Transformation Groups and Lie Algebras

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Authors: Gorbatsevich , V. From the reviews: "This volume consists of two parts. Part I is devoted to a systematic development of the theory of Lie groups. The Lie algebras are studied only in connection with Lie groups, i. Neither the structural theory of the Lie groups and Lie algebras nor a systematic study of the topology of Lie groups form the subject of this volume.

Lie Groups and Lie Algebras I

On the other hand, Part I contains a very interesting chapter on generalizations of Lie groups including very recent results. We find here Lie groups over non-archimedian fields, formal groups, infinite dimensional Lie groups and also analytic loops. Part II deals on an advanced level with actions of Lie groups on manifolds and includes subjec ts like Lie groups actions on manifolds, transitive actions, actions of compact Lie groups on low-dimensional manifolds.

Though the authors state that the geometry and topology of Lie groups is almost entirely beyond the scope of this survey, one can learn a lot in these directions.

Both parts are very nicely written and can be strongly recommended. From my viewpoint, the volume is perfectly fit to serve as such a source This is a hand- rather than a textbook. On the whole, it is quite a pleasure, after making yourself comfortable in that favourite office armchair of yours, just to keep the volume gently in your hands and browse it slowly and thoughtfully; and after all, what more on Earth can one expect of any book? JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.

Mathematics Algebra. What questions do they ask for which Lie groups or algebras will be of any help?

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And if a geometer reads this, how if at all do you use Lie theory? How is the representation theory of Lie algebras useful in differential geometry? Here is a brief answer: Lie groups provide a way to express the concept of a continuous family of symmetries for geometric objects. Most, if not all, of differential geometry centers around this.

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Its Lie algebra is the real line. Lie groups are named after Norwegian mathematician Sophus Lie , who laid the foundations of the theory of continuous transformation groups. Proof of the uniquness of decomposition, Schur's lemma 9b. 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. As a concluding remark, let me note that a general principle is that when certain symmetries are implict in a given context e.

By differentiating the Lie group action, you get a Lie algebra action, which is a linearization of the group action. As a linear object, a Lie algebra is often a lot easier to work with than working directly with the corresponding Lie group. Whenever you do different kinds of differential geometry Riemannian, Kahler, symplectic, etc.

It is possible to learn each particular specific geometry and work with the specific Lie group and algebra without learning anything about the general theory. However, it can be extremely useful to know the general theory and find common techniques that apply to different types of geometric structures. Moreover, the general theory of Lie groups and algebras leads to a rich assortment of important explicit examples of geometric objects.

I consider Lie groups and algebras to be near or at the center of the mathematical universe and among the most important and useful mathematical objects I know. As far as I can tell, they play central roles in most other fields of mathematics and not just differential geometry. I don't think we introduce Lie groups and algebras properly to our students. They are missing from most if not all of the basic courses. Except for the orthogonal and possibly the unitary group, they are not mentioned much in differential geometry courses.

They are too often introduced to students in a separate Lie group and algebra course, where everything is discussed too abstractly and too isolated from other subjects for my taste. Other groups give higher dimensional hyperbolic spaces e. These are some of the most celebrated Riemannian manifolds in mathematics.

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Now as the preceding discussion hopefully makes clear , lots of these spaces are known by other names that don't involve Lie theory, and can be studied in a non-Lie-theoretic way. But the Lie-theoretic perspective provides a unifying, and frequently clarifying, point of view. For example, cohomological or function-theoretic invariants of these spaces can often be described and computed via Lie theoretic tools e. As a concluding remark, let me note that a general principle is that when certain symmetries are implict in a given context e.

In geometry, the symmetry groups that appear of a space, or perhaps of its universal cover are very often Lie groups. And so a little knowledge of Lie theory can turn into a powerful tool for investigating a given geometric situation. Trying to understand and work on these conjectures was my own motivation for learning Lie theory. Lie's motivation for studying Lie groups and Lie algebras was the solution of differential equations. Lie algebras arise as the infinitesimal symmetries of differential equations, and in analogy with Galois' work on polynomial equations, understanding such symmetries can help understand the solutions of the equations.

