Tootfinder

Fractional-charge hadrons and leptons to tell the Standard Model group apart
Rodrigo Alonso, Despoina Dimakou, Mia West
https://arxiv.org/abs/2404.03438 ht…

Fractional-charge hadrons and leptons to tell the Standard Model group apart
The gauge group of strong and electroweak interactions in Nature could be any of the four that share the same Lie algebra, $SU(3)_c\times SU(2)_L\times U(1)_Y/Z_p\equiv G_p$ with $Z_p=\left\{Z_6,Z_3,Z_2,Z_1\right\}$. Each of these cases allows in its spectrum for the matter fields of the SM but also for new distinctive representations, e.g. under the assumption that $q_L$ possesses the minimum possible hypercharge in Nature, $G_p$ allows for particles with a multiple of $p\,e/6$ for electric ch…

Non-abelian extensions of relative Rota-Baxter Lie algebras and Wells type exact sequences
Qinxiu Sun, Qianwen Zhu
https://arxiv.org/abs/2404.02752 https:/…

Non-abelian extensions of relative Rota-Baxter Lie algebras and Wells type exact sequences
In this paper, we explore non-abelian extensions of relative Rota-Baxter Lie algebras and classify the non-abelian extensions by introducing the non-abelian second cohomology group. We also study the inducibility of a pair of automorphisms about a non-abelian extension of relative Rota-Baxter Lie algebras and derive the Wells type exact sequences. Finally, we investigate the inducibility problem of pairs of derivations about an abelian extension of relative Rota-Baxter Lie algebras and give an …

Cyclic Higgs bundles and the Toledo invariant
Oscar Garc\'ia-Prada, Miguel Gonz\'alez
https://arxiv.org/abs/2403.00415 https://

Cyclic Higgs bundles and the Toledo invariant
Let $G$ be a complex semisimple Lie group and $\mathfrak g$ its Lie algebra. In this paper, we study a special class of cyclic Higgs bundles constructed from a $\mathbb Z$-grading $\mathfrak g = \bigoplus_{j=1-m}^{m-1}\mathfrak g_j$ by using the natural representation $G_0 \to GL(\mathfrak g_1 \oplus \mathfrak g_{1-m})$, where $G_0 \le G$ is the connected subgroup corresponding to $\mathfrak g_0$. The resulting Higgs pairs include $G^{\mathbb R}$-Higgs bundles for $G^{\mathbb R} \le G$ a real f…

This https://arxiv.org/abs/1902.06293 has been replaced.
link: https://scholar.google.com/scholar?q=a

Equivariant Principal Bundles over the 2-Sphere
In this paper, we introduce the classification of equivariant principal bundles over the 2-sphere. Isotropy representations provide tools for understanding the classification of equivariant principal bundles. We consider a $Γ$-equivariant principal $G$-bundle over $S^2$ with structural group $G$ a compact connected Lie group, and $Γ\subset SO(3)$ a finite group acting linearly on $S^2.$ We prove that the equivariant 1-skeleton $X \subset S^2$ over the singular set can be classified by means o…

A Universal Kinematical Group for Quantum Mechanics
Gerald A. Goldin, David H. Sharp
https://arxiv.org/abs/2404.18274 https://arxiv.o…

A Universal Kinematical Group for Quantum Mechanics
In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms. Unitary representations of this group describe a wide variety of quantum systems, and have predicted previously unsuspected possibilities; notably, anyons and nonabe…

The Nucleus of a Compact Lie Group, and Support of Singularity Categories
Thomas Peirce
https://arxiv.org/abs/2405.00457 https://arxi…

The Nucleus of a Compact Lie Group, and Support of Singularity Categories
In this paper we adapt the notion of the nucleus defined by Benson, Carlson, and Robinson to compact Lie groups in non-modular characteristic. We show that it describes the singularities of the projective scheme of the cohomology of its classifying space. A notion of support for singularity categories of ring spectra (in the sense of Greenlees and Stevenson) is established, and is shown to be precisely the nucleus in this case, consistent with a conjecture of Benson and Greenlees for finite gro…

This https://arxiv.org/abs/2403.16411 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

A Geometric Perspective on Fusing Gaussian Distributions on Lie Groups
Stochastic inference on Lie groups plays a key role in state estimation problems such as; inertial navigation, visual inertial odometry, pose estimation in virtual reality, etc. A key problem is fusing independent concentrated Gaussian distributions defined at different reference points on the group. In this paper we approximate distributions at different points in the group in a single set of exponential coordinates and then use classical Gaussian fusion to obtain the fused posteriori in those…

This https://arxiv.org/abs/2308.02991 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Controllability of discrete-time linear systems on lie groups with finite semisimple center
In this paper we stated a condition for the controllability of discrete-time linear systems for the case when the Lie group has finite semisimple center and provided a example in the Lie group $SL_2(\mathbb{R})$.

This https://arxiv.org/abs/2310.13180 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Note on the group of vertical diffeomorphisms of a principal bundle, and its relation to the Frölicher-Nijenhuis bracket
The group of vertical diffeomorphisms of a principal bundle forms the generalised action Lie groupoid associated to the bundle. The former is generated by the group of maps with value in the structure group, which is also the group of bisections of the groupoid. The corresponding Lie algebra of general vertical vector fields is generated by maps with value in the Lie algebra of the structure group. The bracket on these maps, induces by the bracket of vertical vector fields, is an ``extended" br…

The structure of locally conformally product Lie algebras
Viviana del Barco, Andrei Moroianu
https://arxiv.org/abs/2404.17956 https://

The structure of locally conformally product Lie algebras
A locally conformally product (LCP) structure on a compact conformal manifold is a closed non-exact Weyl connection (i.e.~a linear connection which is locally but not globally the Levi-Civita connection of Riemannian metrics in the conformal class), with reducible holonomy. A left-invariant LCP structure on a compact quotient $Γ\backslash G$ of a simply connected Riemannian Lie group $(G,g)$ with Lie algebra $\mathfrak{g}$ can be characterized in terms of a closed 1-form $θ\in\mathfrak{g}^*$ …

The Nucleus of a Compact Lie Group, and Support of Singularity Categories
Thomas Peirce
https://arxiv.org/abs/2405.00457 https://arxi…

The Nucleus of a Compact Lie Group, and Support of Singularity Categories
In this paper we adapt the notion of the nucleus defined by Benson, Carlson, and Robinson to compact Lie groups in non-modular characteristic. We show that it describes the singularities of the projective scheme of the cohomology of its classifying space. A notion of support for singularity categories of ring spectra (in the sense of Greenlees and Stevenson) is established, and is shown to be precisely the nucleus in this case, consistent with a conjecture of Benson and Greenlees for finite gro…

