang, Yunhe Sheng
Summary: This paper introduces the concept of quadratic Rota-Baxter Lie algebras of arbitrary weight and explores their applications in Lie bialgebras and Lie groups. It also establishes several correspondences between different mathematical structures.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Mathematics
Jianzhi Han, Haisheng Li, Yukun Xiao
Summary: This paper investigates the structure of cocommutative vertex bialgebras. It is proven that for a general vertex bialgebra V, the set of group-like elements G(V) is naturally an abelian semigroup, and the set of primitive elements P(V) is a vertex Lie algebra. The main results show that a cocommutative vertex bialgebra V can be decomposed as V = V-g + V1, where V-g is a connected component containing a group-like element g, V-1 is a vertex subbialgebra isomorphic to VP(V), and V-g is a V1-module. It is also shown that cocommutative connected vertex bialgebras are isomorphic to vertex Lie algebras. Additionally, when G(V) is a group and lies in the center of V, the coalgebra structure of V is explicitly determined as V = V-P(V)⨁C[G(V)].
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Andrey Lazarev, Yunhe Sheng, Rong Tang
Summary: We study L infinity-algebras governing triangular L infinity-bialgebras and homotopy relative Rota-Baxter Lie algebras, and establish a mapping between them. Our formulas are derived from a functorial approach to Voronov's construction of higher derived brackets, which is independently significant.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Ming Chen, Jiefeng Liu, Yao Ma
Summary: Based on the differential graded Lie algebra controlling deformations of an n-Lie algebra with a representation (called an n-LieRep pair), we construct a Lie n-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on n-LieRep pairs. The notion of an n-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order m deformations to order m + 1 deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on n-LieRep pairs and those on (n + 1)-LieRep pairs by certain linear functions.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Mathematics, Applied
Chengming Bai, Li Guo, Yunhe Sheng
Summary: This paper explores the relationships among the widely applied notions of Lie bialgebras, Manin triples, classical r-matrices, and O-operators of Lie algebras. It introduces the concept of coherent homomorphisms to understand these notions as classes of objects and maps consistently. Additionally, the paper extends these concepts to endo Lie algebras and establishes a correspondence with the category of pre-Lie algebras.
FORUM MATHEMATICUM
(2022)
Article
Physics, Multidisciplinary
Vladislav G. Kupriyanov, Richard J. Szabo
Summary: We present general definitions of semi-classical gauge transformations in noncommutative gauge theories in string theory backgrounds using techniques based on symplectic embeddings. Our technique produces novel explicit constructions that are simpler than previous approaches, and we demonstrate its applicability in various examples in noncommutative field theory and gravity. Additionally, we show that our symplectic embeddings naturally define a P (infinity)-structure on almost Poisson manifolds, leading to a new approach for defining noncommutative gauge theories beyond the gauge sector and the semi-classical limit based on A (infinity)-algebras.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Mathematics, Applied
Naihuan Jing, Fei Kong, Haisheng Li, Shaobin Tan
Summary: This paper continues the previous study on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. The authors investigate the notion of right H-comodule nonlocal vertex algebra and provide constructions for deformations of vertex algebras and coordinated quasi modules for smash product nonlocal vertex algebras. An example of quantum vertex algebras is also given using deformations of vertex algebras associated with non-degenerate even lattices.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2022)
Article
Mathematics, Applied
Johan Alm, Sergei Merkulov
JOURNAL OF NONCOMMUTATIVE GEOMETRY
(2015)
Article
Mathematics
Ricardo Campos, Sergei Merkulov, Thomas Willwacher
DUKE MATHEMATICAL JOURNAL
(2016)
Article
Physics, Mathematical
Anton Khoroshkin, Sergei Merkulov, Thomas Willwacher
LETTERS IN MATHEMATICAL PHYSICS
(2016)
Article
Mathematics
Sergei A. Merkulov
ALGEBRA & NUMBER THEORY
(2008)
Article
Mathematics
Sergei A. Merkulov
BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY
(2011)
Article
Physics, Mathematical
S. A. Merkulov
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2010)
Article
Mathematics
Sergei Merkulov, Bruno Vallette
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2009)
Article
Mathematics, Applied
M. Markl, S. Merkulov, S. Shadrin
JOURNAL OF PURE AND APPLIED ALGEBRA
(2009)
Article
Physics, Mathematical
Sergei Merkulov, Thomas Willwacher
LETTERS IN MATHEMATICAL PHYSICS
(2014)
Article
Physics, Mathematical
Sergei Merkulov, Thomas Willwacher
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2018)
Article
Physics, Mathematical
Sergei Merkulov
LETTERS IN MATHEMATICAL PHYSICS
(2020)
Article
Mathematics
Assar Andersson, Sergei Merkulov
Summary: This study focuses on the homotopy theory of wheeled props controlling Poisson structures on formal graded finite-dimensional manifolds, proving that the Grothendieck-Teichmuller group acts faithfully and homotopy nontrivially on this wheeled prop. The homotopy theory is then applied to the deformation complex of an arbitrary Kontsevich formality map, with calculations of the full cohomology group in terms of the cohomology of a certain graph complex introduced earlier by Kontsevich and studied by Willwacher.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Physics, Mathematical
SA Merkulov
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2006)
Article
Mathematics
SA Merkulov
COMPOSITIO MATHEMATICA
(2005)
