Article
Mathematics
Qingfeng Ma, Lina Song, You Wang
Summary: This paper studies non-abelian extensions of pre-Lie algebras. It introduces the concept of non-abelian cohomology and bimultipliers for pre-Lie algebras, and establishes a correspondence between isomorphic classes of non-abelian extensions and homotopy classes of homomorphisms. Furthermore, it classifies non-abelian extensions of pre-Lie algebras using Maurer-Cartan elements.
COMMUNICATIONS IN ALGEBRA
(2023)
Article
Mathematics
Lei Du, Youjun Tan
Summary: For a Lie coalgebra with a Lie comodule, a cohomology can be constructed using a subcomplex of the Chevalley-Eilenberg cochain complex of the dual Lie algebra. It is shown that the first (resp. the second) order cohomology group corresponds to the space of outer coderivations (resp. the space of equivalent classes of abelian extensions of the Lie coalgebra by its comodule).
COMMUNICATIONS IN ALGEBRA
(2021)
Article
Mathematics
Pietro Corvaja, Andrei S. Rapinchuk, Jinbo Ren, Umberto M. Zannier
Summary: By proving that a linear group Gamma subset of GL(n)(K) is virtually solvable when boundedly generated by semi-simple elements, we conclude that infinite S-arithmetic subgroups of absolutely almost simple anisotropic algebraic groups over number fields are never boundedly generated. This proof relies on Laurent's theorem from Diophantine geometry and properties of generic elements.
INVENTIONES MATHEMATICAE
(2022)
Article
Mathematics, Applied
Lei Du, Youjun Tan
Summary: For a Lie comodule M of a Lie coalgebra C, we demonstrate the existence of a representation of diagonal coderivations and Lie coalgebra automorphisms on the second cohomology group H-c(2)(M, C), which allows us to construct obstruction classes for the extensibility of pairs of coderivations and Lie coalgebra automorphisms, as well as exact sequences of Wells type for abelian extensions of Lie coalgebras.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics
Xueru Wu, Yao Ma, Liangyun Chen
Summary: This paper discusses the representation problem of .2 and A as Lie triple systems. By constructing third-order cohomology classes, a Lie algebra with representation is obtained, and the obstruction classes for extending the derivations of A and .2 to .2 tilde are studied. An application is also discussed.
ELECTRONIC RESEARCH ARCHIVE
(2022)
Article
Mathematics
Qiang Li, Lili Ma
Summary: This paper establishes the representations and cohomologies of Hom-delta-Jordan Lie supertriple systems. As an application, Nijenhuis operators and abelian extensions of Hom-delta-Jordan Lie supertriple systems are explored. The infinitesimal deformation generated by a Nijenhuis operator is obtained, and the sufficient and necessary condition for the equivalence of abelian extensions of Hom-delta-Jordan Lie supertriple systems is derived.
Article
Mathematics, Applied
Satyendra Kumar Mishra, Apurba Das, Samir Kumar Hazra
Summary: This paper investigates non-abelian extensions of a Rota-Baxter Lie algebra gT with another Rota-Baxter Lie algebra hS. The non-abelian cohomology H2nab(gT, hS) is defined to classify such extensions. Furthermore, it is shown that the obstruction for a pair of Rota-Baxter automorphisms to be induced by an automorphism lies in the cohomology group H2nab(gT, hS). The Wells short-exact sequence in the context of Rota-Baxter Lie algebras is obtained along with the relationship between these results and abelian extensions.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Mathematics
Ye Liu, Tan Nhat Tran, Masahiko Yoshinaga
Summary: The study introduces and examines a G-Tutte polynomial associated with finite generated abelian groups and an abelian group G, which carries topological and enumerative information regarding abelian Lie group arrangements. The G-Tutte polynomial is a generalization of various polynomial concepts and arithmetic models, showing differences with arithmetic Tutte polynomials.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Astronomy & Astrophysics
A. Mironov, V Mishnyakov, A. Morozov
Summary: W-representation is a method for defining an exact partition function by integrating Ward identities on unity, commonly used in various eigenvalue matrix models. This method can also be extended to monomial generalized Kontsevich models, introducing ordered P-exponentials for non-commuting operators of different gradings.
Article
Physics, Particles & Fields
Lev Astrakhantsev, Ilya Bakhmatov, Edvard T. Musaev
Summary: The standard fermionic T-duality field transformation rules require fermionic isometries to anticommute, leading to complexification of the Killing spinors and complex valued dual backgrounds. However, by generalizing the field transformations to include non-anticommuting fermionic isometries, the resulting backgrounds are shown to be solutions of double field theory. Explicit examples of non-abelian fermionic T-dualities that produce real backgrounds are provided, with some examples able to be bosonic T-dualized into usual supergravity solutions while others are genuinely non-geometric. Comparison with an alternative treatment based on sigma models on supercosets shows consistency.
JOURNAL OF HIGH ENERGY PHYSICS
(2021)
Article
Mathematics, Applied
Bachuki Mesablishvili
Summary: The descent cohomology theory of monads includes Serre's non-abelian Galois cohomology theory as a special case, as shown in the research.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics
Qiang Li, Lili Ma
Summary: This paper aims to discuss Lie color triple systems and establish their cohomology theory. It further investigates 1-parameter formal deformations and abelian extensions of Lie color triple systems using cohomology.