This https://arxiv.org/abs/2307.00403 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Amenability of finite energy path and loop groups
It is shown that the groups of finite energy (that is, Sobolev class $H^1$) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit
Hanno Sahlmann, Waleed Sherif
https://arxiv.org/abs/2405.00661

Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit
We consider 3-dimensional Euclidean gravity in the weak coupling limit of Smolin and show that it is BF-theory with $\text{U(1)}^3$ as a Lie group. The theory is quantised using loop quantum gravity methods. The kinematical degrees of freedom are truncated, on account of computational feasibility, by fixing a graph and deforming the algebra of the holonomies to impose a cutoff on the charge vectors. This leads to a quantum theory related to $\text{U}_q \text{(1)}^3$ BF-theory. The effect of imp…

Automorphisms of Multiplicative Lie algebra Extensions
Dev Karan Singh, Shiv Datt Kumar
https://arxiv.org/abs/2403.19662 https://arxi…

Automorphisms of Multiplicative Lie algebra Extensions
In this paper, we discuss the inducibility problem for automorphisms of multiplicative Lie algebra extensions and show that obstruction to the inducibility of pairs lies in the second cohomology group of multiplicative Lie algebras. We also establish the Wells type exact sequence for multiplicative Lie algebras, which relates automorphism groups with the second cohomology group of multiplicative Lie algebras.

Generic differentiability and $P$-minimal groups
Will Johnson
https://arxiv.org/abs/2404.17234 https://arxiv.org/pdf/2404.17234

Generic differentiability and $P$-minimal groups
We prove generic differentiability in $P$-minimal theories, strengthening an earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and Pillay's $P$-minimal analogue of Pillay's conjectures on o-minimal groups. Specifically, let $G$ be an $n$-dimensional definable group in a highly saturated model $M$ of a $P$-minimal theory. Then there is an open definable subgroup $H \subseteq G$ such that $H$ is compactly dominated by $H/H^{00}$, and $H/H^{00}$ is a $p$-adic Lie group of the …

This https://arxiv.org/abs/2302.00145 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Controllability of discrete-time linear systems on solvable Lie groups
The objective of this paper is to study the controllability of discrete-time linear control systems in solvable Lie groups. In the special case of nilpotent Lie groups, a necessary and sufficient condition for controllability is established. Furthermore, the class of discrete-time linear systems in the two-dimensional affine Lie group is constructed and a condition for controllability of these systems is also stated.

Multifunctorial Equivariant Algebraic K-Theory
Donald Yau
https://arxiv.org/abs/2404.02794 https://arxiv.org/pdf/2404.02794

Multifunctorial Equivariant Algebraic K-Theory
A central question in equivariant algebraic K-theory asks whether there exists an equivariant K-theory machine from genuine symmetric monoidal G-categories to orthogonal G-spectra that preserves equivariant algebraic structures. We answer this question positively by constructing an enriched multifunctor K from the G-categorically enriched multicategory of O-pseudoalgebras to the symmetric monoidal category of orthogonal G-spectra, for a compact Lie group G and a 1-connected pseudo-commutative G…

Automorphism groups of parafermion vertex operator algebras: general case
Ching Hung Lam, Xingjun Lin, Hiroki Shimakura
https://arxiv.org/abs/2404.17096 ht…

Automorphism groups of parafermion vertex operator algebras: general case
We complete the program for determining the full automorphism groups of all parafermion vertex operator algebras associated with simple Lie algebras and positive integral levels. We show that the full automorphism group of the parafermion vertex operator algebra is isomorphic to the automorphism group of the associated root system for the remaining cases: (i) the level is at least $3$; (ii) the level is $2$ and the simple Lie algebra is non simply laced.

This https://arxiv.org/abs/2306.02237 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Frobenius distributions of low dimensional abelian varieties over finite fields
Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_q$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre--Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we …

This https://arxiv.org/abs/2307.00788 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Positive mass gap of quantum Yang-Mills Fields
We construct a 4-dimensional quantum field theory on a Hilbert space, dependent on a simple Lie Algebra of a compact Lie group, that satisfies Wightman's axioms. This Hilbert space can be written as a countable sum of non-separable Hilbert spaces, each indexed by a non-trivial, inequivalent irreducible representation of the Lie Algebra. In each component Hilbert space, a state is given by a triple, a space-like rectangular surface $S$ in $\mathbb{R}^4$, a measurable section of the Lie Algebra…

This https://arxiv.org/abs/1612.08942 has been replaced.
link: https://scholar.google.com/scholar?q=a

Proper affine actions: a sufficient criterion
For a semisimple real Lie group $G$ with an irreducible representation $ρ$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $ρ$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $ρ(G)$ and that is free, nonabelian and acts properly discontinuously on $V$. This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available…

This https://arxiv.org/abs/2401.08489 has been replaced.
link: https://scholar.google.com/scholar?q=a

Adjoints in symmetric squares of Lie algebra representations
Given finite-dimensional complex representations $V$ and $V'$ of a simply-connected semisimple compact Lie group $G$, we determine the dimension of the $G$-invariant subspace of $\mathrm{adj}(G)\otimes V\otimes V'$, of $\mathrm{adj}(G)\otimes S^2 V$, and of $\mathrm{adj}(G)\otimesΛ^2 V$, where $\mathrm{adj}(G)$ is the adjoint representation. In other words we derive the multiplicity with which summands of $\mathrm{adj}(G)$ appear in a tensor product $V \otimes V'$ or (anti)symmetric square $S^…

Efficient Reachable Sets on Lie Groups Using Lie Algebra Monotonicity and Tangent Intervals
Akash Harapanahalli, Samuel Coogan
https://arxiv.org/abs/2403.16214

Efficient Reachable Sets on Lie Groups Using Lie Algebra Monotonicity and Tangent Intervals
In this paper, we efficiently compute overapproximated reachable sets for control systems evolving on Lie groups, building off results from monotone systems theory and geometric integration theory. We propose to consider intervals living in the Lie algebra, which through the exponential map, describe real sets on the Lie group. A local equivalence between the original system and a system evolving on the Lie algebra allows existing interval reachability techniques to apply in the tangent space. …

This https://arxiv.org/abs/2401.17463 has been replaced.
link: https://scholar.google.com/scholar?q=a

A Group Theoretic Metric for Robot State Estimation Leveraging Chebyshev Interpolation
We propose a new metric for robot state estimation based on the recently introduced $\text{SE}_2(3)$ Lie group definition. Our metric is related to prior metrics for SLAM but explicitly takes into account the linear velocity of the state estimate, improving over current pose-based trajectory analysis. This has the benefit of providing a single, quantitative metric to evaluate state estimation algorithms against, while being compatible with existing tools and libraries. Since ground truth data g…

This https://arxiv.org/abs/2308.15971 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Harmonic Morphisms on Lie groups and Minimal Conformal Foliations of Codimension two
Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple subgroup of G generating a left-invariant conformal foliation F of codimension two on G. We then show that the foliation F is minimal. This means that locally the leaves of F are fibres of a complex-valued harmonic morphism. In the Riemannian case, we prove that if the metric restricted to K is biinvariant then F is totally geodesic.

