From: fmeissch@eduserv2.rug.ac.be (Frank Meisschaert) Subject: Re: Who is Lie? Date: 21 Apr 1999 13:01:34 GMT Newsgroups: sci.math Keyswords: Lie algebras, intro Main Night (mainnight@aol.com) wrote: : >'Lie Algebra', : >appearantly more advanced than abstract algebra. : This prompts me to ask [possible (probably) foolishly), exactly how high DOES : algebra go? all i know of is elementary algebra, linear algebra, abstract : algebra, and now Lie algebra. Of course there's also vector algebra, though i : doubt that counts (correct me if i'm wrong). : The best person to ask this question would probably be Erdo"s, but (sigh) : he's no more. Lie algebra's are objects defined in abstract algebra(the theory of calculation). Generally an algebra(mathematical object) is a vector space on which some sort of multiplication (of vectors, with the result also a vector) which can be either associative or not. It depends on the author which of the following classifications are made : algebra <-> non-associative algebra , or algebra <-> associative algebra ; this is just a matter of naming convention. Now a Lie algebra is (the best known) an algebra which is non-associative, but of which the multiplication law satisfies the antisymmetry condition and the Jacobi associativity. Generally, for other non-associative algebra's, other restrictions on the multiplication law are stated. Abstractly a Lie algebra is associated with derivations (or derivation operators). Practically Lie algebra's are mostly used in physics where they are used to describe (continuous) symmetries. It is important to make a distinction between the two meanings of algebra. As a synonym for calculation theory, one has the classification elementary algebra (calculate with symbolic variables) linear algebra (matrix theory), abstract algebra (all of it, but abstractly defined). As a mathematical object (defined above) it can be a Grassman, Clifford, Lie, general matrix, group, Jordan, ... algebra. In computer science one has also partial algebra's which are also mathematical objects, but I don't know in which extent it is an algebra defined as above. Hope this 'abstract nonsense' helps. Frank Meisschaert PS: 'abstract nonsense' is a term my professor uses for explicitely stated trivialities. ============================================================================== From: jdolan@math.ucr.edu (james dolan) Subject: Re: Who is Lie? Date: Fri, 23 Apr 1999 00:01:40 GMT Newsgroups: sci.math thomas womack wrote: -Ah, this might be the right group to ask this question: - -What are Lie algebras *for*, and where did they come from? - -Groups, rings, fields, vector spaces ... are fairly natural objects to -consider if you start with the natural numbers and play around with -subsets and with modulus operations; algebraic number fields and the -like are natural once you start looking at solutions of polynomials. - -But I've not seen a thing in 'nature' which looks like a Lie algebra, -and the course I've done on them began by presenting them as a series -of axioms, then moved to looking at them as the commutator operation -in various fields of matrices, and finally defined 'semi-simple' and -classified the semi-simple ones with root systems and Dynkin diagrams -in the traditional way. It never suggested a context in which the Lie -algebra results made sense in a 'more natural' field. - -I suppose simple groups don't have very many applications, but they -feel like an interesting point in a well-developed theory which -started in a comprehensible place. Essentially, I can't see why you'd -be led to consider a Lie algebra. it's still amazing to me after all these years how anyone could be so evil or so stupid as to teach a whole course about lie algebras without explaining where they came from. the answers i've seen you get here so far have been pretty pathetic too, spewing crap like "they're important in physics". lie algebras have just _one_ main source of motivation: lie algebra theory = group theory + differential calculus. that is, whenever you have a group to which differential calculus can be applied (in the sense that the multiplication and inverse operations of the group belong to a class of functions to which differential calculus can be applied), a lie algebra can be extracted from the group, and the whole group can almost be recovered from knowledge of it's lie algebra. furthermore it's often worthwhile to extract the lie algebra from the group in this way because the lie algebra is often easier to work with than the group is, mainly because the lie algebra lives in the computationally tractable world of linear algebra. so, if you understand why group theory is important and why differential calculus is important, then you should be able to understand why lie algebra theory is important. (unfortunately though it looks from some of your other comments like no one ever explained to you the real reason why group theory is important, either.) ----- Posted via Deja.com, The People-Powered Information Exchange ----- ------ http://www.deja.com/ Discussions * Ratings * Communities ------ ============================================================================== From: parendt@nmt.edu (Paul Arendt) Subject: Re: Who is Lie? Date: 24 Apr 1999 03:42:07 