arXiv subjects

Lina Song

Publications and source records attributed to Lina Song.

14 recordsLinked to original sources

arXiv2026

Mock-pre-Lie bialgebras

In this paper, we systematically develop the theory of mock-pre-Lie bialgebras from multiple perspectives. We introduce the notion of a phase space of a mock-Lie algebra, and show that a mock-Lie algebra admits a phase space if and only if it is sub-adjacent to a mock-pre-Lie algebra. We introduce the notions of Manin triples of mock-pre-Lie algebras and mock-pre-Lie bialgebras, and prove the equivalences between mock-pre-Lie bialgebras, Manin triples of mock-pre-Lie algebras, certain matched pairs of mock-pre-Lie algebras, certain matched pairs of mock-Lie algebras and phase spaces of a mock-Lie algebra, which lays a theoretical foundation for subsequent research. Next, we investigate coboundary mock-pre-Lie bialgebras, and derive an analogue of the classical Yang-Baxter equation. In addition, we introduce two important special classes of mock-pre-Lie bialgebras: quasi-triangular mock-pre-Lie bialgebras and factorizable mock-pre-Lie bialgebras. We show that quasi-triangular mock-pre-Lie bialgebras naturally induce relative Rota-Baxter operators of weight -1. Finally, we provide a new perspective for the study of triangular and factorizable mock-pre-Lie bialgebras by introducing the concept of quadratic Rota-Baxter mock-pre-Lie algebras of arbitrary weight.

math.RA↗

arXiv2026

Kimodo: Scaling Controllable Human Motion Generation

High-quality human motion data is becoming increasingly important for applications in robotics, simulation, and entertainment. Recent generative models offer a potential data source, enabling human motion synthesis through intuitive inputs like text prompts or kinematic constraints on poses. However, the small scale of public mocap datasets has limited the motion quality, control accuracy, and generalization of these models. In this work, we introduce Kimodo, an expressive and controllable kinematic motion diffusion model trained on 700 hours of optical motion capture data. Our model generates high-quality motions while being easily controlled through text and a comprehensive suite of kinematic constraints including full-body keyframes, sparse joint positions/rotations, 2D waypoints, and dense 2D paths. This is enabled through a carefully designed motion representation and two-stage denoiser architecture that decomposes root and body prediction to minimize motion artifacts while allowing for flexible constraint conditioning. Experiments on the large-scale mocap dataset justify key design decisions and analyze how the scaling of dataset size and model size affect performance.

cs.CV↗

arXiv2025

SONIC: Supersizing Motion Tracking for Natural Humanoid Whole-Body Control

Despite the rise of billion-parameter foundation models trained across thousands of graphical processing units (GPUs), similar scaling gains have not been shown for humanoid control. Current neural controllers for humanoids remain modest in size, target a limited set of behaviors, and are trained on a handful of GPUs. We show that scaling model capacity, data, and compute yields a generalist humanoid controller capable of natural, robust whole-body movements. We position motion tracking as a scalable task for humanoid control, leveraging dense supervision from diverse motion-capture data to acquire human motion priors without manual reward engineering. We build a foundation model for motion tracking by scaling along three axes: network size (1.2M to 42M parameters), dataset volume (100M+ frames from 700 hours of motion capture), and compute (21k GPU hours). Beyond demonstrating the benefits of scale, we further show downstream utility through a real-time kinematic planner that bridges motion tracking to tasks such as navigation, enabling natural and interactive control, as well as a unified token space that supports virtual reality (VR) teleoperation and vision-language-action (VLA) models with a single policy. Through this interface, we demonstrate autonomous VLA-driven whole-body loco-manipulation requiring coordinated hand and foot placement. Scaling motion tracking exhibits favorable properties: performance improves steadily with compute and data diversity, and learned policies generalize to unseen motions, establishing motion tracking at scale as a practical foundation for humanoid control.

cs.RO↗

arXiv2024

Deformation maps of Quasi-twilled associative algebras

In this paper, we introduce two types of deformation maps of quasi-twilled associative algebras. Each type of deformation maps unify various operators on associative algebras. Right deformation maps unify modified Rota-Baxter operators of weight $\lambda$, derivations, homomorphisms and crossed homomorphisms. Left deformation maps unify relative Rota-Baxter operators of weight 0, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of associative algebras. Furthermore, we give the controlling algebra and the cohomology of these two types of deformation maps. On the one hand, we obtain some existing results for modified Rota-Baxter operators of weight $\lambda$, derivations, homomorphisms, crossed homomorphisms, relative Rota-Baxter operators of weight 0, twisted Rota-Baxter operators and Reynolds operators. On the other hand, we also obtain some new results, such as the controlling algebra of a modified Rota-Baxter operator of weight $\lambda$ on an associative algebra, the controlling algebra and the cohomology of a deformation map of a matched pair of associative algebras.

math.RA↗

arXiv2023

Deformations and cohomologies of embedding tensors on 3-Lie algebras

In this paper, first we introduce the notion of an embedding tensor on a 3-Lie algebra, which naturally induces a 3-Leibniz algebra. Using the derived bracket, we construct a Lie 3-algebra, whose Maurer-Cartan elements are embedding tensors. Consequently, we obtain the $L_{\infty}$-algebra that governs deformations of embedding tensors. We define the cohomology theory for embedding tensors on 3-Lie algebras. As applications, we show that if two formal deformations of an embedding tensor on a 3-Lie algebra are equivalent, then their infinitesimals are in the same cohomology class in the second cohomology group. Moreover, an order n deformation of an embedding tensor is extendable if and only if the obstruction class, which is in the third cohomology group, is trivial.