On some relationships between the center and the derived subalgebra in Poisson (2-3)-algebras
P. Ye. Minaiev, O. O. Pypka, I. V. Shyshenko
https://arxiv.org/abs/2404.17741

On some relationships between the center and the derived subalgebra in Poisson (2-3)-algebras
One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/ζ(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $n$-groups, associative algebras, Lie algebras, Lie $n$-algebras, Lie rings, L…

This https://arxiv.org/abs/2307.13087 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Jointly Equivariant Dynamics for Interacting Particles
Let a finite set of interacting particles be given, together with a symmetry Lie group $G$. Here we describe all possible dynamics that are jointly equivariant with respect to the action of $G$. This is relevant e.g., when one aims to describe collective dynamics that are independent of any coordinate change or external influence. We particularize the results to some key examples, i.e. for the most basic low dimensional symmetries that appear in collective dynamics on manifolds.

Integro-differential diffusion equations on graded Lie groups
Joel E. Restrepo, Michael Ruzhansky, Berikbol T. Torebek
https://arxiv.org/abs/2402.14125 htt…

Integro-differential diffusion equations on graded Lie groups
We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on $L^p(\mathbb{G})$ $(1

Lagrangian subvarieties of hyperspherical varieties related to $G_2$
Nikolay Kononenko
https://arxiv.org/abs/2404.19033 https://arxiv…

Lagrangian subvarieties of hyperspherical varieties related to $G_2$
We consider two $S$-dual hyperspherical varieties of the group $G_2 \times \text{SL}(2)$: an equivariant slice for $G_2$, and the symplectic representation of $G_2 \times \text{SL}_2$ in the odd part of the basic classical Lie superalgebra $\mathfrak{g}(3)$. For these varieties we check the equality of numbers of irreducible components of their Lagrangian subvarieties (zero levels of the moment maps of Borel subgroups' actions) conjectured in arXiv:2310.19770.

This https://arxiv.org/abs/2402.19174 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On an $n$-ary generalization of the Lie representation
We continue our study, initiated in our prior work with Richard Stanley, of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra known as the free Fillipov $n$-algebra with $k$ brackets. Our ultimate aim is to determine the multiplicities of the irreducible representations in this representation. This had been done for the ordinary Lie representation ($n=2$ case) by Kraskiewicz and Weyman. The $k=2$ case was handled in our p…

This https://arxiv.org/abs/2402.02990 has been replaced.
link: https://scholar.google.com/scholar?q=a

Poisson-Lie analogues of spin Sutherland models revisited
Some generalizations of spin Sutherland models descend from `master integrable systems' living on Heisenberg doubles of compact semisimple Lie groups. The master systems represent Poisson--Lie counterparts of the systems of free motion modeled on the respective cotangent bundles and their reduction relies on taking quotient with respect to a suitable conjugation action of the compact Lie group. We present an enhanced exposition of the reductions and prove rigorously for the first time that the …

A post-group theoretic perspective on the operator-valued S-transform in free probability
Kurusch Ebrahimi-Fard, Timoth\'e Ringeard
https://arxiv.org/abs/2402.16450

A post-group theoretic perspective on the operator-valued S-transform in free probability
We investigate the algebraic structure underlying Voiculescu's S-transform in the setting of operator-valued free probability. We show that its twisted factorisation property gives rise to post-groups, crossed morphisms, as well as pre- and post-Lie algebras.

This https://arxiv.org/abs/2312.04191 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Subsets of groups with context-free preimages
We study subsets $E$ of finitely generated groups where the set of all words over a given finite generating set that lie in $E$ forms a context-free language. We call these sets recognisably context-free. They are invariant of the choice of generating set and a theorem of Muller and Schupp fully classifies when the set $\{1\}$ can be recognisably context-free. We extend Muller and Schupp's result to show that a group $G$ admits a finite recognisably context-free subset if and only if $G$ is vir…

This https://arxiv.org/abs/2011.11669 has been replaced.
link: https://scholar.google.com/scholar?q=a

On piecewise hyperdefinable groups
The aim of this paper is to generalize and improve two of the main model-theoretic results of "Stable group theory and approximate subgroups" by E. Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the stabilizer theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies.

Almost gauge-invariant states and the ground state of Yang-Mills theory
A. Mariani
https://arxiv.org/abs/2402.16743 https://arxiv.org…

Almost gauge-invariant states and the ground state of Yang-Mills theory
We consider the problem of the explicit description of the gauge-invariant subspace of pure lattice gauge theories in the Hamiltonian formulation, where the gauge group is either a compact Lie group or a finite group. The latter case is particularly interesting for quantum simulation. A basis of states where configurations are grouped according to their holonomies is shown to have several advantages over other descriptions. Using this basis, we compute some properties of interest for some non- …

This https://arxiv.org/abs/2209.01603 has been replaced.
link: https://scholar.google.com/scholar?q=a

Exterior powers of a parabolic Springer sheaf on a Lie algebra
We compute the exterior powers, with respect to the additive convolution on the general linear Lie algebra, of a parabolic Springer sheaf corresponding to a maximal parabolic subgroup of type (1, n -- 1). They turn out to be isomorphic to the semisimple perverse sheaves attached by the Springer correspondence to the exterior powers of the permutation representation of the symmetric group.

