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 <> nonassociative 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
nonassociative, but of which the multiplication law satisfies the
antisymmetry condition and the Jacobi associativity. Generally, for other
nonassociative 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 'semisimple' and
classified the semisimple 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 welldeveloped 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.)
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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 easytoread 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 unrelatedlooking 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 oneparameter 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 (smallerdimensional 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 3D space where the axes are {x,y,y'}: the
above ODE is a 2D surface in this space: a plane which cuts a 45
degree line in a constantx crosssection 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 xaxis: 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 higherdimensional 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 xaxis 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 xdirection 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.