Tootfinder

A Recursive Lie-Group Formulation for the Second-Order Time Derivatives of the Inverse Dynamics of parallel Kinematic Manipulators
Andreas Mueller, Shivesh Kumar, Thomas Kordik
https://arxiv.org/abs/2507.14274

A Recursive Lie-Group Formulation for the Second-Order Time Derivatives of the Inverse Dynamics of parallel Kinematic Manipulators
Series elastic actuators (SEA) were introduced for serial robotic arms. Their model-based trajectory tracking control requires the second time derivatives of the inverse dynamics solution, for which algorithms were proposed. Trajectory control of parallel kinematics manipulators (PKM) equipped with SEAs has not yet been pursued. Key element for this is the computationally efficient evaluation of the second time derivative of the inverse dynamics solution. This has not been presented in the lite…

Exterior differential systems on Lie algebroids and the invariant inverse problem of the calculus of variations
Tom Mestdag, Kenzo Yasaka
https://arxiv.org/abs/2507.14678

Exterior differential systems on Lie algebroids and the invariant inverse problem of the calculus of variations
We extend the theory of exterior differential systems from manifolds and their tangent bundles to Lie algebroids. In particular, we define the concept of an integral manifold of such an exterior differential system. We support our developments with several examples, including an application to dynamical systems with a symmetry group and to the invariant inverse problem of the calculus of variations.

Cohomology of $p$-adic Chevalley groups
Andrea Dotto, Bao V. Le Hung
https://arxiv.org/abs/2507.13500 https://arxiv.org/pdf/2507.1350…

Cohomology of $p$-adic Chevalley groups
Let $G$ be a split connected reductive group over the ring of integers of a finite unramified extension $K$ of $\mathbf{Q}_p$. Under a standard assumption on the Coxeter number of $G$, we compute the cohomology algebra of $G(\mathcal{O}_K)$ and its Iwahori subgroups, with coefficients in the residue field of $K$. Our methods involve a new presentation of some graded Lie algebras appearing in Lazard's theory of saturated $p$-valued groups, and a reduction to coherent cohomology of the flag varie…

Rota-Baxter operators on compact simple Lie groups and algebras
Saveliy V. Skresanov
https://arxiv.org/abs/2506.14324 https://arxiv.o…

Rota-Baxter operators on compact simple Lie groups and algebras
A Rota-Baxter operator on a Lie group $ G $ is a smooth map $ B : G \to G $ such that $ B(g)B(h) = B(gB(g)hB(g)^{-1}) $ for all $ g, h \in G $. This concept was introduced in 2021 by Guo, Lang and Sheng as a Lie group analogue of Rota-Baxter operators of weight 1 on Lie algebras. We show that the only Rota-Baxter operators on compact simple Lie groups are the trivial map and the inverse map. A similar description for Rota-Baxter operators of weight 1 on compact simple Lie algebras is provided.

Lie Group Theory of Multipole Moments and Shape of Stationary Rotating Fluid Bodies
Sergei M. Kopeikin (University of Missouri-Columbia, USA)
https://arxiv.org/abs/2505.14712

Lie Group Theory of Multipole Moments and Shape of Stationary Rotating Fluid Bodies
We present a rigorous framework for determining the equilibrium configurations of uniformly rotating, self-gravitating fluid bodies. This work addresses the classical challenge of modeling rotational deformation in celestial objects such as stars and planets. By integrating foundational theory with modern mathematical tools, we develop a unified formalism that enhances the precision and generality of shape modeling in astrophysical contexts. Our method applies Lie group theory to vector flows a…

Deformations of quasi-Hamiltonian spaces
Jean-Philippe Burelle, Mohamed Moussadek Maiza, Maxence Mayrand
https://arxiv.org/abs/2505.16689 https://

Deformations of quasi-Hamiltonian spaces
We introduce a notion of deformations of quasi-Hamiltonian $G$-spaces to Hamiltonian $G$-spaces and provide several examples. In particular, we show that the double $G \times G$ of a Lie group, viewed as a quasi-Hamiltonian $G \times G$-space, deforms smoothly to the cotangent bundle $T^*G$. Likewise, any conjugacy class of $G$ sufficiently close to the identity deforms to a coadjoint orbit. We further show that the moduli space of flat $G$-connections on a compact oriented surface with boundar…

Generalized False Vacuum Skyrme model
L. A. Ferreira, L. R. Livramento
https://arxiv.org/abs/2506.12235 https://arxiv.org/pdf/2506.12…

Generalized False Vacuum Skyrme model
We propose a generalization of the False Vacuum Skyrme model for any simple compact Lie groups $G$ that leads to Hermitian symmetric spaces. The Skyrme field that in the original theory takes its values in $SU(2)$ Lie group, now takes its values in $G$, while the remaining fields correspond to the entries of a symmetric, positive, and invertible $\dim G \times \dim G$-dimensional matrix $h$. This model is also an extension of the generalized BPS Skyrme model. We prove that the global minima cor…

Flag-transitive point-primitive quasi-symmetric $2$-designs and exceptional groups of Lie type
Jianbing Lu
https://arxiv.org/abs/2506.10266 https://…

Flag-transitive point-primitive quasi-symmetric $2$-designs and exceptional groups of Lie type
Let $\mathcal{D}$ be a non-trivial quasi-symmetric $2$-design with two block intersection numbers $x=0$ and $2\leq y\leq10$, and suppose that $G$ is an automorphism group of $\mathcal{D}$. If $G$ is flag-transitive and point-primitive, then it is known that $G$ is either of affine type or almost simple type. In this paper, we show that the socle of $G$ cannot be a finite simple exceptional group of Lie type.

