arXiv subjects

Ruibin Zhang

Publications and source records attributed to Ruibin Zhang.

33 records · Page 2Linked to original sources

arXiv2020

Fast-Tracker: A Robust Aerial System for Tracking Agile Target in Cluttered Environments

This paper proposes a systematic solution that uses an unmanned aerial vehicle (UAV) to aggressively and safely track an agile target. The solution properly handles the challenging situations where the intent of the target and the dense environments are unknown to the UAV. Our work is divided into two parts: target motion prediction and tracking trajectory planning. The target motion prediction method utilizes target observations to reliably predict the future motion of the target considering dynamic constraints. The tracking trajectory planner follows the traditional hierarchical workflow.A target informed kinodynamic searching method is adopted as the front-end, which heuristically searches for a safe tracking trajectory. The back-end optimizer then refines it into a spatial-temporal optimal and collision-free trajectory. The proposed solution is integrated into an onboard quadrotor system. We fully test the system in challenging real-world tracking missions.Moreover, benchmark comparisons validate that the proposed method surpasses the cutting-edge methods on time efficiency and tracking effectiveness.

cs.RO↗

arXiv2019

Schur-Weyl duality for certain infinite dimensional $\rm{U}_q(\mathfrak{sl}_2)$-modules

Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $\mathfrak{sl}_2$ at generic $q$. We show that for any $r\ge 1$, the endomorphism algebra of $M\otimes V^{\otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $\rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $\rm{TLB}_r(q,Q)$ as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.

math.RT↗

arXiv2018

Degenerate quantum general linear groups

Given any pair of positive integers m and n, we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of gl(m+n). We study its structure and develop a highest weight representation theory. The finite dimensional simple modules are classified in terms of highest weights, which are essentially characterised by m+n-2 nonnegative integers and two arbitrary nonzero scalars. In the special case with m=2 and n=1, an explicit basis is constructed for each finite dimensional simple module. For all m and n, the degenerate quantum group has a natural irreducible representation acting on C(q)^(m+n). It admits an R-matrix that satisfies the Yang-Baxter equation and intertwines the co-multiplication and its opposite. This in particular gives rise to isomorphisms between the two module structures of any tensor power of C(q)^(m+n) defined relative to the co-multiplication and its opposite respectively. A topological invariant of knots is constructed from this R-matrix, which reproduces the celebrated HOMFLY polynomial. Degenerate quantum groups of other classical types are briefly discussed.

math.QA↗

arXiv2018

Drinfeld realisations and vertex operator representations of quantum affine superalgebras

Drinfeld realisations are constructed for the quantum affine superalgebras of the series ${\rm\mathfrak{osp}}(1|2n)^{(1)}$,${\rm\mathfrak{sl}}(1|2n)^{(2)}$ and ${\rm\mathfrak{osp}}(2|2n)^{(2)}$. By using the realisations, we develop vertex operator representations and classify the finite dimensional irreducible representations for these quantum affine superalgebras.

math.QA↗

arXiv2017

Borelic pairs for stratified algebras

We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley-Lieb and Jones algebras. For this, we develop and apply a general theory of algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of König. Another application of the theory leads to a proof that Auslander-Dlab-Ringel algebras admit exact Borel subalgebras.

math.RT↗

arXiv2017

Vertex operator representations of quantum affine superalgebras

Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1 irreducible integrable highest weight representations and classify the finite dimensional irreducible representations of Uq(g). This makes essential use of the Drinfeld realisation for Uq(g), and quantum correspondences between affine Kac-Moody superalgebras, developed in earlier papers.

math.QA↗

arXiv2016

Invariants of the special orthogonal group and an enhanced Brauer category

We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group $\text{SO}_m(\mathbb{C})$, given the FFT for $\text{O}_m(\mathbb{C})$. We then define, by means of a presentation with generators and relations, an enhanced Brauer category $\widetilde{\mathcal{B}}(m)$ by adding a single generator to the usual Brauer category $\mathcal{B}(m)$, together with four relations. We prove that our category $\widetilde{\mathcal{B}}(m)$ is actually (and remarkably) {\em equivalent} to the category of representations of $\text{SO}_m$ generated by the natural representation. The FFT for $\text{SO}_m$ amounts to the surjectivity of a certain functor $\mathcal{F}$ on $\text{Hom}$ spaces, while the Second Fundamental Theorem for $\text{SO}_m$ says simply that $\mathcal{F}$ is injective on $\text{Hom}$ spaces. This theorem provides a diagrammatic means of computing the dimensions of spaces of homomorphisms between tensor modules for $\text{SO}_m$ (for any $m$). These methods will be applied to the case of the orthosymplectic Lie algebras $\text{osp}(m|2n)$, where the super-Pfaffian enters, in a future work.