GMT Newsgroups: sci.math >thomas womack wrote: > >-What are Lie algebras *for*, and where did they come from? >- >-... I've not seen a thing in 'nature' which looks like a Lie algebra, > ... stevel@coastside.net (Steve Leibel) wrote: >When I was in math grad school I attended an informal talk about this >subject, and the story was that Sophus Lie and another mathematician, >whose name I don't recall at the moment, together proposed to tackle a >general theory of groups. They flipped a coin or went through some kind >of decision making process. The other guy got the discrete groups, and >Lie got the continuous groups, which after he was all done with his theory >were forever more known as Lie groups. The other guy was Felix Klein. There is an easy-to-read book about them: Felix Klein and Sophus Lie, by I. M. Yaglom (trans. S. Sossinsky), Birkhauser, 1988 james dolan wrote: >lie algebras have just _one_ main source of motivation: lie algebra >theory = group theory + differential calculus. that is, whenever you >have a group to which differential calculus can be applied (in the >sense that the multiplication and inverse operations of the group >belong to a class of functions to which differential calculus can be >applied), a lie algebra can be extracted from the group, and the whole >group can almost be recovered from knowledge of it's lie algebra. Or, you can start with things using differential calculus, and often find out that there are Lie groups around! Lie's original motivation for studying Lie groups was actually ordinary differential equations. Most of the different techniques used to solve O.D.E.'s are actually seen to be the same technique when viewed in the right way: the existence of continuous symmetries of the differential equations are ultimately the reason why we can solve them at all (if we can). Because I'm lazy, I'll just include an old post: >>As another aside, it's amusing to note that Sophus Lie himself (of >>Lie group fame) became interested in Lie groups (he didn't call them >>that) as a tool to solve (ordinary) differential equations !!! Many >>of the unrelated-looking techniques for doing so, like solving exact >>or homogeneous equations, or taking Fourier or other transforms, >>all all unified by this: they're really the same technique. They only >>look different because the underlying differential equation possesses >>a different Lie group as a symmetry. Oz wrote: >If you can keep it mindblowingly simple could you expand on this a bit? In a nutshell, if an ODE has a one-parameter Lie group as a symmetry, it is overspecified in a sense, and can be simplified by taking advantage of this (reducing the number of variables by one). The first step is to visualize the space where your ODE lives: make axes out of all your variables (independent and dependent), and add on axes which signify all relevant derivatives of dependent variables. Your ODE is then some hypersurface (smaller-dimensional manifold) in this space. I think an example would do nicely here: let's say we have the simple ODE y' = y where y' = dy/dx. Draw a 3-D space where the axes are {x,y,y'}: the above ODE is a 2-D surface in this space: a plane which cuts a 45 degree line in a constant-x cross-section of the {y,y'} plane. OK? (You've probably solved this in your head already: y = C exp(x), where C is any number. But we're going to see, from symmetry reasons, why this is so easy to solve.) Now, this ain't no ordinary surface, since the y' axis is special. Imagine looking down the y' axis at our surface, and drawing contours of constant y' upon it (here, they are also lines parallel to the x-axis: lines of constant y). There are little tangent vectors induced on this surface, which represent the direction solutions are going to "travel." What I mean is, the vector at each point on the surface represents the slope of the line which is a solution through that point (yes, it's unique). However, this is a projection artifact, as we are still looking down the y' axis: the vectors are really tangent to the surface (as opposed to being parallel to the y'=constant plane). Along each of the contours we've drawn, we will see vectors all pointing the same way: that's what we mean by a contour of constant y'. So an ODE is really a "surface" in some higher-dimensional space (which includes derivatives as axes), with arrows all over it, and always tangent to it. In the example above, the surface always looks the same if we move along the x-axis by any amount. This is the geometrical picture of the " y' = y " equation's invariance under the transformation x -> x + a for all real numbers "a". (This symmetry is the simplest Lie group there is: the real numbers under addition.) So we expect that moving in the x-direction by any amount takes us from one solution onto another solution. And that's exactly what happens: redefine C in the solution above, and rewrite the solutions as the three families y = + exp(x + a) y = 0 y = - exp(x + a) where "a" is any real number. There is a _lot_ to be said about how the solutions split into these three families, why it was so easy to solve (the symmetry direction was already a coordinate), and how to use the representations of the group to look for solutions, but I hope this shows the basic idea.