Level structures on parahoric torsors and complete integrability
Georgios Kydonakis, Lutian Zhao
https://arxiv.org/abs/2506.12302 https://

Level structures on parahoric torsors and complete integrability
For a smooth complex algebraic curve $X$ and a reduced effective divisor $D$ on $X$, we introduce a notion of $D$-level structure on parahoric $\mathcal{G}_{\boldsymbol θ}$-torsors over $X$, for any connected complex reductive Lie group $G$. A moduli space of parahoric $\mathcal{G}_{\boldsymbol θ}$-torsors equipped with a $D$-level structure is constructed and we identify a canonical moment map with respect to the action of a level group on this moduli space. This action extends to a Poisson …

Convergence of Normal Form Power Series for Infinite-Dimensional Lie Pseudo-Group Actions
Peter J. Olver, Masoud Sabzevari, Francis Valiquette
https://arxiv.org/abs/2506.08869

Convergence of Normal Form Power Series for Infinite-Dimensional Lie Pseudo-Group Actions
We prove the convergence of normal form power series for suitably nonsingular analytic submanifolds under a broad class of infinite-dimensional Lie pseudo-group actions. Our theorem is illustrated by a number of examples, and includes, as a particular case, Chern and Moser's celebrated convergence theorem for normal forms of real hypersurfaces. The construction of normal forms relies on the equivariant moving frame method, while the convergence proof is based on the realization that the normal …

Lie algebras generated by reflections in types BCD
Christopher M. Drupieski, Jonathan R. Kujawa
https://arxiv.org/abs/2506.01198 https://

Lie algebras generated by reflections in types BCD
We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of even-signed permutations), viewed as Lie algebras via the commutator bracket, and determine the structure of the Lie subalgebras generated by the sets of reflections.

Cohomology and Extensions of $C_p$-Green Functors of Lie Type
Tarik Anowar, Satyendra Kumar Mishra, Ripan Saha
https://arxiv.org/abs/2507.07962 https://

Cohomology and Extensions of $C_p$-Green Functors of Lie Type
We develop a theory of $C_p$-Green functors of Lie type, unifying the axiomatic framework of Green functors with the structure of Lie algebras under the action of a cyclic group $C_p$ of prime order. Extending classical notions from representation theory and topology, we define tensor and exterior products, introduce an equivariant Chevalley-Eilenberg cohomology, and construct cup products that endow the cohomology with a graded Green functor of Lie type structure. A key result establishes a co…

A Study of the Spectral Sequence for Locally Free Isometric Actions of Abelian Lie Groups
Pawe{\l} Ra\'zny
https://arxiv.org/abs/2506.12422 https://

A Study of the Spectral Sequence for Locally Free Isometric Actions of Abelian Lie Groups
We give an upper bound on the number of the page on which the spectral sequence corresponding to a locally free isometric action of an abelian Lie group degenerates. We give examples showing that these bounds are indeed sharp. Finally, we further justify the study of this sequence by exhibiting a potential application to the study of harmonic forms.

On stable Cartan subgroups of Lie groups
Parteek Kumar, Arunava Mandal, Shashank Vikram Singh
https://arxiv.org/abs/2507.07027 https://

On stable Cartan subgroups of Lie groups
Let $G$ be a connected real Lie group with associated Lie algebra $\mathfrak g$, and let ${\rm Aut}(G)$ be the group of (Lie) automorphisms of $G$. It is noted here that, given a super-solvable subgroup $Γ\subset {\rm Aut}(G)$ of semisimple automorphisms, there exists a $Γ$-stable Cartan subgroup, by using a result of Borel and Mostow. We characterize the $Γ$-stable Cartan subgroups (with induced action) in the quotient group modulo a $Γ$-stable closed normal subgroup as the images of the $…

IRS on orbits and the case of $SL(2, \mathbb{R} ) \ltimes \mathbb{R} ^2$
Manoj Choudhuri, C. R. E. Raja
https://arxiv.org/abs/2506.09723 https://

IRS on orbits and the case of $SL(2, \mathbb{R} ) \ltimes \mathbb{R} ^2$
Let $G$ be a connected Lie group and $\text{Sub}_G$ be the space of closed subgroups of $G$ equipped with the Chabauty topology. In this article, we investigate the existence of invariant random subgroups of $G$ supported on various orbits of the conjugation action of $G$ on $\text{Sub}_G$. We also find the orbit closure for each of these orbits in the case of the Lie group $G=\SL(2,\mathbb{R})\ltimes \mathbb{R}^2$.