Convergence of Kinetic Langevin Monte Carlo on Lie groups
Lingkai Kong, Molei Tao
https://arxiv.org/abs/2403.12012 https://arxiv.org/…

Convergence of Kinetic Langevin Monte Carlo on Lie groups
Explicit, momentum-based dynamics for optimizing functions defined on Lie groups was recently constructed, based on techniques such as variational optimization and left trivialization. We appropriately add tractable noise to the optimization dynamics to turn it into a sampling dynamics, leveraging the advantageous feature that the momentum variable is Euclidean despite that the potential function lives on a manifold. We then propose a Lie-group MCMC sampler, by delicately discretizing the resul…

A Geometric Perspective on Fusing Gaussian Distributions on Lie Groups
Yixiao Ge, Pieter van Goor, Robert Mahony
https://arxiv.org/abs/2403.16411 https://<…

A Geometric Perspective on Fusing Gaussian Distributions on Lie Groups
Stochastic inference on Lie groups plays a key role in state estimation problems, such as inertial navigation, visual inertial odometry, pose estimation in virtual reality, etc. A key problem is fusing independent concentrated Gaussian distributions defined at different reference points on the group. In this paper we approximate distributions at different points in the group in a single set of exponential coordinates and then use classical Gaussian fusion to obtain the fused posteriori in those…

This https://arxiv.org/abs/2310.15356 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Lie Group Variational Collision Integrators for a Class of Hybrid Systems
The problem of 3-dimensional, convex rigid-body collision over a plane is fully investigated; this includes bodies with sharp corners that is resolved without the need for nonsmooth convex analysis of tangent and normal cones. In particular, using nonsmooth Lagrangian mechanics, the equations of motion and jump equations are derived, which are largely dependent on the collision detection function. Following the variational approach, a Lie group variational collision integrator (LGVCI) is system…

This https://arxiv.org/abs/2311.04070 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the Post-Lie Structure in SISO Affine Feedback Control Systems
The main objective of this work is to show that the single-input, single-output (SISO) affine feedback group, a transformation group in the context of the affine feedback interconnection of Chen-Fliess series, is a post-group in the sense of Bai, Guo, Sheng and Tang.

Invariant divisors and equivariant line bundles
Boris Kruglikov, Eivind Schneider
https://arxiv.org/abs/2404.19439 https://arxiv.org/…

Invariant divisors and equivariant line bundles
Scalar relative invariants play an important role in the theory of group actions on a manifold as their zero sets are invariant hypersurfaces. Relative invariants are central in many applications, where they often are treated locally since an invariant hypersurface may not be a locus of a single function. Our aim is to establish a global theory of relative invariants. For a Lie algebra $\mathfrak{g}$ of holomorphic vector fields on a complex manifold $M$, any holomorphic $\mathfrak{g}$-invari…

This https://arxiv.org/abs/2401.15456 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Product Mixing in Compact Lie Groups
If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and Sós from 1985. For th…

Spherical $p$-group complexes arising from finite groups of Lie type
Kevin Iv\'an Piterman
https://arxiv.org/abs/2403.07489 https://

Spherical $p$-group complexes arising from finite groups of Lie type
We show that the $p$-group complex of a finite group $G$ is homotopy equivalent to a wedge of spheres of dimension at most $n$ if $G$ contains a self-centralising normal subgroup $H$ which is isomorphic to a group of Lie type and Lie rank $n$ in characteristic $p$. If in addition every order-$p$ element of $G$ induces an inner or field automorphism on $H$, the $p$-group complex of $G$ is $G$-homotopy equivalent to a spherical complex obtained from the Tits building of $H$. We also prove that …

This https://arxiv.org/abs/2403.08296 has been replaced.
link: https://scholar.google.com/scholar?q=a

Quasinormability and property $(Ω)$ for spaces of smooth and ultradifferentiable vectors associated with Lie group representations
We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fréchet space $E$ are quasinormable if $E$ is so. A similar result is shown to hold for the linear topological invariant $(Ω)$. In the ultradifferentiable case, our results particularly apply to spaces of Gevrey vectors of Beurling type. As an application, we study the quasinormability and the property $(Ω)$ for a broad class of Fréchet spaces of smooth and ultradiffe…

Simplicial volume of manifolds fibering with connected structure group
Thorben Kastenholz
https://arxiv.org/abs/2404.14818 https://ar…

Simplicial volume of manifolds fibering with connected structure group
In this note we investigate the simplicial volume of fiber bundles with connected structure group. We are able to show that if the structure group is either compact or a Lie group, or if the fiber is aspherical that the simplicial volume of the total space agrees with the simplicial volume of the trivial bundle.

This https://arxiv.org/abs/1610.09621 has been replaced.
link: https://scholar.google.com/scholar?q=a

Hamiltonian analysis of the BFCG theory for a strict Lie 2-group
We perform a complete Hamiltonian analysis of the BFCG action for a general Lie 2-group by using the Dirac procedure. We show that the resulting dynamical constraints eliminate all local degrees of freedom which implies that the BFCG theory is a topological field theory.

Relating Real and P-adic Kazhdan-Lusztig Polynomials
Leticia Barchini, Peter E. Trapa
https://arxiv.org/abs/2402.17956 https://arxiv.…

Relating Real and P-adic Kazhdan-Lusztig Polynomials
Fix an integral semisimple element $λ$ in the Lie algebra $\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the centralizer of $λ$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$ eigenspace of $\mathrm{ad}(λ)$ in $\mathfrak{g}$. Under a natural hypothesis (which is always satisfied for classical subgroups of $\mathrm{GL}(n)$), we embed the closure of each $L$ orbit on $\mathfrak{g}(-1)$ into the closure of an orbit of a symmetric subgroup $K$ containing $L$ on a par…

Perturbative Expansion of Yang-Baxter Operators
Emanuele Zappala
https://arxiv.org/abs/2403.09796 https://arxiv.org/pdf/2403.09796

Perturbative Expansion of Yang-Baxter Operators
We study the deformations of a wide class of Yang-Baxter (YB) operators arising from Lie algebras. We relate the higher order deformations of YB operators to Lie algebra deformations. We show that the obstruction to integrating deformations of self-distributive (SD) objects lie in the corresponding Lie algebra third cohomology group, and interpret this result in terms of YB deformations. We show that there are YB operators that admit integrable deformations (i.e. that can be deformed infinitely…

This https://arxiv.org/abs/2311.04070 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the Post-Lie Structure in SISO Affine Feedback Control Systems
The main objective of this work is to show that the single-input, single-output (SISO) affine feedback group, a transformation group in the context of the affine feedback interconnection of Chen-Fliess series, is a post-group in the sense of Bai, Guo, Sheng and Tang.

Deformations and extensions of modified $\lambda$-differential Lie-Yamaguti algebras
Wen Teng
https://arxiv.org/abs/2403.17015 https://

Deformations and extensions of modified $λ$-differential Lie-Yamaguti algebras
In this paper, we first introduce the concept and representations of modified $λ$-differential Lie-Yamaguti algebras. We then establish the cohomology of a modified $λ$-differential Lie-Yamaguti algebra with coefficients in a representation. As applications, we investigate the formal deformations and abelian extensions of modified $λ$-differential Lie-Yamaguti algebras by using the second cohomology group.

This https://arxiv.org/abs/2312.07710 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The classifying element for quotients of Fermat curves
Suppose $C$ is a cyclic Galois cover of the projective line branched at the three points $0$, $1$, and $\infty$. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of $C$ in terms of a basis which is well-suited to studying the action of the absolute Galois group of $\mathbb{Q}$.