math.RA↗

arXiv2022

Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras

In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a Lie algebra to the semidirect product Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an $L_\infty$-algebra whose Maurer-Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted $L_\infty$-algebra that controls deformations of a given crossed homomorphism on 3-Lie algebras.

math.RA↗

arXiv2021

The full cohomology, abelian extensions and formal deformations of Hom-pre-Lie algebras

The main purpose of this paper is to provide a full cohomology of a Hom-pre-Lie algebra with coefficients in a given representation. This new type of cohomology exploit strongly the Hom-type structure and fits perfectly with simultaneous deformations of the multiplication and the homomorphism defining a Hom-pre-Lie algebra. Moreover, we show that its second cohomology group classifies abelian extensions of a Hom-pre-Lie algebra by a representation.

math-ph↗

arXiv2021

On Hom-pre-Poisson algebras

In this paper, first we discuss Hom-pre-Poisson algebras and their relationships with Hom-Poisson algebra. Then we introduce the notion of a Hom-pre-Gerstenhaber algebra and show that a Hom-pre-Gerstenhaber algebra gives rise to a Hom-Gerstenhaber algebra. Moreover, we consider Hom-dendriform formal deformations of Hom-zinbiel algebras and show that Hom-pre-Poisson algebras are the corresponding semi-classical limits. Furthermore, we consider Hom-O-operators on Hom-Poisson algebras and study their relationships with Hom-pre-Poisson algebras. Finally, we define the notion of dual-Hom-pre-Poisson algebra and show that a Hom-average-operator on a Hom-Poisson algebra naturally gives rise to a dual-Hom-pre-Poisson algebra.

math.RA↗

arXiv2019

On Hom-pre-Lie bialgebras

In this paper we introduce the notion of Hom-pre-Lie bialgebra in the general framework of the cohomology theory for Hom-Lie algebras. We show that Hom-pre-Lie bialgebras, standard Manin triples for Hom-pre-Lie algebras and certain matched pairs of Hom-pre-Lie algebras are equivalent. Due to the usage of the cohomology theory, it makes us successfully study the coboundary Hom-pre-Lie bialgebras. The notion of Hom-s-matrix is introduced, by which we can construct Hom-pre-Lie bialgebras naturally. Finally we introduce the notions of Hom-O-operators on Hom-pre-Lie algebras and Hom-L-dendriform algebras, by which we construct Hom-s-matrices.

math.RA↗

arXiv2019

Representations and cohomologies of Hom-pre-Lie algebras

In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of Hom-pre-Lie algebras in term of the cohomology theory of Hom-Lie algebras. As applications, we study linear deformations of Hom-pre-Lie algebras, which are characterized by the second cohomology groups of Hom-pre-Lie algebras with the coefficients in the regular representation. The notion of a Nijenhuis operator on a Hom-pre-Lie algebra is introduced which can generate trivial linear deformations of a Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a Hom-pre- Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an $\huaO$-operator on a Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure.

math.RA↗

arXiv2017

On non-abelian extensions of 3-Lie algebras

In this paper, we study non-abelian extensions of 3-Lie algebras through Maurer-Cartan elements. We show that there is a one-to-one correspondence between isomorphism classes of non-abelian extensions of 3-Lie algebras and equivalence classes of Maurer-Cartan elements in a DGLA. The structure of the Leibniz algebra on the space of fundamental objects is also analyzed.

math.RT↗

arXiv2017

Cohomology characterizations of non-abelian extensions of Hom-Lie algebras

In this paper, first we show that under the assumption of the center of h being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra g by a regular Hom-Lie algebra h are in one-to-one correspondence with Hom-Lie algebra morphisms from g to Out(h). Then for a general Hom-Lie algebra morphism from g to Out(h), we construct a cohomology class as the obstruction of existence of a non-abelian extension that induce the given Hom-Lie algebra morphism.

math.RA↗

arXiv2017

Cohomologies, deformations and extensions of n-Hom-Lie algebras

In this paper, first we give the cohomologies of an $n$-Hom-Lie algebra and introduce the notion of a derivation of an $n$-Hom-Lie algebra. We show that a derivation of an $n$-Hom-Lie algebra is a $1$-cocycle with the coefficient in the adjoint representation. We also give the formula of the dual representation of a representation of an $n$-Hom-Lie algebra. Then, we study $(n-1)$-order deformation of an $n$-Hom-Lie algebra. We introduce the notion of a Hom-Nijenhuis operator, which could generate a trivial $(n-1)$-order deformation of an $n$-Hom-Lie algebra. Finally, we introduce the notion of a generalized derivation of an $n$-Hom-Lie algebra, by which we can construct a new $n$-Hom-Lie algebra, which is called the generalized derivation extension of an $n$-Hom-Lie algebra.

math.RA↗

arXiv2016

Derivation Hom-Lie 2-algebras and non-abelian extensions of Hom-Lie algebras

In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).

math.RA↗