This https://arxiv.org/abs/2412.18698 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Strong factorization of ultradifferentiable vectors associated with compact Lie group representations
We show a strong factorization theorem of Dixmier-Malliavin type for ultradifferentiable vectors associated with compact Lie group representations on sequentially complete locally convex Hausdorff spaces. In particular, this solves a conjecture by Gimperlein et al. [J. Funct. Anal. 262 (2012), 667-681] for analytic vectors in the case of compact Lie groups.

This https://arxiv.org/abs/2506.04668 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCV_…

Feature-Based Lie Group Transformer for Real-World Applications
The main goal of representation learning is to acquire meaningful representations from real-world sensory inputs without supervision. Representation learning explains some aspects of human development. Various neural network (NN) models have been proposed that acquire empirically good representations. However, the formulation of a good representation has not been established. We recently proposed a method for categorizing changes between a pair of sensory inputs. A unique feature of this approa…

This https://arxiv.org/abs/2405.12717 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A note on the Thom morphism for the classifying space of certain Lie groups and gauge groups
We give a complete description of which non-torsion generators are not in the image of the Thom morphism from complex cobordism to integral cohomology for the classifying space of exceptional Lie groups except for E_8. We then show that the Thom morphism is not surjective for the classifying space of the gauge group of a principal E_7-bundle over the four-dimensional sphere. We use the results to detect nontrivial elements in the kernel of the reduced Thom morphism for Lie groups and their clas…

Depth one part of Tannakian groups of filtrations
Payman Eskandari
https://arxiv.org/abs/2507.11717 https://arxiv.org/pdf/2507.11717

Depth one part of Tannakian groups of filtrations
Let $(F_r M)_{r\in\mathbb{Z}}$ be a finite filtration on an object $M$ of a neutral Tannakian category $\mathbf{T}$ in characteristic zero. Let $u(M)=u^F(M)$ be the Lie algebra of the subgroup of the Tannakian fundamental group of $M$ that acts trivially on the associated graded $Gr^FM$. The filtration $F_\bullet M$ induces a filtration on the internal Hom $\underline{Hom}(M,M)$, which in turn induces a filtration $F_\bullet u(M)$ on $u(M)$. This filtration on $u(M)$ is concentrated in negative…

Representations of the group of two-diagonal triangular matrices
Dmitry Fuchs, Alexandre Kirillov
https://arxiv.org/abs/2507.03769 https://

Representations of the group of two-diagonal triangular matrices
Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between points of $\widehat G$ and $G$-orbits in $g*$. For many Lie groups it gives the answers to all major problems of representation theory in terms of coadjoint orbits. Formally, the notions and statements of the orbit method make sense when $G$ is infinite-dimensi…

Admitted symmetries of Backward Stochastic Differential Equations
Anas Ouknine, Paul Lescot
https://arxiv.org/abs/2506.10650 https://…

Admitted symmetries of Backward Stochastic Differential Equations
In this article, we introduce the concept of admitted Lie group of transformations for both backward stochastic differential equations (BSDEs) and forward backward stochastic differential equations (FBSDEs), following the approach of Meleshko et al. An application to BSDE is presented.

Introducing a vertex polynomial invariant for embedded graphs
Qi Yan, Qingying Deng, Metrose Metsidik
https://arxiv.org/abs/2506.07522 https://

Introducing a vertex polynomial invariant for embedded graphs
The ribbon group action extends geometric duality and Petrie duality by defining two embedded graphs as twisted duals precisely when they lie within the same orbit under this group action. Twisted duality yields numerous novel properties of fundamental graph polynomials. In this paper, we resolve a problem raised by Ellis-Monaghan and Moffatt [Trans. Amer. Math. Soc. 364 (2012), 1529--1569] for vertex counts by introducing the vertex polynomial: a generating function quantifying vertex distribu…

Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$
Cristina Draper, Alberto Elduque, Mikhail Kochetov
https://arxiv.org/abs/2507.03762

Special pure gradings on simple Lie algebras of types $E_6$, $E_7$, $E_8$
A group grading on a semisimple Lie algebra over an algebraically closed field of characteristic zero is special if its identity component is zero; it is pure if at least one of its components, other than the identity component, contains a Cartan subalgebra. We classify special pure gradings on Lie algebras of types $E_6$, $E_7$, $E_8$ up to equivalence and up to isomorphism. To this end, we use quadratic forms over the field of two elements to show that there are exactly three equivalence clas…

This https://arxiv.org/abs/2201.12881 has been replaced.
link: https://scholar.google.com/scholar?q=a

Oscillating singular integral operators on graded Lie groups revisited
In this work, we extend the Euclidean theory of oscillating singular integrals due to Fefferman and Stein in \cite{Fefferman1970,FeffermanStein1972} to arbitrary graded Lie groups. Our approach reveals the strong compatibility between the geometric measure theory of a graded Lie group and the Fourier analysis associated with Rockland operators. Our criteria are presented in terms of the oscillating Fefferman condition of the kernel of the operator and its group Fourier transform. One of the nov…

Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups
Alexandru Chirvasitu
https://arxiv.org/abs/2507.04065

Generic infinite generation, fixed-point-poor representations and compact-element abundance in disconnected Lie groups
The semidirect product $\mathbb{G}=\mathbb{L}\rtimes \mathbb{K}$ attached to a compact-group action on a connected, simply-connected solvable Lie group has a dense set of compact elements precisely when the $s\in \mathbb{K}$ operating on $\mathbb{L}$ fixed-point-freely constitute a dense set. This (along with a number of alternative equivalent characterizations) extends the Wu's analogous result for connected Lie $\mathbb{K}$, and also provides ample supplies of examples of almost-connected Lie…

Invariant Hilbert spaces of distribution vectors in Lie group representations
Ingrid Beltita, Daniel Beltita
https://arxiv.org/abs/2507.05988 https://

Invariant Hilbert spaces of distribution vectors in Lie group representations
For every unitary irreducible representation of a nilpotent Lie group we prove that the representation Hilbert space is the only nonzero invariant Hilbert space of distribution vectors.