This https://arxiv.org/abs/2309.07282 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Centralizer Rigidity near Elements of the Weyl Chamber Flow
In this paper, we prove centralizer rigidity near an element of the Weyl chamber flow on a semisimple Lie group. We show that a volume preserving perturbation of an element of the Weyl chamber flow on a quotient $G/Γ$ of an $\mathbb{R}$-split, simple Lie group $G$ either has centralizer of dimension $0$ or $1$, or is smoothly conjugate to an element of the Weyl chamber flow. We also acquire a general condition for the centralizer of a partially hyperbolic diffeomorphism to be a Lie group.

This https://arxiv.org/abs/2305.06997 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_grqc_…

Spatially homogeneous teleparallel spacetimes with four-dimensional groups of motions
We study metric teleparallel geometries, which can either be defined through a Lorentzian metric and flat, metric-compatible affine connection, or a tetrad and a flat spin connection, which are invariant under the transitive action of a four-dimensional Lie group on their spatial equal-time hypersurfaces. There are three such group actions, and their corresponding spatial hypersurfaces belong to the Bianchi types II, III and IX, respectively. For each of these three symmetry groups, we determin…

This https://arxiv.org/abs/2312.01765 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Infinitesimal rational actions
For any $k$-group scheme of finite type $G$ acting rationally on a $k$-variety $X$, if the action is generically free then the dimension of $\mathrm{Lie}(G)$ is upper bounded by the dimension of the variety. We show that this is the only obstruction when $k$ is a perfect field and $G$ is infinitesimal commutative trigonalizable. We also give necessary conditions to have faithful rational actions of infinitesimal commutative trigonalizable group schemes on varieties.

This https://arxiv.org/abs/2401.16725 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

Exploiting Equivariance in the Design of Tracking Controllers for Euler-Poincare Systems on Matrix Lie Groups
The trajectory tracking problem is a fundamental control task in the study of mechanical systems. A key construction in tracking control is the error or difference between an actual and desired trajectory. This construction also lies at the heart of observer design and recent advances in the study of equivariant systems have provided a template for global error construction that exploits the symmetry structure of a group action if such a structure exists. Hamiltonian systems are posed on the co…

This https://arxiv.org/abs/2403.08296 has been replaced.
link: https://scholar.google.com/scholar?q=a

Quasinormability and property $(Ω)$ for spaces of smooth and ultradifferentiable vectors associated with Lie group representations
We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fréchet space $E$ are quasinormable if $E$ is so. A similar result is shown to hold for the linear topological invariant $(Ω)$. In the ultradifferentiable case, our results particularly apply to spaces of Gevrey vectors of Beurling type. As an application, we study the quasinormability and the property $(Ω)$ for a broad class of Fréchet spaces of smooth and ultradiffe…

Existence of a global fundamental solution for H\"ormander operators
Mattia Galeotti
https://arxiv.org/abs/2403.19619 https://ar…

Existence of a global fundamental solution for Hörmander operators
We prove that for a simply connected manifold $M$ - and a vast class of non-simply connected manifolds - the existence of a fundamental solution for a differential operator $\mathcal L=\sum_{α\in\mathbb N^q} r_α\cdot X^α$ of finite degree over $M$, follows via a saturation method from the existence of a fundamental solution for the associated "lifted" operator over a group $G$. Given some smooth complete vector fields $X_1,\dots, X_q$ on $M$, suppose that they generate a finite dimensional L…

Strong generalized holomorphic principal bundles
Debjit Pal, Mainak Poddar
https://arxiv.org/abs/2404.18113 https://arxiv.org/pdf/240…

Strong generalized holomorphic principal bundles
We introduce the notion of a strong generalized holomorphic (SGH) fiber bundle and develop connection and curvature theory for an SGH principal $G$-bundle over a regular generalized complex (GC) manifold, where $G$ is a complex Lie group. We develop a de Rham cohomology for regular GC manifolds, and a Dolbeault cohomology for SGH vector bundles. Moreover, we establish a Chern-Weil theory for SGH principal $G$-bundles under certain mild assumptions on the leaf space of the GC structure. We also …

This https://arxiv.org/abs/2102.05339 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Formanek-Procesi group with base a right-angled Artin group: Residual nilpotence and Lie algebra
We investigate the Lie algebra of the Formanek-Procesi group FP(H) with base group H a right-angled Artin group. We show that the Lie algebra gr(FP(H)) has a presentation that is dictated by the group presentation. Moreover we show that FP(H) is a Magnus group.

Lusztig varieties and Macdonald polynomials
Arun Ram
https://arxiv.org/abs/2402.17935 https://arxiv.org/pdf/2402.17935

Lusztig varieties and Macdonald polynomials
This paper uses Lusztig varieties to give central elements of the Iwahori-Hecke algebra corresponding to unipotent conjugacy classes in the finite Chevalley group $GL_n(\mathbb{F}_q)$. We explain how these central elements are related to Macdonald polynomials and how this provides a framework for generalizing integral form and modified Macdonald polynomials to Lie types other than $GL_n$. The key steps are to recognize (a) that counting points in Lusztig varieties is equivalent to computing tra…

This https://arxiv.org/abs/2304.08705 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The chain control set of discrete-time linear system on the affine two-dimensional Lie group
In this paper, we present conditions for the existence and uniqueness of chain control sets of discrete-time linear systems on the affine two-dimensional Lie group. More specifically, we prove that these chain control sets are given by the union of an infinite number of control sets with empty interiors.

This https://arxiv.org/abs/2401.15456 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Product Mixing in Compact Lie Groups
If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and Sós from 1985. For th…

This https://arxiv.org/abs/2212.11857 has been replaced.
link: https://scholar.google.com/scholar?q=a

Aut(Fn) actions on representation spaces
J. Wiegold conjectured that if n>2 and G is a finite simple group, then the action of Aut(F_n) on Epi(F_n,G) is transitive. In this note we consider analogous questions where G is a compact Lie group, a non-compact simple analytic group or a simple algebraic group.

