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Such structures were already considered in [ 5 , 11 , 10 ] and [ 9 ], among others. For the rest of the paper we will assume that all variable Lie algebras are smooth. One has a similar definition of the kernel functions if one takes another smooth family of polarisations together with a smooth family of Malcev bases. In this section, we state the main theorem of the paper which will be proved in the next section.
The proof of Theorem 3. Therefore the result is true for the function g. We start this section with an example which will demonstrate the use of a variable group and its variable algebra in the retract construction. In this subsection, we will consider the free two-step nilpotent Lie group on four generators. There are two preliminary steps to check. The algorithm used to build the retract function f respects the semi-norms defining the topology of our function spaces. From the proof of [ 8 , Theorem 1. The authors would like to thank the referees for suggestions and comments which helped to improve the presentation of the paper.
Skip to main content Skip to sections. Advertisement Hide. Download PDF. Open Access. First Online: 01 December Definition 2. In this case, there are no Ludwig-Zahir indices, i. The function F has the Schwartz space property, i. Remarks 2. Theorem 3. Remark 3. Proof It suffices to apply the Fourier inversion formula. Remark 4. Let us focus on the manifold M again. The co-adjoint orbits and the Pukanszky layers of the orbit spaces of the group G have been determined in the paper [ 4 ]. Using 4. Definition 5.
Theorem 5. Acknowledgements The authors would like to thank the referees for suggestions and comments which helped to improve the presentation of the paper. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group. An example of a simply connected group is the special unitary group SU 2 , which as a manifold is the 3-sphere.
The rotation group SO 3 , on the other hand, is not simply connected. See Topology of SO 3. The failure of SO 3 to be simply connected is intimately connected to the distinction between integer spin and half-integer spin in quantum mechanics. Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups. The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group.
We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows. The formal similarity of this formula with the one valid for the exponential function justifies the definition. The exponential map relates Lie group homomorphisms. In other words, the following diagram commutes , [Note 1]. In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.
The exponential map from the Lie algebra to the Lie group is not always onto , even if the group is connected though it does map onto the Lie group for connected groups that are either compact or nilpotent. For example, the exponential map of SL 2, R is not surjective. One important aspect of the study of Lie groups is their representations, that is, the way they can act linearly on vector spaces. In physics, Lie groups often encode the symmetries of a physical system.
The way one makes use of this symmetry to help analyze the system is often through representation theory. One important example of such a system is the Hydrogen atom. This space, therefore, constitutes a representation of SO 3.
These representations have been classified and the classification leads to a substantial simplification of the problem , essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation. The case of a connected compact Lie group K including the just-mentioned case of SO 3 is particularly tractable. The irreducible representations, in turn, were classified by Hermann Weyl. The classification is in terms of the "highest weight" of the representation.
The classification is closely related to the classification of representations of a semisimple Lie algebra. One can also study in general infinite-dimensional unitary representations of an arbitrary Lie group not necessarily compact. According to the most authoritative source on the early history of Lie groups Hawkins, p.
Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of to the fall of " that led to the theory's creation ibid. Some of Lie's early ideas were developed in close collaboration with Felix Klein. Lie stated that all of the principal results were obtained by But during the s all his papers except the very first note were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe ibid , p.
In a young German mathematician, Friedrich Engel , came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume Theorie der Transformationsgruppen , published in , , and The term groupes de Lie first appeared in French in in the thesis of Lie's student Arthur Tresse.
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Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi , on the theory of partial differential equations of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the s, generating enormous interest in France and Germany Hawkins, p. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries.
The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations , the driving conception was of a theory capable of unifying, by the study of symmetry , the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled.
Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory , but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures , the indefinite integrals required to express solutions. Additional impetus to consider continuous groups came from ideas of Bernhard Riemann , on the foundations of geometry, and their further development in the hands of Klein.
Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing , who in published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen The composition of continuous finite transformation groups Hawkins, p.
Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups i. Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis.
What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra Lie himself called them "infinitesimal groups".
It can be defined because Lie groups are smooth manifolds, so have tangent spaces at each point. The Lie algebra of any compact Lie group very roughly: one for which the symmetries form a bounded set can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting since the Lie bracket is identically zero ; the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups?
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It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A n , B n , C n and D n , which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E 8 is the largest of these. Lie groups are classified according to their algebraic properties simple , semisimple , solvable , nilpotent , abelian , their connectedness connected or simply connected and their compactness.
A first key result is the Levi decomposition , which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup. The identity component of any Lie group is an open normal subgroup , and the quotient group is a discrete group. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center.
Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. This can be used to reduce some problems about Lie groups such as finding their unitary representations to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.
Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional.
Unitary Representations of Nilpotent Super Lie Groups
The simplest way to define infinite-dimensional Lie groups is to model them locally on Banach spaces as opposed to Euclidean space in the finite-dimensional case , and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not Banach manifolds.
Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold. The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix Lie in Lie group.
On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough for example, failure of the exponential map to be onto a neighborhood of the identity.
It is the "nice enough" that is not universally defined.
The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity. From Wikipedia, the free encyclopedia. Group that is also a differentiable manifold with group operations that are smooth. Classical groups. Simple Lie groups. Other Lie groups. Lie algebras. Exponential map Adjoint representation group algebra.
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Killing form Index. Semisimple Lie algebra. Dynkin diagrams Cartan subalgebra Root system Weyl group. Real form Complexification. Homogeneous spaces.
Closed subgroup Parabolic subgroup Symmetric space Hermitian symmetric space Restricted root system. Representation theory. Lie group representation Lie algebra representation. Lie groups in physics. Basic notions. Subgroup Normal subgroup Quotient group Semi- direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics.
Finite groups. Discrete groups Lattices. Topological and Lie groups. Algebraic groups. Linear algebraic group Reductive group Abelian variety Elliptic curve. Not to be confused with Group of Lie type. See also: Table of Lie groups and List of simple Lie groups. Main article: Lie group—Lie algebra correspondence.
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