An $O(n$)-Algorithm for the Higher-Order Kinematics and Inverse Dynamics of Serial Manipulators using Spatial Representation of Twists
Andreas Mueller
https://arxiv.org/abs/2506.10686

An $O(n$)-Algorithm for the Higher-Order Kinematics and Inverse Dynamics of Serial Manipulators using Spatial Representation of Twists
Optimal control in general, and flatness-based control in particular, of robotic arms necessitate to compute the first and second time derivatives of the joint torques/forces required to achieve a desired motion. In view of the required computational efficiency, recursive $O(n)$-algorithms were proposed to this end. Aiming at compact yet efficient formulations, a Lie group formulation was recently proposed, making use of body-fixed and hybrid representation of twists and wrenches. In this paper…

Cotangent Bundles as Coadjoint Orbits and Asymptotic Character Formulas
Michael Gjertsen, Alexander Schmeding
https://arxiv.org/abs/2506.10519 https://

Cotangent Bundles as Coadjoint Orbits and Asymptotic Character Formulas
The present article presents geometric quantization on cotangent bundles as a special instance of Kirillov's orbit method. To this end, the cotangent bundle is realized as a coadjoint orbit of an infinite-dimensional Lie group constructed from the diffeomorphism group. We also develop two asymptotic character formulas by employing the tangent groupoid.

Localization of strings on group manifolds
Sameer Murthy, Edward Witten
https://arxiv.org/abs/2506.20028 https://arxiv.org/pdf/2506.2…

Localization of strings on group manifolds
We compute the partition function of the WZW model with target a compact Lie group $G$ by adapting a method used by Choi and Takhtajan to compute the heat kernel of the group manifold. The basic idea is to compute the partition function of a supersymmetric version of the WZW model using a form of supersymmetric localization and then use the fact that, since the fermions of the supersymmetric WZW model are actually decoupled from the bosons, this also determines the partition function of the pur…

Classification of Equivariant Legendrian Embeddings of Rational Homogeneous Spaces into Nilpotent Orbits
Minseong Kwon
https://arxiv.org/abs/2507.03932 htt…

Classification of Equivariant Legendrian Embeddings of Rational Homogeneous Spaces into Nilpotent Orbits
For a complex semi-simple Lie algebra, every nilpotent orbit in its projectivization comes with a complex contact structure. For each nilpotent orbit, we classify projective Legendrian subvarieties that are homogeneous under the actions of their stabilizers in the adjoint group. In particular, we present a classification of equivariant Legendrian embeddings of rational homogeneous spaces into adjoint varieties.

The Optimality of a Nested Generalized Pairwise Group Testing Procedure
Yaakov Malinovsky, Viktor Skorniakov
https://arxiv.org/abs/2506.15797 https://

The Optimality of a Nested Generalized Pairwise Group Testing Procedure
We study the problem of identifying defective units in a finite population of \( n \) units, where each unit \( i \) is independently defective with known probability \( p_i \). This setting is referred to as the \emph{Generalized Group Testing Problem}. A testing procedure is called optimal if it minimizes the expected number of tests. It has been conjectured that, when all probabilities \( p_i \) lie within the interval \( \left[1 - \frac{1}{\sqrt{2}},\, \frac{3 - \sqrt{5}}{2} \right] \), the…

Structural and rigidity properties of Lie skew braces
Marco Damele, Andrea Loi
https://arxiv.org/abs/2507.06214 https://arxiv.org/pdf…

Structural and rigidity properties of Lie skew braces
We investigate structural and rigidity properties of \emph{Lie skew braces} (LSBs), objects essentially known in the literature as \emph{post--Lie groups}, obtained by endowing a manifold with two compatible group laws that share the same identity element. LSBs extend skew left braces, which are central to the study of non-involutive set-theoretic solutions of the Yang--Baxter equation, to the smooth category. Our first main result shows that, for every connected LSB $(G,\cdot,\circ)$, linearit…

Action of the Witt algebra on categorified quantum groups
Jernej Grlj, Aaron D. Lauda
https://arxiv.org/abs/2507.01877 https://arxiv.…

Action of the Witt algebra on categorified quantum groups
We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type A case, we show that this action induces an action of the positive Witt algebra on $\mathfrak{gl}_n$-foams, recovering the action of Qi, Robert, Sussan, and Wagner. We also show that this construction is compatible with the trace decategorification, inducing the action of the positive Witt algebra on the current algebra.