Moduli of codimension two linear sections of subadjoint varieties
Yingqi Liu
https://arxiv.org/abs/2403.17230 https://arxiv.org/pdf/2…

Moduli of codimension two linear sections of subadjoint varieties
Let $G$ be a simple algebraic group of type $F_{4}$, $E_{6}$, $E_{7}$ or $E_{8}$, and let $\mathfrak{g}$ be its Lie algebra. The adjoint variety $X_{ad} \subseteq \mathbb{P} \mathfrak{g}$ is defined as the unique closed orbit of the adjoint action of $G$ on $\mathbb{P}\mathfrak{g}$. $X_{ad}$ is a Fano contact manifold covered by lines in $\mathbb{P} \mathfrak{g}$. The subadjoint variety $S \subseteq \mathbb{P} W$ is denoted by the variety of lines on $X_{ad}$ through a fixed point $x$, where $W…

This https://arxiv.org/abs/2212.03775 has been replaced.
link: https://scholar.google.com/scholar?q=a

Semisimple elements and the little Weyl group of real semisimple $Z_m$-graded Lie algebras
We consider the semisimple orbits of a Vinberg $θ$-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one corresponding to the restricted roots of a Cartan subspace, the other corresponding to the complex reflections in the (little) Weyl group. The semisimple orbits have representatives in a finite number of sets that correspond to reflection subgroups of the (little) Weyl group…

Moduli of codimension two linear sections of subadjoint varieties
Yingqi Liu
https://arxiv.org/abs/2403.17230 https://arxiv.org/pdf/2…

Moduli of codimension two linear sections of subadjoint varieties
Let $G$ be a simple algebraic group of type $F_{4}$, $E_{6}$, $E_{7}$ or $E_{8}$, and let $\mathfrak{g}$ be its Lie algebra. The adjoint variety $X_{ad} \subseteq \mathbb{P} \mathfrak{g}$ is defined as the unique closed orbit of the adjoint action of $G$ on $\mathbb{P}\mathfrak{g}$. $X_{ad}$ is a Fano contact manifold covered by lines in $\mathbb{P} \mathfrak{g}$. The subadjoint variety $S \subseteq \mathbb{P} W$ is denoted by the variety of lines on $X_{ad}$ through a fixed point $x$, where $W…

This https://arxiv.org/abs/2310.09598 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Adelic Descent for Equivariant Elliptic Cohomology
We define $k$-rationalized $G$-equivariant elliptic cohomology, for a field of characteristic zero $k$ and a compact Lie group $G$, via adelic descent. We also give adelic descriptions of rationalized $G$-equivariant singular cohomology and K-theory. This completes a program first proposed by Roşu. These descriptions are then used to obtain comparison results with periodic cyclic homology theories defined via derived algebraic geometry.

The Table of the Structure Constants for the Complex Simple Lie Algebra of Type E_6 and Chevalley Commutator Formulas in the Chevalley Group of Type E_6 over a Field
Anna I. Polovinkina, Sergey G. Kolesnikov
https://arxiv.org/abs/2403.10038

The Table of the Structure Constants for the Complex Simple Lie Algebra of Type E_6 and Chevalley Commutator Formulas in the Chevalley Group of Type E_6 over a Field
This article is the third in the series. It is devoted the calculation of the structure constants for the complex simple Lie algebra of type E_6 and Chevalley commutator formulas.

The G\"odel Universe as the Lie Group with left-invariant Lorentz metric
V. N. Berestovskii
https://arxiv.org/abs/2404.06866 https://

The Gödel Universe as the Lie Group with left-invariant Lorentz metric
The author studies the Gödel Universe as the Lie group with left-invariant Lorentz metric. The expressions for timelike and isotropic geodesics in elementary functions are found by methods of geometric theory of optimal control for the search of geodesics on Lie groups with left-invariant (sub-)Lorentz metrics. It is proved that the Gödel Universe has no closed timelike or isotropic geodesics.

This https://arxiv.org/abs/2303.07157 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Computing equivariant matrices on homogeneous spaces for Geometric Deep Learning and Automorphic Lie Algebras
We develop an elementary method to compute spaces of equivariant maps from a homogeneous space $G/H$ of a Lie group $G$ to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact…

Refining the grading of irreducible Lie colour algebra representations
Mitchell Ryan
https://arxiv.org/abs/2403.02855 https://arxiv.o…

Refining the grading of irreducible Lie colour algebra representations
We define the refining extension of representations for Lie colour algebras which simultaneously extends the representations and refines the grading to a larger grading group.We show that the refining extension provides a general method for deriving all finite-dimensional irreducible Lie colour algebra representations from those for Lie superalgebras. This method yields a bijection between equivalence classes but, despite this, Lie colour algebras maintain a non-trivial representation theory di…

The Table of the Structure Constants for the Complex Simple Lie Algebra of Type E_6 and Chevalley Commutator Formulas in the Chevalley Group of Type E_6 over a Field
Anna I. Polovinkina, Sergey G. Kolesnikov
https://arxiv.org/abs/2403.10038

The Table of the Structure Constants for the Complex Simple Lie Algebra of Type E_6 and Chevalley Commutator Formulas in the Chevalley Group of Type E_6 over a Field
This article is the third in the series. It is devoted the calculation of the structure constants for the complex simple Lie algebra of type E_6 and Chevalley commutator formulas.

This https://arxiv.org/abs/2201.08740 has been replaced.
link: https://scholar.google.com/scholar?q=a

A Hörmander-Mikhlin theorem for high rank simple Lie groups
We establish regularity conditions for $L_p$-boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural Hörmander-Mikhlin criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard and de la Salle for …

Refining the grading of irreducible Lie colour algebra representations
Mitchell Ryan
https://arxiv.org/abs/2403.02855 https://arxiv.o…

Refining the grading of irreducible Lie colour algebra representations
We define the refining extension of representations for Lie colour algebras which simultaneously extends the representations and refines the grading to a larger grading group.We show that the refining extension provides a general method for deriving all finite-dimensional irreducible Lie colour algebra representations from those for Lie superalgebras. This method yields a bijection between equivalence classes but, despite this, Lie colour algebras maintain a non-trivial representation theory di…

On Commuting automorphisms of nilpotent Lie algebras
Shushma Rani, Niranjan Nehra, Rohit Garg
https://arxiv.org/abs/2404.12998 https://

On Commuting automorphisms of nilpotent Lie algebras
We investigate the commuting automorphisms of nilpotent Lie algebras $L$ with coclass $\leq 3$. Our examination exposes the conditions under which the set of commuting automorphisms of $L$ forms a subgroup within its automorphism group.

Tensor K-matrices for quantum symmetric pairs
Andrea Appel, Bart Vlaar
https://arxiv.org/abs/2402.16676 https://arxiv.org/pdf/2402.16…

Tensor K-matrices for quantum symmetric pairs
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, $U_q(\mathfrak{g})$ its quantum group, and $U_q(\mathfrak{k}) \subset U_q(\mathfrak{g})$ a quantum symmetric pair subalgebra determined by a distinguished Lie algebra automorphism $θ$. We introduce a category $W_θ$ of \emph{weight} $U_q(\mathfrak{k})$-modules, which is acted upon by the category of weight $U_q(\mathfrak{g})$-modules via tensor products. We construct a universal tensor K-matrix $\mathbb{K}$ (that is, a solution of a refl…

This https://arxiv.org/abs/2208.00140 has been replaced.
link: https://scholar.google.com/scholar?q=a

Line bundles on the moduli space of Lie algebroid connections over a curve
We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid ($ \mathcal{L} $) connections on a compact Riemann surface $X$ of genus $g \,\geq\, 3$. A smooth compactification of the moduli space of $\mathcal{L}$-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group o…

Polyhedral products, graph products and p-central series
Taras Panov, Temurbek Rahmatullaev
https://arxiv.org/abs/2402.11556 https://…

Polyhedral products, graph products and p-central series
We relate polyhedral products of spaces to graph products in algebraic categories via the loop homology and Lie algebras associated with central series. By way of application, we describe the restricted Lie algebra associated with the lower 2-central series of a right-angled Coxeter group and identify its universal enveloping algebra with the loop homology of the Davis-Januszkiewicz space.