ELECTRONIC RESEARCH ARCHIVE
(2022)
Article
Mathematics, Applied
Alexey Basalaev, Andrei Ionov
Summary: This study investigates the non-abelian maximal group of symmetries of a polynomial and the Hochschild cohomology of its equivariant matrix factorizations. It introduces a pairing on the cohomology and proves it to be a Frobenius algebra.
JOURNAL OF GEOMETRY AND PHYSICS
(2022)
Article
Astronomy & Astrophysics
Sourav Roychowdhury, Prasanta K. Tripathy
Summary: This paper examines the Klebanov-Tseytlin background and its non-Abelian T-dual geometry, analyzing Penrose limits along various null geodesics. It is found that the Klebanov-Tseytlin geometry does not admit any pp-wave solutions, but the T-dual background does. The study also comments on the potential gauge theory dual for the pp-wave background.
Article
Mathematics
Shanshan Liu, Abdenacer Makhlouf, Lina Song
Summary: The main purpose of this paper is to study the cohomology of Hom-pre-Lie algebras with coefficients in a given representation, and to show its classification properties.
ELECTRONIC RESEARCH ARCHIVE
(2022)
Article
Mathematics
Matija Bucic, Richard Montgomery
Summary: This article improves upon previous research by showing that any n-vertex graph can be decomposed into O(n log* n) cycles and edges.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu
Summary: The paper investigates the relationship between linear optimization degree and geometric structure. By analyzing the geometric structure of the conormal variety of an affine variety, the Chern-Mather classes of the given variety can be completely determined. Additionally, the paper shows that these bidegrees coincide with the linear optimization degrees of generic affine sections.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
William Chan, Stephen Jackson, Nam Trang
Summary: Under the determinacy hypothesis, this paper completely characterizes the existence of nontrivial maximal almost disjoint families for specific cardinals kappa, considering the ideals of bounded subsets and subsets of cardinality less than kappa.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhenguo Liang, Zhiyan Zhao, Qi Zhou
Summary: This paper investigates the reducibility of the one-dimensional quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form. It provides a description and upper bound for the growth of the Sobolev norms of the solution, and demonstrates the optimality of the upper bound.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhao Yu Ma, Yair Shenfeld
Summary: This study provides a new approach to understanding the extremal cases of Stanley's inequalities by establishing a connection between the combinatorics of partially ordered sets and the geometry of convex polytopes.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurent Laurent, Rosa M. Miro-Roig
Summary: This paper discusses the problem of constructing matrices of linear forms of constant rank by focusing on vector bundles on projective spaces. It introduces important examples of classical Steiner bundles and Drezet bundles, and uses the classification of globally generated vector bundles to describe completely the indecomposable matrices of constant rank up to six.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Nicoletta Cantarini, Fabrizio Caselli, Victor Kac
Summary: In this paper, we construct a duality functor in the category of continuous representations to study the Lie superalgebra E(4, 4). By constructing a specific type of Lie conformal superalgebra, we obtain that E(4, 4) is its annihilation algebra. Furthermore, we also obtain an explicit realization of E(4, 4) on a supermanifold in the process of studying.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Rotem Assouline, Bo'az Klartag
Summary: This article studies the horocyclic Minkowski sum of two subsets in the hyperbolic plane and its properties. It proves an inequality relating the area of the subsets when they are Borel-measurable, and provides a connection to other inequalities.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Alessio Porretta
Summary: This article discusses Fokker-Planck equations driven by Levy processes in the entire Euclidean space, under the influence of confining drifts, similar to the classical Ornstein-Ulhenbeck model. A new PDE method is introduced to obtain exponential or sub-exponential decay rates of zero average solutions as time goes to infinity, under certain diffusivity conditions on the Levy process, including the fractional Laplace operator as a model example. The approach relies on long-time oscillation estimates of the adjoint problem and applies to both local and nonlocal diffusions, as well as strongly or weakly confining drifts.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Weichao Qian, Yong Li, Xue Yang
Summary: In this paper, we investigate the persistence of resonant invariant tori in Hamiltonian systems with high-order degenerate perturbation, and prove a quasiperiodic Poincare theorem under high degeneracy, answering a long-standing conjecture on the persistence of resonant invariant tori in general situations.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Julius Ross, David Witt Nystroem
Summary: This article extends Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to F-subharmonicity, and applies it to the interpolation problem of convex functions and convex sets, introducing a new notion of harmonic interpolation.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Airi Takeuchi, Lei Zhao
Summary: In this article, we explore the connection between several integrable mechanical billiards in the plane through conformal transformations. We discuss the equivalence of free billiards and central force problems, as well as the correspondence between integrable Hooke-Kepler billiards. We also investigate the integrability of Kepler billiards and Stark billiards, and the relationship between billiard systems and Euler's two-center problems.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Damiano Rossi
Summary: In this study, we prove new results in generalised Harish-Chandra theory by providing a description of the Brauer-Lusztig blocks using the p-adic cohomology of Deligne-Lusztig varieties. We then propose new conjectures for finite reductive groups by considering geometric analogues of the p-local structures. Our conjectures coincide with the counting conjectures for large primes, thanks to a connection established between p-structures and their geometric counterparts. Finally, we simplify our conjectures by reducing them to the verification of Clifford theoretic properties.
ADVANCES IN MATHEMATICS
(2024)