Group gradings on classical Lie superalgebras
Caio De Naday Hornhardt, Mikhail Kochetov
https://arxiv.org/abs/2506.23037 https://arxi…

Group gradings on classical Lie superalgebras
We classify, up to isomorphism, the group gradings on the non-exceptional classical simple Lie superalgebras, except for type A(1,1), over an algebraically closed field of characteristic zero. To this end, we study graded-simple and graded-superinvolution-simple associative superalgebras satisfying the descending chain condition on graded left superideals, which allows us to classify abelian group gradings on finite-dimensional simple and superinvolution-simple associative superalgebras over an…

Group schemes and their Lie algebras over a symmetric tensor category
Dave Benson, Julia Pevtsova
https://arxiv.org/abs/2507.02031 https://

Group schemes and their Lie algebras over a symmetric tensor category
We investigate the theory of affine group schemes over a symmetric tensor category, with particular attention to the tangent space at the identity. We show that this carries the structure of a restricted Lie algebra, and can be viewed as the degree one distributions on the group scheme, or as the right invariant derivations on the coordinate ring. In the second half of the paper, we illustrate the theory in the particular case of the symmetric tensor category $\mathsf{Ver}_4^+$ in characteristi…

Twisted Bredon-Illman Cohomology is a Morita Invariant
Carla Farsi, Laura Scull, Jordan Watts
https://arxiv.org/abs/2507.03091 https://

Twisted Bredon-Illman Cohomology is a Morita Invariant
We show that the twisted Bredon-Illman cohomology defined by Mukherjee-Mukherjee applied to compact Lie group action groupoids is Morita-invariant. This cohomology uses coefficient systems twisted over the discrete tom Dieck equivariant fundamental groupoid. To show Morita invariance, we use bibundles to transfer coefficient systems from one groupoid to another Morita equivalent one. This generalises results of Pronk-Scull on ordinary Bredon-Illman cohomology by removing both the finite isotrop…

Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons in $D^4$
Hank Chen
https://arxiv.org/abs/2506.05785 https:/…

Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons in $D^4$
This is a continuation of the first paper (arXiv:2501.06486) of this series, where the framework for the combinatorial quantization of the 4d 2-Chern-Simons theory with an underlying compact structure Lie 2-group $\mathbb{G}$ was laid out. In this paper, we continue our quest and characterize additive module *-functors $ω:\mathfrak{C}_q(\mathbb{G}^{Γ^2})\rightarrow\mathsf{Hilb}$, which serve as a categorification of linear *-functionals (ie. a state) on a $C^*$-algebra. These allow us to cons…

Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces
Kazuki Kannaka, Toshiyuki Kobayashi
https://arxiv.org/abs/2507.03476

Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces
Let $X=G/H$ be a homogeneous space of a Lie group $G$. When the isotropy subgroup $H$ is non-compact, a discrete subgroup $Γ$ may fail to act properly discontinuously on $X$. In this article, we address the following question: in the setting where $G$ and $H$ are reductive Lie groups and $Γ\backslash X$ is a standard quotient, to what extent can one deform the discrete subgroup $Γ$ while preserving the proper discontinuity of the action on $X$? We provide several classification results, in…

Generic graded contractions of Lie algebras
Mikhail V. Kochetov, Serhii D. Koval
https://arxiv.org/abs/2506.00610 https://arxiv.org/p…

Generic graded contractions of Lie algebras
We study generic graded contractions of Lie algebras from the perspectives of group cohomology, affine algebraic geometry and monoidal categories. We show that generic graded contractions with a fixed support are classified by a certain abelian group, which we explicitly describe. Analyzing the variety of generic graded contractions as an affine algebraic variety allows us to describe which generic graded contractions define graded degenerations of a given graded Lie algebra. Using the interpre…

The Difference between the Left and Right Invariant Extended Kalman Filter
Yixiao Ge, Giulio Delama, Martin Scheiber, Alessandro Fornasier, Pieter van Goor, Stephan Weiss, Robert Mahony
https://arxiv.org/abs/2507.04568

The Difference between the Left and Right Invariant Extended Kalman Filter
The extended Kalman filter (EKF) has been the industry standard for state estimation problems over the past sixty years. The Invariant Extended Kalman Filter (IEKF) is a recent development of the EKF for the class of group-affine systems on Lie groups that has shown superior performance for inertial navigation problems. The IEKF comes in two versions, left- and right- handed respectively, and there is a perception in the robotics community that these filters are different and one should choose …

Counting abelian number fields with restricted ramification type
Julie Tavernier
https://arxiv.org/abs/2507.00448 https://arxiv.org/p…

Counting abelian number fields with restricted ramification type
We count abelian number fields ordered by arbitrary height function whose generator of tame inertia is restricted to lie in a given subset of the Galois group, and find an explicit formula for the leading constant. We interpret our results as a version of the Batyrev-Manin conjecture on $BG$ and rephrase our result on number fields with restricted ramification type in terms of integral points on $BG$. We also prove that such number fields are equidistributed with respect to suitable collections…