The binary actions of simple groups of Lie type of characteristic 2
Nick Gill, Pierre Guillot, Martin W. Liebeck
https://arxiv.org/abs/2402.08357 https://<…

The binary actions of simple groups of Lie type of characteristic 2
Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $Γ(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For $t\in \mathcal{C}$, we define the component group of $t$ to be the subgroup of $G$ generated by all vertices in $Γ(\mathcal{C})$ that lie in the connected component of the graph that contains $t$. We classify the component groups of all involutions in simple …

Geometry of infinite dimensional Cartan Developments
Johanna Michor, Peter W. Michor
https://arxiv.org/abs/2404.05416 https://arxiv.o…

Geometry of infinite dimensional Cartan Developments
The Cartan development takes a Lie algebra valued 1-form satisfying the Maurer-Cartan equation on a simply connected manifold $M$ to a smooth mapping from $M$ into the Lie group. In this paper this is generalized to infinite dimensional $M$ for infinite dimensional regular Lie groups. The Cartan development is viewed as a generalization of the evolution map of a regular Lie group. The tangent mapping of a Cartan development is identified as another Cartan development.

WEAK $G$-IDENTITIES FOR THE PAIR $(M_2( \mathbb{C}),sl_2( \mathbb{C}))$
Ramon C\'odamo, Plamen Koshlukov
https://arxiv.org/abs/2402.13986 https://

WEAK $G$-IDENTITIES FOR THE PAIR $(M_2( \mathbb{C}),sl_2( \mathbb{C}))$
In this paper we study algebras acted on by a finite group $G$ and the corresponding $G$-identities. Let $M_2( \mathbb{C})$ be the $2\times 2$ matrix algebra over the field of complex numbers $ \mathbb{C}$ and let $sl_2( \mathbb{C})$ be the Lie algebra of traceless matrices in $M_2( \mathbb{C})$. Assume that $G$ is a finite group acting as a group of automorphisms on $M_2( \mathbb{C})$. These groups were described in the Nineteenth century, they consist of the finite subgroups of $PGL_2( \mathb…

On quasi-homomorphism rigidity for lattices in simple algebraic groups
Guillaume Dumas (UCBL, ICJ, EDPA)
https://arxiv.org/abs/2403.16500 https://

On quasi-homomorphism rigidity for lattices in simple algebraic groups
Property $(TTT)$ was introduced by Ozawa as a strengthening of Kazhdan's property $(T)$ and Burger and Monod's property $(TT)$. In this paper, we improve Ozawa's result by showing that any simple algebraic group of rank $\geq 2$ over a local field has property $(TTT)$. We also show that lattices in a second countable locally compact group inherits property $(TTT)$. Finally, we study to what extent Lie groups with infinite center fail to have properties $(TT)$ and $(TTT)$.

This https://arxiv.org/abs/2402.08361 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Brauer--Galois height zero conjecture
Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a certain elementary abelian $p$-subgroup of the Galois group and propose a Galois version of Brauer's height zero conjecture for principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when $G$ does not involve certain groups of Lie type of small…

Torsors on moduli spaces of principal $G$-bundles
Indranil Biswas, Swarnava Mukhopadhyay
https://arxiv.org/abs/2404.12877 https://arx…

Torsors on moduli spaces of principal $G$-bundles
Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $\mathfrak{g}$, and let $\mathcal{M}^0_{G}$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $κ$ on $C$ which is even in case $\dim{\mathfrak{g}}$ is odd. We show that there is a nonempty Zariski open substack ${\mathcal U}_κ$ of $\mathcal{M}^0_{G}$ such that $H^i(C,\, \text{ad}(E_G)\otimesκ) \,=\, 0$, $i\,=\, 1,\, 2$, for all $E_G\,\in\, {…

This https://arxiv.org/abs/2009.08526 has been replaced.
link: https://scholar.google.com/scholar?q=a

The equivariant cohomology for semidirect product actions
The rational Borel equivariant cohomology for actions of a compact connected Lie group is determined by restriction of the action to a maximal torus. We show that a similar reduction holds for any compact Lie group $G$ when there is a closed subgroup $K$ such that the cohomology of the classifying space $BK$ is free over the cohomology of $BG$ for field coefficients. We study the particular case when $G$ is a semi-direct product and $K$ is its maximal elementary abelian 2-subgroup for cohomolog…

This https://arxiv.org/abs/2401.14062 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Minimal doubling for small subsets in compact Lie groups
We prove a sharp bound for the minimal doubling of a small measurable subset of a compact connected Lie group. Namely, let $G$ be a compact connected Lie group of dimension $d_G$, we show that for for all measurable subsets $A$, we have $$μ_G(A^2) \geq \left(2^{d_G-d_H} - Cμ_G(A)^{\frac{2}{d_G-d_H}}\right)μ_G(A)$$ where $d_H$ is the maximal dimension of a proper closed subgroup $H$ and $C > 0$ is a dimensional constant. This settles a conjecture of Breuillard and Green, and recovers and impr…

Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms
Bas Janssens, Milan Niestijl
https://arxiv.org/abs/2404.06110

Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms
Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations $(\overlineρ, \mathcal{H})$ of the Lie group $\mathrm{Diff}_c(M)$ of compactly supported diffeomorphisms of a smooth manifold $M$ that satisfy a so-called generalized positive energy condition. In particular, this captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by $\overlineρ$. We show that if $M$ is connected and …

This https://arxiv.org/abs/2401.04548 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Generalised Killing Spinors on Three-Dimensional Lie Groups
For each $n\in\mathbb{N}$, we construct a Riemannian spin manifold carrying a generalised Killing spinor with $n$ distinct eigenvalues, providing the first known examples of such objects for $n>4$. For each left-invariant metric on a three-dimensional Lie group, we explicitly find all invariant generalised Killing spinors. We find many new examples with three distinct eigenvalues, and we prove that, in some cases, this is the generic situation.