math.RT↗

arXiv2015

The first fundamental theorem of invariant theory for the orthosymplectic supergroup

We give an elementary proof of the first fundamental theorem of the invariant theory for the orthosymplectic supergroup by generalising the method of Atiyah, Bott and Patodi to the supergroup context. We use methods from super-algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space. In this way, a super Schur-Weyl-Brauer duality is established between the orthosymplectic supergroup of superdimension $(m|2n)$ and the Brauer algebra with parameter $m-2n$. The result may be interpreted in terms of the relevant Harish-Chandra super pair action (over the complex field), or equivalently, the orthosymplectic Lie supergroup over the infinite dimensional Grassmann algebra. We also state a corresponding theorem for the orthosymplectic Lie superalgebra, which involves an extra invariant generator, the super-Pfaffian.

math.RT↗

arXiv2012

Invariant integration on orthosymplectic and unitary supergroups

The orthosymplectic supergroup OSp(m|2n) and unitary supergroup U(p|q) are studied following a new approach that starts from Harish-Chandra pairs and links the sheaf-theoretical supermanifold approach of Berezin and others with the differential geometry approach of Rogers and others. The matrix elements of the fundamental representation of the Lie supergroup G are expressed in terms of functions on the product supermanifold G_0 x R^{0|N}, with G_0 the underlying Lie group and N the odd dimension of G. This product supermanifold is isomorphic to the supermanifold of G. This leads to a new expression for the standard generators of the corresponding Lie superalgebra g as invariant derivations on G. Using these results a new and transparent formula for the invariant integrals on OSp(m|2n) and U(p|q) is obtained.

math-ph↗

arXiv2011

The second fundamental theorem of invariant theory for the orthogonal group

Let $V=\C^n$ be endowed with an orthogonal form and $G=\Or(V)$ be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism $ν:B_r(n)\to\End_G(V^{\otimes r})$, where $B_r(n)$ is the $r$-string Brauer algebra with parameter $n$. However the kernel of $ν$ has remained elusive. In this paper we show that, in analogy with the case of $\GL(V)$, for $r\geq n+1$, $ν$ has kernel which is generated by a single idempotent element $E$, and we give a simple explicit formula for $E$. Using the theory of cellular algebras, we show how $E$ may be used to determine the multiplicities of the irreducible representations of $\Or(V)$ in $V^{\ot r}$. We also show how our results extend to the case where $\C$ is replaced by an appropriate field of positive characteristic, and comment on quantum analogues of our results.

math.GR↗

arXiv2010

Detecting Cohomology for Lie Superalgebras

In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras. These results are used later to affirmatively answer questions, which were originally posed in \cite{BKN1} and \cite{BaKN}, about realizing support varieties for Lie superalgebras via rank varieties constructed for the smaller detecting subalgebras.

math.RT↗

arXiv2010

Topological Vortices in Chiral Gauge Theory of Graphene

Generation mechanism of energy gaps between conductance and valence bands is at the centre of the study of graphene material. Recently Chamon, Jackiw, et al. proposed a mechanism of using a Kekulé distortion background field $% φ$ and its induced gauge potential $A_{i}$ to generate energy gaps. In this paper various vortex structures inhering in this model are studied. Regarding $φ$ as a generic background field rather than a fixed Nielson-Oleson type distribution, we have found two new types of vortices on the graphene surface --- the velocity field vortices and the monopole-motion induced vortices --- from the inner structure of the potential $A_{i}$. These vortex structures naturally arise from the motion of the Dirac fermions instead of from the background distortion field.

cond-mat.mes-hall↗

arXiv2009

Spherical functions on homogeneous superspaces

Homogeneous superspaces arising from the general linear supergroup are studied within a Hopf algebraic framework. Spherical functions on homogeneous superspaces are introduced, and the structures of the superalgebras of the spherical functions on classes of homogeneous superspaces are described explicitly.

math.RT↗

arXiv2007

Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules

For each integer $n\ge 1$, we demonstrate that a $(2n+1)$-dimensional generalized MICZ-Kepler problem has an $\mr{Spin}(2, 2n+2)$ dynamical symmetry which extends the manifest $\mr{Spin}(2n+1)$ symmetry. The Hilbert space of bound states is shown to form a unitary highest weight $\mr{Spin}(2, 2n+2)$-module which occurs at the first reduction point in the Enright-Howe-Wallach classification diagram for the unitary highest weight modules. As a byproduct, we get a simple geometric realization for such a unitary highest weight $\mr{Spin}(2, 2n+2)$-module.

math-ph↗

arXiv2007

Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus

We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.

math.RT↗