This https://arxiv.org/abs/2505.23488 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Moduli spaces of twisted equivariant G-bundles over a curve
Let $X$ be a compact Riemann surface, $Γ$ a finite group of automorphisms of $X$ and $G$ a connected reductive complex Lie group with center $Z$. If we equip this data with a homomorphism $θ:Γ\to\text{Aut}(G)$ and a 2-cocycle $c:Γ\timesΓ\to Z$, there is a notion of $(θ,c)$-twisted $Γ$-equivariant $G$-bundle over $X$. The aim of this paper is to construct a coarse moduli space of isomorphism classes of polystable $(θ,c)$-twisted equivariant $G$-bundles over $X$, according to the definiti…

$q$-deformed Howe duality for orthosymplectic Lie superalgebras
Jeong Bae, Jae-Hoon Kwon
https://arxiv.org/abs/2506.05959 https://arx…

$q$-deformed Howe duality for orthosymplectic Lie superalgebras
We give a $q$-analogue of Howe duality associated to a pair $(\mathfrak{g},G)$, where $\mathfrak{g}$ is an orthosymplectic Lie superalgebra and $G=O_\ell, Sp_{2\ell}$. We define explicitly a commuting action of a quantized enveloping algebra of $\mathfrak{g}$ and the $\imath$quantum group of $\mathfrak{so}_\ell, \mathfrak{sp}_{2\ell}$ on a $q$-deformed supersymmetric space, and describe its semisimple decomposition whose classical limit recovers the $(\mathfrak{g},G)$-duality. As special cases,…

Pluriclosed metrics on compact semisimple Lie groups
Jorge Lauret, Facundo Montedoro
https://arxiv.org/abs/2506.21725 https://arxiv.o…

Pluriclosed metrics on compact semisimple Lie groups
Given a compact semisimple Lie group G and a maximal torus T of G, we give an explicit description of all left and Ad(T)-invariant pluriclosed Hermitian structures on G in terms of the corresponding root system. They depend on 2d+1 parameters in the irreducible case, where dim(T)=2d.

On geometric hydrodynamics and infinite dimensional magnetic systems
Levin Maier
https://arxiv.org/abs/2506.00544 https://arxiv.org/p…

On geometric hydrodynamics and infinite dimensional magnetic systems
In this article, we combine V. Arnold's celebrated approach via the Euler--Arnold equation -- describing the geodesic flow on a Lie group equipped with a right-invariant metric~\cite{Arnold66} -- with his formulation of the motion of a charged particle in a magnetic field~\cite{ar61}. We introduce the \emph{magnetic Euler--Arnold equation}, which is the Eulerian form of the magnetic geodesic flow for an infinite-dimensional magnetic system on a Lie group endowed with a right-invariant metric an…

Examples of scalar extension Hopf algebroids over a universal enveloping algebra
Martina Stoji\'c, Zoran \v{S}koda
https://arxiv.org/abs/2506.03125 htt…

Examples of scalar extension Hopf algebroids over a universal enveloping algebra
We present several related examples of Hopf algebroids over a universal enveloping algebra which are of the scalar extension Hopf algebroid type and explain their origin in Lie and algebraic group theory.

Gaussian integrals on symmetric spaces (the complex case and beyond)
Salem Said
https://arxiv.org/abs/2505.22672 https://arxiv.org/pd…

Gaussian integrals on symmetric spaces (the complex case and beyond)
The present work is concerned with Gaussian integrals on simply-connected non-positively-curved Riemannian symmetric spaces. It is motivated by the aim of explicitly finding the high-rank limit of these integrals for each of the eleven families of classical Riemannian symmetric spaces. To begin, it deals with the easier complex case (where the isometry group admits a complex Lie group structure). To go beyond this case, it introduces a variational characterisation of the high-rank limit, as the…

Universal property of framed $G$-disc algebras
Aleksandar Miladinovi\'c
https://arxiv.org/abs/2506.02699 https://arxiv.org/pdf/25…

Universal property of framed $G$-disc algebras
Given a compact Lie group $G$ and its finite subgroup $H$ we prove that the $\infty$-category of $G/H$-framed $G$-disc algebras taking values in a $G$-symmetric monoidal category $\underline{\mathcal{C}}^{\otimes}$ is equivalent to the $\infty$-category of $V$-framed $H$-disc algebras (where $V$ is an $H$-representation) which take values in $\underline{\mathcal{C}}^{\otimes}_H$, the underlying $H$-symmetric monoidal subcategory of $\underline{\mathcal{C}}^{\otimes}$. We will use this construct…

Pointwise-relatively-compact subgroups and trivial-weight-free representations
Alexandru Chirvasitu
https://arxiv.org/abs/2506.18861 https://

Pointwise-relatively-compact subgroups and trivial-weight-free representations
A pointwise-elliptic subset of a topological group is one whose elements all generate relatively-compact subgroups. A connected locally compact group has a dense pointwise-elliptic subgroup if and only if it is an extension by a compact normal subgroup of a semidirect product $\mathbb{L}\rtimes \mathbb{K}$ with connected, simply-connected Lie $\mathbb{L}$, compact Lie $\mathbb{K}$, with the commutator subgroup $\mathbb{K}'$ acting on the Lie algebra $Lie(\mathbb{L})$ with no trivial weights. Th…

A Proof of the Hodge Conjecture for a Special Class of Calabi-Yau 5-Folds via Exceptional Group Constraints
Dongzhe Zheng
https://arxiv.org/abs/2506.17754 …