This https://arxiv.org/abs/2311.08208 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Modular Representations of Truncated current Lie algebras
In this paper we consider the structure and representation theory of truncated current algebras $\mathfrak{g}_m = \mathfrak{g}[t]/(t^{m+1})$ associated to the Lie algebra $\mathfrak{g}$ of a standard reductive group over a field of positive characteristic. We classify semisimple and nilpotent elements and describe their associated support varieties. Next, we prove various Morita equivalences for reduced enveloping algebras, including a reduction to nilpotent $p$-characters, analogous to a famou…

Averaging operators on groups, racks and Leibniz algebras
Apurba Das
https://arxiv.org/abs/2403.06250 https://arxiv.org/pdf/2403.0625…

Averaging operators on groups, racks and Leibniz algebras
This paper considers averaging operators on various algebraic structures and studies the induced structures. We first introduce the notion of an averaging operator on a group $G$ and show that it induces a rack structure. Moreover, the given group structure and the induced rack structure form a group-rack. We observe that any pointed group-rack can be embedded into an averaging group. We show that the differentiation of a smooth pointed averaging operator on a Lie group gives rise to an averagi…

Probabilistic Generation of Finite Almost Simple Groups
Jason Fulman, Daniele Garzoni, Robert M. Guralnick
https://arxiv.org/abs/2403.17291 https://…

Probabilistic Generation of Finite Almost Simple Groups
We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to infinity). This is new even if G is simple. Together with results of Lucchini and Burness--Guralnick--Harper, this proves a conjecture of Lucchini and has an applic…

Quantisation of semisimple real Lie groups
Kenny De Commer
https://arxiv.org/abs/2404.05397 https://arxiv.org/pdf/2404.05397

Quantisation of semisimple real Lie groups
We provide a novel construction of quantized universal enveloping $*$-algebras of real semisimple Lie algebras, based on Letzter's theory of quantum symmetric pairs. We show that these structures can be `integrated', leading to a quantization of the group C$^*$-algebra of an arbitrary semisimple algebraic real Lie group.

A renormalised description of meromorphic connections over curves
Rapha\"el Belliard
https://arxiv.org/abs/2402.13362 https://ar…

A renormalised description of meromorphic connections over curves
The Renormalisation Group (RG) is a systematic procedure used to regularise divergences appearing as artefacts when constructing solutions to a large class of differential problems, whether perturbatively or not. This paper is devoted to performing it in the non-perturbative context of flat sections of meromorphic connections in holomorphic principal bundles over a base complex curve, with reductive and complex structure Lie groups. Using the interpretation of Kunuhiro of RG in terms of envelop…

The prescribed Ricci curvature problem on 5-dimensional nilpotent Lie groups
Marius Landry Foka, Romain Pefoukeu Nimpa, Salomon Joseph Mbatakou, Michel Bertrand Ngaha Djiadeu, Thomas Bouetou Bouetou
https://arxiv.org/abs/2403.08402

The prescribed Ricci curvature problem on 5-dimensional nilpotent Lie groups
In this paper, using the Milnor-type theorem technique, we provide on each nilpotent five dimensional Lie group, some global existence result of a pair (g, c) consisting of a left-invariant Riemannian metric g and a positive constant c such that Ric(g) =cT, where Ric(g) is the Ricci curvature of g and T a given left-invariant symmetric (0, 2)-tensor field.

A renormalised description of meromorphic connections over curves
Rapha\"el Belliard
https://arxiv.org/abs/2402.13362 https://ar…

A renormalised description of meromorphic connections over curves
The Renormalisation Group (RG) is a systematic procedure used to regularise divergences appearing as artefacts when constructing solutions to a large class of differential problems, whether perturbatively or not. This paper is devoted to performing it in the non-perturbative context of flat sections of meromorphic connections in holomorphic principal bundles over a base complex curve, with reductive and complex structure Lie groups. Using the interpretation of Kunuhiro of RG in terms of envelop…

This https://arxiv.org/abs/2302.07471 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A characterization of the alpha-connections on the statistical manifold of multivariate normal distributions
We study a statistical manifold $(\mathcal{N}, g^F, \nabla^{A}, \nabla^{A*})$ of multivariate normal distributions, where $g^F$ is the Fisher metric and $\nabla^{A}$ is the Amari-Chentsov connection and $\nabla^{A*}$ is its conjugate connection. We will show that it admits a solvable Lie group structure and moreover the Amari-Chentsov connection $\nabla^{A}$ on $(\mathcal{N}, g^F)$ will be characterized by the conjugate symmetry, i.e., a curvatures identity $R=R^*$ of a connection $\nabla$ and …

This https://arxiv.org/abs/2311.15976 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Quantitative Selberg's Lemma
We show that an arithmetic lattice $Γ$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $δ(v)$ where $v = μ(G/Γ)$ is the co-volume of the lattice. We prove that $δ$ is polynomial in general and poly-logarithmic under GRH. We then show that this poly-logarithmic bound is almost optimal, by constructing certain lattices with torsion elements of order $\sim \frac{\log v}{\log \log v}$.

This https://arxiv.org/abs/2311.08512 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Abelian gauge-like groups of $L_\infty$-algebras
Given a finite type degree-wise nilpotent $L_\infty$-algebra, we construct an abelian group that acts on the set of Maurer-Cartan elements of the given $L_\infty$-algebra so that the quotient by this action becomes the moduli space of equivalence classes of Maurer-Cartan elements. Specializing this to degree-wise nilpotent dg Lie algebras, we find that the associated ordinary gauge group of the dg Lie algebra with the Baker-Campbell-Hausdorff multiplication might be substituted by the underlyin…

This https://arxiv.org/abs/2404.10266 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Irreducible components in Hochschild cohomology of flag varieties
Let $G$ be a simple, simply-connected complex algebraic group with Lie algebra $\mathfrak{g}$, and $G/B$ the associated complete flag variety. The Hochschild cohomology $HH^\bullet(G/B)$ is a geometric invariant of the flag variety related to its generalized deformation theory and has the structure of a $\mathfrak{g}$-module. We study this invariant via representation-theoretic methods; in particular, we give a complete list of irreducible subrepresentations in $HH^\bullet(G/B)$ when $G=SL_n(\m…

This https://arxiv.org/abs/2402.02530 has been replaced.
link: https://scholar.google.com/scholar?q=a

Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces
Given a real semisimple connected Lie group $G$ and a discrete torsion-free subgroup $Γ< G$ we prove a precise connection between growth rates of the group $Γ$, polyhedral bounds on the joint spectrum of the ring of invariant differential operators, and the decay of matrix coefficients. In particular, this allows us to completely characterize temperedness of $L^2(Γ\backslash G)$ in this general setting.