Lie Theory Research Articles

Schur–Weyl quasi-duality and (co)triangular Hopf quasigroups

Schur–Weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic and plays a central role in many parts of algebraic Lie theory. In this paper, we will introduce the notion of Schur–Weyl quasi-duality and study it. For this, generally, we consider a braided vector space (V,c) and its braided Lie algebra Endk(V)(−). Then, we can construct its braided enveloping algebra U(Endk(V)(−)), which is a connected braided c-cocommutative Hopf algebra. Let H be a triangular Hopf quasigroup with bijective antipode and B be a cotriangular Hopf quasigroup with bijective antipode. Let V be any finite dimensional vector space in the category LQ(H,R)(B,σ) of generalized Long quasimodules. We show that (U((EndkV)(−))⋆H⋆B,kSn,V⊗n) is a Schur–Weyl quasi-duality under suitable conditions.

Parabolic double cosets in Coxeter groups

Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group.

A Family of New Borel Subalgebras of Quantum Groups

We construct a family of right coideal subalgebras of quantum groups, which have the property that all irreducible representations are one-dimensional, and which are maximal with this property. The obvious examples for this are the standard Borel subalgebras expected from Lie theory, but in a quantum group there are many more. Constructing and classifying them is interesting for structural reasons, and because they lead to unfamiliar induced (Verma-)modules for the quantum group. The explicit family we construct in this article consists of quantum Weyl algebras combined with parts of a standard Borel subalgebra, and they have a triangular decomposition. Our main result is proving their Borel subalgebra property. Conversely we prove under some restrictions a classification result, which characterizes our family. Moreover we list for Uq(4) all possible triangular Borel subalgebras, using our underlying results and additional by-hand arguments. This gives a good working example and puts our results into context.

Potential systems of a Buckley–Leverett equation: Lie point symmetries and conservation laws

In this work, we study a Buckley–Leverett equation of two-phase flow in porous media from the point of view of the Lie theory. We find that for some functions the equation has abundant exact solutions expressible in terms of Jacobi elliptic functions and consequently has many solutions in the form of periodic waves or solitary waves. We determine low-order conservation laws by using the multiplier method. From conservations laws, we construct the potential systems. From these systems we deduce potential equations. We study these equations from the point of view of Lie theory. Moreover, for these equations some nontrivial conservation laws are constructed by using the multipliers method.

A completely integrable case in the complex Lorenz equations

By application of the Lie theory of extended groups and for the parameter values σ=1/2, b=1, r1= e^2/2, r2=e/2, e arbitrary we prove that the system of the complex Lorenz equations is algebraically completely integrable. The respective general exact solution i$ expressed by means of Jacobian elliptic functions

Manif: A micro Lie theory library for state estimation in robotics applications

Manif: A micro Lie theory library for state estimation in robotics applications

Analyzing Controllability of Bilinear Systems on Symmetric Groups: Mapping Lie Brackets to Permutations

Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the design of optimal control laws, stabilization of unstable systems, and minimal realization of input-output relations. Tools from Lie theory have been adopted to establish controllability conditions for bilinear systems, and the most notable development was the Lie algebra rank condition (LARC). However, the application of the LARC may be computationally expensive for high-dimensional systems. In this paper, we present an alternative and effective algebraic approach to investigate controllability of bilinear systems. The central idea is to map Lie bracket operations of the vector fields governing the system dynamics to permutation multiplications on a symmetric group, so that controllability and controllable submanifolds can be characterized by permutation cycles. The method is further applicable to characterize controllability of systems defined on undirected graphs, such as multi-agent systems with controlled couplings between agents and Markov chains with tunable transition rates between states, which in turn reveals a graph representation of controllability through the graph connectivity.

On cohomologies and algebraic $K$-theory of Lie $p$-superalgebras

An enveloping associative superalgebra $\Lambda [L,\alpha ,\beta ]$ and its groups of cohomologies are defined and it is proved that there exists Frobenius multiplication of the Quillen algebraic $K$-functors of $\Lambda [L,\alpha ,\beta ]$. These results generalize corresponding results for Lie $p$-algebras which were proved by the author earlier.