A Proof of the Hodge Conjecture for a Special Class of Calabi-Yau 5-Folds via Exceptional Group Constraints
This paper provides a complete proof of the (1,1)-type Hodge conjecture for a special class of 5-dimensional Calabi-Yau manifolds ($h^{1,1}=56$), while establishing and implementing a novel methodology based on exceptional group constraints. The core idea of this methodology is to replace the "geometric construction" that traditional methods rely on -- which becomes infeasible in high-dimensional cases -- with the "algebraic rigidity" of Lie algebra representation theory. We prove that by intro…

A counter example for the Homogeneity Conjecture
Ming Xu, Shaoqiang Deng
https://arxiv.org/abs/2506.09471 https://arxiv.org/pdf/2506.…

A counter example for the Homogeneity Conjecture
We construct a counter example to show that the Homogeneity Conjecture, first proposed by J.A. Wolf in 1962, is not true. To be precise, we prove that on the Lie group Sp(2), there exists a left invariant Riemannian metric and a cyclic subgroup Γ of order (2n+1), such that the left translation of each element of Γ on Sp(2) is a Clifford-Wolf translation, but the Riemannian quotient Γ\Sp(2) is not homogeneous.

Compound Burgers-KdV Soliton Behaviour: Refraction, Reflection and Fusion
Darryl D. Holm, Ruiao Hu, Oliver D. Street, Hanchun Wang
https://arxiv.org/abs/2505.17026

Compound Burgers-KdV Soliton Behaviour: Refraction, Reflection and Fusion
We consider a coupled PDE system between the Burgers equation and the KdV equation to model the interactions between `bore'-like structures and wave-like solitons in shallow water. Two derivations of the resulting Burgers-swept KdV system are presented, based on Lie group symmetry and reduced variational principles. Exact compound soliton solutions are obtained, and numerical simulations show that the Burgers and KdV momenta tend toward a balance at which the coupled system reduces to the integ…

Multiple rational normal forms in Lie theory
Dmitriy Voloshyn
https://arxiv.org/abs/2506.01530 https://arxiv.org/pdf/2506.01530

Multiple rational normal forms in Lie theory
We study the decomposition of a generic element $g \in G$ of a connected reductive complex algebraic group $G$ in the form $g = N(g) B(g) \bar{u} N(g)^{-1}$ where $N: G \dashrightarrow \mathcal{N}_-$ and $B : G \dashrightarrow \mathcal{B}_+$ are rational maps onto a unipotent subgroup $\mathcal{N}_-$ and a Borel subgroup $\mathcal{B}_+$ opposite to $\mathcal{N}_-$, and $\bar{u}$ is a representative of a Weyl group element $u$. We introduce a class of rational Weyl group elements that give rise …

This https://arxiv.org/abs/2309.03765 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csRO_…

Equivariant Symmetries for Inertial Navigation Systems
This paper investigates the problem of inertial navigation system (INS) filter design through the lens of symmetry. The extended Kalman filter (EKF) and its variants have been the staple of INS filtering for 50 years. However, recent advances in inertial navigation systems have exploited matrix Lie group structure to design stochastic filters and state observers that have been shown to display superior performance compared to classical solutions. In this work, we explore various symmetries of i…

Introduction to moduli spaces and Dirac geometry
Eckhard Meinrenken
https://arxiv.org/abs/2506.04150 https://arxiv.org/pdf/2506.04150…

Introduction to moduli spaces and Dirac geometry
Let $G$ be a Lie group, with an invariant metric on its Lie algebra $\mathfrak{g}$. Given a surface $Σ$ with boundary, and a collection of base points $\mathcal{V}\subset Σ$ meeting every boundary component, the moduli space (representation variety) $\mathcal{M}_G(Σ,\mathcal{V})$ carries a distinguished `quasi-symplectic' 2-form. We shall explain the finite-dimensional construction of this 2-form and discuss its basic properties, using quasi-Hamiltonian techniques and Dirac geometry. This ar…

Moduli spaces of twisted equivariant G-bundles over a curve
Guillermo Barajas
https://arxiv.org/abs/2505.23488 https://arxiv.org/pdf/…

Moduli spaces of twisted equivariant G-bundles over a curve
Let $X$ be a compact Riemann surface, $Γ$ a finite group of automorphisms of $X$ and $G$ a connected reductive complex Lie group with center $Z$. If we equip this data with a homomorphism $θ:Γ\to\text{Aut}(G)$ and a 2-cocycle $c:Γ\timesΓ\to Z$, there is a notion of $(θ,c)$-twisted $Γ$-equivariant $G$-bundle over $X$. The aim of this paper is to construct a coarse moduli space of isomorphism classes of polystable $(θ,c)$-twisted equivariant $G$-bundles over $X$, according to the definiti…

Conjugator length of locally compact groups of Euclidean isometries
Yuri Santos Rego, Petra Schwer
https://arxiv.org/abs/2507.01268 https://

Conjugator length of locally compact groups of Euclidean isometries
We consider locally compact subgroups $H$ of the full isometry group $\mathrm{Isom}(\mathbb{E}^n)$ of Euclidean $n$-space which respect the splitting into an orthogonal and a translation subgroup. We prove that the conjugator length function of such groups grows linearly. Our theorem applies, in particular, to the Lie group $\mathrm{Isom}(\mathbb{E}^n)$ itself but also to affine Coxeter groups and to split crystallographic groups.

