Nonexistence of Surjective Lie Homomorphism Between Complex Special and General Linear Groups

Weights for symmetric and general linear groups

Suppose that G is a locally compact group, that \({\Gamma}\) is a discrete, finitely generated group, and that $$\varphi\colon G \longrightarrow \Gamma$$ is an ‘abstract’ surjective homomorphism. We are interested in conditions which imply that \({\varphi}\) is automatically continuous. We obtain a complete answer to this question in the case where G is a topologically finitely generated locally compact abelian group or an almost connected Lie group. In these two cases the well-known structure theory for such groups G leads quickly to a solution. The question becomes much more difficult if one assumes only that G is a locally compact group. This leads to interesting questions about normal subgroups in infinite products and in ultraproducts. Łos’ theorem, the solution of the 5th Hilbert problem, and recent results by Nikolov–Segal can be combined to answer the question.

Automatic detection of feather defects using Lie group and fuzzy Fisher criterion for shuttlecock production

We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analytic Lie group. However, we prove that the Lie group structure on the group of real analytic diffeomorphisms is in general not real analytic in our sense.

We investigate a natural Lie group structure that correlations possess, interpreted as a special case of the quotient manifold structure for correlation matrices. The one‐dimensional setting of the general theory gives rise to additional structure in the form of a natural associative multiplication of correlations making the space of correlations into a Lie group. We explicitly compute left‐invariant vector fields, exponential and logarithmic mappings, and ultimately, a closed‐form distance formula between correlations. The Lie group formalism is then applied to an application in anomalous correlation detection. The advantage of the Lie group method illustrated with time series data of stock prices.

It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o $o$ -minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452) in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group G $G$ is Lie isomorphic to a group definable in an o $o$ -minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately, we cannot achieve a full characterization. Assuming that a Lie group G $G$ has its Levi subgroups with finite center, we prove that in order for G $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452).

This paper explores the problem of position and speed estimation of a target using satellite-based measurements in a Lie Group Kalman-like filter. The filter employs an intrinsic formulation of a nearly constant velocity model based on the Frenet-Serret frame, embedded in a Lie Group structure. It stands as a more suitable model for the kinematics of a target in space than the usual Euclidean model since it induces a banana-shaped distribution instead of the typical ellipsoidal provided by Gaussian schemes. The paper presents numeric experiments using a real application dataset to evaluate the proposed filtering scheme’s performance against the standard Euclidean representation approach. A differential GNSS solution provided by commercial software furnishes the ground truth. The results indicate a better performance of the proposed Lie Groups filtering scheme.

In this study, the authors address a two‐dimensional (2D) shape registration problem on data with anisotropic‐scale deformation and noise. First, the model is formulated under the iterative closest point (ICP) framework, which is one of the most popular methods for shape registration. To overcome the effect of noise, the expectation maximisation algorithm is used to improve the model. Then, the structure of Lie groups is adopted to parameterise the proposed model, which provides a unified framework to deal with the shape registration problems. Such representation makes it possible to introduce some suitable constraints to the model, which improves the robustness of the algorithm. Thereby, the 2D shape registration problem is turned to an optimisation problem on the matrix Lie group. Furthermore, a sequence of quadratic programming is designed to approximate the solution for the model. Finally, several comparative experiments are carried out to validate that the authors’ algorithm performs well in terms of robustness, especially in the presence of outliers.

The survey deals with the investigations reviewed inReferativnyi Zhurnal “Matematika” between 1977–1981. In the survey there are reflected the investigations on the structure of Lie groups and Lie algebras, on their finite-dimensional linear representations and universal enveloping algebras, on the theory of invariants and Lie groups of transformations, and also on continuous and discrete subgroups of Lie groups.

This article gives a local answer to the coquecigrue problem for Leibniz algebras, that is, the problem of finding a generalization of the (Lie) group structure such that Leibniz algebras are the corresponding tangent algebra structure. Using links between Leibniz algebra cohomology and Lie rack cohomology, we generalize the integration of a Lie algebra into a Lie group by proving that every Leibniz algebra is isomorphic to the tangent Leibniz algebra of a local Lie rack. This article ends with an example of a Leibniz algebra integration in dimension 5.

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data.For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.

The Butcher group is a powerful tool to analyse integration methods for ordinary differential equations, in particular Runge--Kutta methods. In the present paper, we complement the algebraic treatment of the Butcher group with a natural infinite-dimensional Lie group structure. This structure turns the Butcher group into a real analytic Baker--Campbell--Hausdorff Lie group modelled on a Fr\'echet space. In addition, the Butcher group is a regular Lie group in the sense of Milnor and contains the subgroup of symplectic tree maps as a closed Lie subgroup. Finally, we also compute the Lie algebra of the Butcher group and discuss its relation to the Lie algebra associated to the Butcher group by Connes and Kreimer.

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

1. Symmetries of vector spaces: 1.1. What is a symmetry? 1.2. Distance is fundamental 1.3. Groups of symmetries 1.4. Bilinear forms and symmetries of spacetime 1.5. Putting the pieces together 1.6. A broader view: Lie groups 2. Complex numbers, quaternions and geometry: 2.1. Complex numbers 2.2. Quaternions 2.3. The geometry of rotations of R3 2.4. Putting the pieces together 2.5. A broader view: octonions 3. Linearization: 3.1. Tangent spaces 3.2. Group homomorphisms 3.3. Differentials 3.4. Putting the pieces together 3.5. A broader view: Hilbert's fifth problem 4. One-parameter subgroups and the exponential map: 4.1. One-parameter subgroups 4.2. The exponential map in dimension one 4.3. Calculating the matrix exponential 4.4. Properties of the matrix exponential 4.5. Using exp to determine L(G) 4.6. Differential equations 4.7. Putting the pieces together 4.8. A broader view: Lie and differential equations 4.9. Appendix on convergence 5. Lie algebras: 5.1. Lie algebras 5.2. Adjoint maps { big `A' and small `a' 5.3. Putting the pieces together 5.4. A broader view: Lie theory 6. Matrix groups over other fields: 6.1. What is a field? 6.2. The unitary group 6.3. Matrix groups over finite fields 6.4. Putting the pieces together 6.5. A broader view of finite groups of Lie type and simple groups Appendix I. Linear algebra facts Appendix II. Paper assignment used at Mount Holyoke College Appendix III. Opportunities for further study Solutions to selected problems Bibliography.

In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.