Lie Theory, from Lie to Borel and Weil

Lie Theory, from Lie to Borel and Weil

Inside and Outside Perspectives on Institutions: An Economic Theory of the Noble Lie

If there exist no incentive or selective mechanisms that make cooperation in large groups incentive-compatible under realistic circumstances, functional social institutions will require subjective preferences to diverge from objective payoffs – a “noble lie.” This implies the existence of irreducible and irreconcilable “inside” and “outside” perspectives on social institutions; that is, between foundationalist and functionalist approaches, both of which have a long pedigree in political economy. The conflict between the two, and the inability in practice to dispense with either, has a number of surprising implications for human organizations, including the impossibility of algorithmic governance, the necessity of discretionary rule enforcement in the breach, and the difficulty of an ethical economics of institutions. Leeson and Suarez argue that “some superstitions, and perhaps many, support self-governing arrangements. The relationship between such scientifically false beliefs and private institutions is symbiotic and socially productive” (2015, 48). This paper stakes out a stronger claim: that something like superstition is essential for any governance arrangement, self- or otherwise. Specifically, we argue that human social structure both requires and maintains a systematic divergence between subjective preferences and objective payoffs, in a way that usually (though in principle does not necessarily) entails “scientifically false beliefs” for at least a subset of agents. We will refer to the basis of such preferences from the perspective of those holding them as an “inside perspective,” as opposed to a functionalist-evolutionary explanation of their existence, which we will call an “outside perspective.” Drawing on the theory of cooperation, we then show that the two perspectives are in principle irreconcilable, discussing some implications of that fact for political economy and the prospects of social organization.

Lie groupoids of mappings taking values in a Lie groupoid

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current Lie groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper \'etale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current Lie groupoid as a current Lie algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators given by postcomposition with a fixed function, between manifolds of $C^\ell$-functions. Under natural hypotheses, these operators turn out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map. These results are new in their generality and of independent interest.

Enveloping Algebras and Geometric Representation Theory

The workshop brought together experts investigating algebraic Lie theory from the geometric and categorical viewpoints.

A Miniature CCA Public Key Encryption Scheme Based on Non-abelian Factorization Problem in Finite Groups of Lie Type

Abstract With the development of Lie theory, Lie groups have attained profound significance in several branches of Mathematics and Physics. In Lie theory, the matrix exponential plays a crucial role between Lie groups and Lie algebras. Meanwhile, as the finite analogue of Lie groups, finite groups of Lie type have potential applications in cryptography due to their unique mathematical structures. In this paper, we first put forward a novel idea of designing cryptosystems based on Lie theory. First of all, combing with discrete logarithm problem and group factorization problem, we proposed several new intractable assumptions based on the matrix exponential in finite groups of Lie type. Subsequently, in analog with Boyen’s scheme (Asiacrypt 2007), we designed a public-key encryption scheme based on the non-abelian factorization problem in finite groups of Lie type. Finally, our proposal was proved to be indistinguishable against adaptively chosen-ciphertext attack in the random oracle model. It is encouraging that our scheme also has the potential to resist against Shor’s quantum algorithm attack.

A proof of the first Kac–Weisfeiler conjecture in large characteristics

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra g \mathfrak {g} . The first predicts the maximal dimension of simple g \mathfrak {g} -modules and in this paper we apply the Lefschetz Principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of g l n ( k ) \mathfrak {gl}_n(k) whenever k k is an algebraically closed field of sufficiently large characteristic p p (depending on n n ). As a consequence we deduce that the conjecture holds for the Lie algebra of an affine algebraic group scheme over any commutative ring, after specialising to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, an alternative, short proof of the first Kac–Weisfeiler conjecture is given for the Lie algebra of a group scheme over a finitely generated ring R ⊆ C R \subseteq \mathbb {C} , after base change to a field of large positive characteristic.