Replaced article(s) found for physics.class-ph. https://arxiv.org/list/physics.class-ph/new
[1/1]:
- Lie Group Theory of Multipole Moments and Shape of Stationary Rotating Fluid Bodies
Sergei M. Kopeikin (University of Missouri-Columbia, USA)

The Exponential of Skew-Symmetric Matrices: A Nearby Inverse and Efficient Computation of Derivatives
Zhifeng Deng, P. -A. Absil, Kyle A. Gallivan, Wen Huang
https://arxiv.org/abs/2506.18302

The Exponential of Skew-Symmetric Matrices: A Nearby Inverse and Efficient Computation of Derivatives
The matrix exponential restricted to skew-symmetric matrices has numerous applications, notably in view of its interpretation as the Lie group exponential and Riemannian exponential for the special orthogonal group. We characterize the invertibility of the derivative of the skew-restricted exponential, thereby providing a simple expression of the tangent conjugate locus of the orthogonal group. In view of the skew restriction, this characterization differs from the classic result on the inverti…

Crowned Lie groups and nets of real subspaces
Daniel Beltita, Karl-Hermann Neeb
https://arxiv.org/abs/2506.16422 https://arxiv.org/pd…

Crowned Lie groups and nets of real subspaces
We introduce the notion of a complex crown domain for a connected Lie group $G$, and we use analytic extensions of orbit maps of antiunitary representations to these domains to construct nets of real subspaces on $G$ that are isotone, covariant and satisfy the Reeh--Schlieder and Bisognano--Wichmann conditions from Algebraic Quantum Field Theory. This provides a unifying perspective on various constructions of such nets.The representation theoretic properties of different crowns are discussed i…

A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I
Z. Sela
https://arxiv.org/abs/2505.22755

A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I
The JSJ decomposition and the Makanin-Razborov diagram were proved to be essential in studying varieties over free groups, semigroups and associative algebras. In this paper we suggest a unified conceptual approach to the applicability of these structures over all these algebraic categories. With a variety over each of these algebraic categories we naturally associate a set of tuples in a free group. Then we show how to associate a Makanin-Razborov diagram with any set of tuples over a free gro…

An anticanonical perspective on G/P Schubert varieties
Changzheng Li, Konstanze Rietsch, Mingzhi Yang
https://arxiv.org/abs/2506.18388 https://

An anticanonical perspective on G/P Schubert varieties
We describe a natural basis of the Cartier class group of an arbitrary Schubert variety $X_{w,P}$ in a flag variety $G/P$ of general Lie type. We then characterise when the Schubert variety is factorial/Fano, along with an explicit formula for the anticanonical line bundle in these cases. We also prove that, for Schubert varieties in simply-laced types (only), being factorial is equivalent to being $Q$-factorial, and is equivalent to the equality of the Betti numbers $b_2(X_{w,P})=b_{2\ell(w)-2…

A finite Linear Dependence of Discrete Series Multiplicities
Kaustabh Mondal, Gunja Sachdeva
https://arxiv.org/abs/2506.00542 https://

A finite Linear Dependence of Discrete Series Multiplicities
Let $G$ be a connected semisimple simply connected Lie group with a compact Cartan subgroup and let $Γ$ be a uniform lattice in $G$. Let $\widehat{G}_d$ denote the set of equivalence classes of unitary discrete series representations of $G$. We prove that for any finite subset of $\widehat{G}_d$ satisfying a certain condition, the associated finite set of discrete series multiplicities in $L^2(Γ\backslash G)$ determines all discrete series multiplicities in $L^2(Γ\backslash G)$. This allows …

Connections between hyperlinearity, stability and character rigidity for higher rank lattices
Alon Dogon, Itamar Vigdorovich
https://arxiv.org/abs/2506.20843

Connections between hyperlinearity, stability and character rigidity for higher rank lattices
Let $Γ$ be an irreducible lattice in a semisimple Lie group of real rank at least $2$. Suppose that $Γ$ has property (T;FD), that is, its finite dimensional representations have a uniform spectral gap. We show that if $Γ$ is (flexibly) Hilbert--Schmidt stable then: $(a)$ infinite central extensions $\widetildeΓ$ of $Γ$ are not hyperlinear, and $(b)$ every character of $Γ$ is either finite-dimensional or induced from the center (character rigidity). As a consequence, a positive answer to t…

Howe duality over finite fields II: explicit stable computation
Sophie Kriz
https://arxiv.org/abs/2506.22983 https://arxiv.org/pdf/25…

Howe duality over finite fields II: explicit stable computation
In this second paper of a series dedicated to type I Howe duality for finite fields, we explicitly decompose the restriction of an oscillator representation of a finite symplectic group to the product of a symplectic and an orthogonal subgroup which are each other's centralizers in terms of G. Lusztig's classification of irreducible representations of finite groups of Lie type in the two so-called stable ranges.