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I'm having trouble getting my head arround the concept. The folks at mathoverflow were too clever to answer me. Can someone explain it to me?

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As someone on mathoverflow asked: could you provide some context? What's your background? –  Raskolnikov Feb 20 '11 at 19:46
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My rule of thumb here is that you should read the Wikipedia article (en.wikipedia.org/wiki/Lie_group) and then ask about anything that confuses you in it. (This could be "all of it," which is fine. The important thing is that you should answer at least two other questions first: do you know what a group is, and do you know what a smooth manifold is?) –  Qiaochu Yuan Feb 20 '11 at 21:06
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Never mind addressing the whole Wikipedia article, I think this question would be greatly improved by addressing the following question. In what way is the answer "A smooth manifold with a group structure so that the group operations are smooth maps" insufficient? Are you having trouble getting your head around this definition, around some particular examples, or around something else (like structure theory or the relationship with Lie algebras)? –  Anton Geraschenko Feb 20 '11 at 21:38
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Perhaps the correct questions are: Why should I care about Lie groups? What are they useful for? Why were they "invented"? What do they represent? An answer of the genre "a smooth manifold +++" is just not interesting from this perspective; I could make up lots of sets of axioms, but why do people care about this specific one? –  Yuval Filmus Feb 21 '11 at 0:05
    
My guess is that Lie groups are one of two frameworks in which it makes sense to study matrix groups, in particular it is well suited for studying these groups' topology and differential geometry of their homogeneous spaces. –  Alexei Averchenko Oct 27 '12 at 15:43

4 Answers 4

Consider the set of $(n\times n)$ matrices that have non-zero determinant. Such a matrix corresponds to a system of linear equations ($n$ equations in $n$ unknowns) that has a unique solution. You can think of the solution as the unique point of intersection between the graph of a function and a horizontal hyperplane. Here it is helpful to think of $n=1$. In other words, the coefficients of the system correspond to a transformation of space: the variables $x_1, \ldots x_n$ are transformed to $\sum a_{ij} x_i$. The set of such transformations form a group: the matrices can be multiplied, each has an inverse, the multiplication is associative, and the identity transformation fixes each point of space.

Intuitively, it is easy to see which transformations are close to one another. They are close if they move points that are nearby to points that are nearby. Arithmetically, if the entries in the matrix are close, then the transformations are close: thus $0.14x + .33y$, is a reasonable approximation to $x/7+y/3$.

Thus the set of invertible $(n\times n)$ matrices is a space of invertible $(n\times n)$ matrices. What is not easy for a layman to see is that its spacial characteristics are defined via the determinant since as a set, the $(n\times n)$ matrices are a subset of $n^2$-space. The non-singular matrices are the pull-back of a regular value of the determinant function. [There is a small lie here: this is true for for matrices of determinant 1, but all non-zero determinant matrices deform onto that smaller space].

One important spacial characteristic is that these matrices form a smooth manifold. This is something that is analogous to the surface of a sphere (which is NOT a lie group), the surface of a torus (which is) or the $3$-dimensional sphere that consists of the set of $(x,y,z,w)$ such that $x^2+y^2+z^2+w^2=1$ (which also happens to be a Lie group).

From these examples, we abstract the idea of a Lie group which is a group (that can be thought of as a set of transformations or symmetries) that has the structure of a smooth manifold --- at small scales it is indistinguishable from ordinary Euclidean space. The multiplication and inversion maps are a differentiable functions. And these multiplications occur between pairs of symmetries --- they should not be confused with the action of the matrices on the vector space which is where I started the discussion.

Examples include the real line, the non-zero real numbers, the circle, the torus, the $3$-sphere, the set of rotations of 3-dimensional space, and the special unitary groups representations of which determine particles in physics.

There are some small problems with the definition that I gave. A smooth manifold is a topological space which is paracompact and Hausdorff (neither definition will play a role in the layman's understanding), and that is covered by coordinate charts with specific properties. I imagine that Wikipedia has the relevant definitions articulated carefully.

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I think that understanding comes through examples. The most fundamental example I believe to be the rotation group. Consider the sphere $S^2\subset \mathbb{R}^3$. The sphere has rotational symmetries. If we rotate the sphere by any angle, the sphere doesn't change.

The collection of all rotations forms a Lie group. The group property basically means that if we rotate the sphere over any angle $\alpha$, after this over an angle $\beta$, it is the same if we would have rotated it in one go (over some different angle). Also any rotation has an inverse (rotating it over the opposite angle). This makes the rotations a group. The "Lie" in Lie group means that these rotations can be done arbitrary small. Many small rotations makes for a big rotation.

Lie groups capture the concept of "continuous symmetries".

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Group theory is the study of symmetry. Lie groups were invented by Lie to study the symmetries of differential equations. You might like the article:

Starrett, John. "Solving differential equations by symmetry groups." Amer. Math. Monthly 114 (2007), no. 9, 778–792. MR2360906 (author's preprint).

It describes how to solve some reasonably calc-2 or calc-4 problems using symmetry and can serve as introduction to what Lie groups do. It also has pretty pictures of smoothly flowing curves.

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it's a group thats a smooth manifold. Technically, the group structure and topology should be related, i.e. $X\times X\to X, (a,b)\mapsto ab^{-1}$ should be smooth. For instance, $\mathbb{R}^n$, the circle, a torus (surface of a doughnut) the three sphere ($\mathbb{R}^3$ with a point at infinity) can all be given the structure of a lie group.

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My understanding is as follows: Lie group - smoothly varying families of symmetries. A differentiable (locally smooth) manifold whereby the group operations are compatible with the smooth structure – Bijective transform mappings maintain symmetry. It is a group object in the category of smooth manifolds. In a compact lie group, the symmetries form a bounded set. –  user7293 Feb 20 '11 at 22:56

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