Analysis of unified error model and simulated parameters calibration for robotic machining based on Lie theory

Abstract In robotic machining process, the kinematic errors of serial structure and compliance errors caused by external cutter-workpiece interactions can result in considerable deviation of the desired trajectory. Therefore, this paper proposes an efficient calibration methodology by establishing a unified error model about kinematic errors and compliance errors based on Lie theory, which simultaneously calibrates the kinematic parameters and joint compliances of a serial machining robot. In this methodology, the propagation law of kinematic errors is investigated by analysis of the kinematic error model, and the corresponding equivalent kinematic error model is thus obtained, in which the joint offset errors are regarded as one source of twist (joint twist and reference configuration twist) errors. On this basis, with the segmentation and modelling of the joint compliance errors caused by the link self-weight and cutting payloads, the unified error model is developed by linear superposition of configuration errors of the robotic end-cutter, calculated from the kinematic errors and compliance errors respectively. Meanwhile, to improve the accuracy of parameters calibration, the observability index is adopted to optimize the calibration configurations so as to eliminate the twist error constraints. The calibrated kinematic parameters and joint compliances are obtained eventually, and used to compensate the kinematic and compliance errors of the serial machining robot. Finally, to validate the effectiveness of the proposed unified error model, simulation analysis is performed using a 6-DOF serial machining robot, namely KUKA KR500. The comparisons among calibrated parameters show that the unified error model is more computationally efficient with optimal calibration configurations, rendering it suitable for the calibration of kinematic parameters and joint compliances in actual machining applications.

On the smoothness of some quotients of Banach-Lie groups

Quotients of Banach-Lie groups are regarded as topological groups with Lie algebra in the sense of Hofmann-Morris on the one hand, and as Q-groups in the sense of Barre-Plaisant on the other hand. For the groups of the type G/N where N⊆G is a pseudo-discrete normal subgroup, their Lie algebra in the sense of Q-groups turns out to be isomorphic to the Lie algebra of G, which is in general merely a dense subalgebra of the Lie algebra of G/N when regarded as a topological group with Lie algebra. The submersion-like behavior of quotient maps of Banach-Lie groups is also investigated. The two aforementioned approaches to the Lie theory of the quotients of Banach-Lie groups thus lead to differing results and the Lie theoretic properties of quotient groups are more accurately described by the Q-group approach than by the approach via topological groups with Lie algebras.

Lie theory of multiplicative tensors

We study tensors on Lie groupoids suitably compatible with the groupoid structure, called {\em multiplicative}. Our main result gives a complete description of these objects only in terms of infinitesimal data. Special cases include the infinitesimal counterparts of multiplicative forms, multivector fields and holomorphic structures, obtained through a unifying and conceptual method. We also give a full treatment of multiplicative vector-valued forms, particularly Nijenhuis operators and related structures.

Differential calculus on multiple products

This paper introduced the calculus of mappings on finite product of spaces, allowing different degrees of differentiability in the different factors. This enables us to prove an important feature in the infinite-dimensional Lie theory, the exponential law in generalized setting for locally convex spaces and for manifolds modelled on locally convex spaces.

A 2-group construction from an extension of the 3-loop group varOmega ^3G

We define a 3-loop group {varOmega ^3G} as a subgroup of smooth maps from a 3-ball to a Lie group G, and then construct a 2-group based on an automorphic action on the Mickelsson–Faddeev extension of {varOmega ^3G}. In this, we follow the strategy of Murray et al. (J Lie Theory 27(4):1151–1177, 2017), who earlier described a similar construction for one-dimensional loops. The three-dimensional situation presented here is further complicated by the fact that the 3-loop group extension is not central.

Moving frames for Lie symmetries reduction of nonholonomic systems

Given a Lie group of finite-dimensional transformations acting on a manifold, there is always an action known as a long-acting group action. This action describes the fundamental basis of the Lie theory connecting groups of symmetry in differential equations. Differential invariants emerge as constants of the action of the prolongation of a group. Elie Cartan extended this in the twentieth century involving the geometry of the action of this group, grounding the so-called moving frame theory. With this theory, various applications are possible and detailed in the literature, such as symmetries of variational problems, conservation laws, invariant differential forms, and group invariant solutions. In order to demonstrate the approach, two nonholonomic constrained mechanical systems are exemplified to obtain either the general closed-solution in explicit form, when possible, or an order reduction provided by the Lie symmetries via moving frames. The first example is a coin with mass m rolling without slipping and takes on an inclined plane (x, y) with angle $$\alpha $$ and nonlinear constraint. The second example is a Chetaev type described by a dog pursuing a man in a plane surface with a nonholonomic restriction. A full detailed analysis is addressed to define the Lie symmetries and the corresponding moving frames obtained in both examples.