I found a nice discussion of some of these ideas in Applications of Lie groups to differential equations by Peter J. I like Deane's answer, and I doubt that I can improve upon it, but here is an attempt. One understanding of fundamental particles is that they are representations of classical Lie groups.

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I think that is reason enough to study them. But more down to earth, the circle is one of the easiest examples of a Lie group to study. Its Lie algebra is the real line. These are really cool examples. The general theory might also be really cool. A very nice example of a use of representation theory is the Hodge theory for Kaehler manifolds as is done e. One can view this as a decomposition of exterior forms and deRham differential under a subgroup which preserves the geometric structure. In this example representation theory helps to organize things and calculations and there are many similar ones in spirit.

Let me also try to exapand Deane Yang's answer and explain the importance of Lie groups in differential geometry. Bernhard Riemann solved the equivalence problem i. Elie Cartan developed a general method for solving such equivalence problems see Cartan's equivalence method or Method of moving frames on wikipedia.

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The notion of Lie group is already explicit there as it represents the symmetries of the geometrical structure one is interested in. This approach was later developed into what is now called Cartan geometry. Informally, these geometries are curved versions of Klein geometries.

The story can be told like this:. Given a geometrical structure, it is often hard theorem that the category of manifolds with this structure is isomorphic to a certain subcategory of the category of appropriate Cartan geometries. Nevertheless, Cartan's approach gives you very general and conceptual view on geometries like Riemannian, conformal, projective, Kaehler, quaternionic Kaehler, hyperKaehler, contact-projective, CR, One can regard the curvature tensor as an element of the tensor product of these and decomposition into irreducible subrepresentations then gives generalizations of Weyl and Ricci curvatures from Riemannian geometry.

The Dirac operator of mathematical physics can be thought of as a deRham differential composed with a projection and an intertwining map between certain representations. In fact even such fancy gadgets as Lie algebra cohomology play their role the keyword being "harmonic curvature". In the end, you see that in order to understand the appearance of Lie groups in geometry, one has to read Klein's program.

The rest is just ingenious technology to allow for nonflat things. Although the title is about Lie algebras, the question body mentions Lie groups, and my answer will deal more with these. As mentioned in other answers, Lie groups show up frequently in geometry as groups of symmetries of geometric objects. Let's look at something a bit more specific. What we have done -- roughly speaking -- is cast aside the manifold and are now working primarily with the group.

Strictly speaking, this is only true of equivariant vector bundles, i.

Jordan-Kronecker invariants of finite-dimensional Lie algebras

The key phrase here is "Borel--Weil--Bott theorem. Here is a concrete example. There is another obvious reason why Lie groups are important in geometry: they are themselves geometric objects namely, manifolds! So you cannot expect to say something about general manifolds that cannot be said about them. Since Lie groups are a relatively well-behaved class of manifolds, one can use them as a test case of or a launch pad to more general results. As has been said, Lie groups are our best theory encoding continuous symmetry. Lie algebra theory, which is the infinitesimal counterpart, is a theory good enough that numerous problems can be solved by look-up, rather than arguing from first principles.

You can look at the history, particularly with Cartan and Weyl; you can look at the examples coming from "commutation relations" people want to study; you can look at representation theory or root systems or the theory of universal enveloping algebras; you can look at string theory or the Langlands philosophy. It has been found very natural to look at the Lie algebra as a linearised object behind the Lie group, and something easier to study. Large subfields of modern differential geometry hardly ever use Lie group theory, e.

Major uses of Lie groups in Riemannian geometry are:. Principal bundles and Chern-Weil theory. Collapsing theory with two sided curvature bounds where local models are nilpotent Lie groups. Kobayashi-Nomizu's two volume "Foundations of Differential Geometry" discusses 1,2,3 extensively. I would like mentioning that also Lie algebras defined over fields of prime characteristic are very useful in several areas. For instance, they have many applications in algebraic geometry in particular, to affine group schemes , group theory for example, to the solution of the restricted Burnside problem or field theory to purely inseparable fields extensions.