Let's use SL(2,5) as a running example:
Three 2-dim generators (with determinant 1) over GF(5) are …
But of course 2 generators are be enough (we achieved these matrices in Chapter 12, Representation Theory):
gap>SL25:=SL(IsMatrixGroup,2,5);;
gap>GeneratorsOfGroup(SL25);
[ [[ Z(5) , 0*Z(5) ], [ 0*Z(5), Z(5)^3 ]],
[[ Z(5)^2, Z(5)^0 ], [ Z(5)^2, 0*Z(5) ]] ]
And the corresponding Lie algebra sl(2,5) has as well 3 generators h,x,y (with trace zero):
gap>mat:=[ [[Z(5)^0,0*Z(5)],[0*Z(5),-Z(5)^0]],
[[0*Z(5),Z(5)^0],[0*Z(5),0*Z(5)]],
[[0*Z(5),0*Z(5)],[Z(5)^0,0*Z(5)]]];;
gap>sl25:=LieAlgebra(GF(5), mat);
<Lie algebra over GF(5)m with 3 generators>
gap>Size(sl25);
125
gap>SemiSimpleType(sl25);
"A1"
with [h,x]=2x, [h,y]=-2y and [x,y]=h. How do we construct them, starting with the matrix group? Anyway it seems that always the projective groups and their covers have the same algebra (let us be exact and say isomorph algebras instead) associated to them. And in the same manner non-isomorph algebras can have the same unique (besides complexified means adding rot if needed?!) universal enveloping, associative algebra. The Lie algebra only determines uniquely the simply connected (universal) cover of a Lie group G.
The universal cover of SL(2, ) has no matrix representation. Every Lie group G has the topology of K × ^n where K is a maximal compact subgroup of G. SO(2) is a maximal compact subgroup of SL(2, ), and SL(2, ) is 3-dimensional, so as a topological space, SL(2, ) is the same as SO(2) × ^2 . It looks like a circle times a plane. SL(2, ) is not simply connected. Its fundamental group is , the integers. This means that its universal cover is an infinite-sheeted cover.
There is as well some talking about the universal enveloping algebra of matrix polynomials. Well, how about the automorphism group of an algebra, composed of mappings of algebra elements? Many things are to be done.
According to [Bak02] the Heisenberg algebra is a Lie algebra. The Heisenberg groups are actually examples for Lie groups not expressable as matrix groups (but as quotients of matrix groups). Both the quotient and the embedding have the same Lie algebra with the canonical commutation relation [p,q]=1.
Should we say that going "a" or "b" measure the momentum and changes the location, and that doing "rot" measures the location and changes the momentum (let's fix the radial momentum component for now)? Or are those algebras two different things?
It seems that the Heisenberg algebra is retrieved by a central extension, starting from the sl(2,5) algebra (and this remark belongs to algebra cohomology, yep).
You know, there are cases where we can express the rot just using "a" and "b", and this finding seems to correspond to the fact that of course you can say rot = ydx-xdy and use for rot an algebra element like:
gap>[[0*Z(5), Z(5)^0],[-Z(5)^0, 0*Z(5)]];
In the general case this additional generator will make a difference though.
And for the resulting group we should anway use a rotation with a periodicity in compliance with torsion.
Ok, and once we are in the mood to contemplate, here comes an idea how to incorporate the morphing groups. When we have two pieces and one piece moves, then the location of the other piece isn't measured totally, but only to some extent. So we shouldn't need a complete rot, but only a morphing along the pieces path. This creates the need for new labels instead of just enforcing the 100% complete Heisenberg picture. Ok, so for the new topologies we have other generators, and the morphing will then transform between them.
Anyway we have the impression that s.th. from the world of gauge theory, spin, symplectic, metaplectic, SO and Poincaré groups are in general what we are looking for.
Well yes, the relativity stuff must still be incorporated yet, you are invited to contribute. It's something about Galilei- versus Lorentz transformation invariance. What does the standard physics read like?
The so-called Poincaré group (the inhomogeneous Lorentz group) is the group of isometries of the Minkowski space, that's to say all A with A+gA = g whereas A+ denotes the adjoint operator of A and g is a non-degenerate (det != 0), semidefinite (positive and negative Eigenvalues) metric like
gap>metric:=[
[-1 0 0]
[ 0 1 0]
[ 0 0 1]
]
with signature 1 (an invariant, it's the sum of Eigenvalues) in this example.
In standard (3+1)-dim physics it is a 10-dimensional Lie group (rotation, boost, +4). Its unitary irreducible representations on Hilbert space are indexed by mass (nonnegative number) and spin (integer or half integer), and are associated with particles in quantum mechanics. And the Minkowski space is four dimensional space ^4 with metric of signature (-1, 1, 1, 1). Well, what is time? Certainly in a 2-body system we'll have to use s.th. from the respectively other piece instead of some global placeholder called 'time', but in its own right we can work with some time coordinate as well of course for a start. Probably the time variable helps eliminating singularities, somehow the stuff gets embedded into a projective space or so. Let's guess that we would get a 2+1(time)+1(rotation)+1(boost) = 5 - dimensional Lie group for 1 piece in the SL(2,5) case. Perhaps let's start with the generators of the Poincaré Lie algebra …
The invariant ring of a group has to do with integrable systems, it's about time that we get comfortable with this field as well.
The maximal Abelian subalgebra is about all observables to be measured simultaneously, and you can describe this as well as the center of the universal enveloping algebra. The latter is consisting of all polynomials of the (Lie algebra) operators.
From any associative algebra we can cook up a Lie algebra by defining
[X,Y] = XY-YX
and the universal enveloping algebra is the reverse process where for any Lie algebra we cook up an associative algebra such that
XY-YX = [X,Y]
. The Casimirs are the generators of the center just mentioned, and for example the Laplace-Beltrami operator will arise in this way (when dealing with harmonic functions).
Universal enveloping algebras are the Lie theoretic analogues of group algebras.
Let L be a finite dimensional Lie algebra with universal enveloping algebra U(L). Every (irreducible) L-module M is also an (irreducible) module for the enveloping algebra U(L). If we are working over an algebraically closed field, then the center Z(L) of U(L) operates on every finite dimensional irreducible L-module via scalars. In the classical context, these scalars separate the irreducible modules, so that each finite dimensional irreducible module is uniquely determined by its central character. This fact depends on Harish-Chandra's Theorem, which relates Z(L) to polynomial invariants of the Weyl group of L.
If we work over fields of positive characteristic (the modular case), the situation changes completely. Among other factors, the difference resides in the fact that the center of the enveloping algebra U(L) tends to be bigger. Given an n-dimensional Lie algebra L one can always find a subalgebra O(L) of Z(L) such that O(L) is a polynomial ring in n variables, and U(L) is a free O(L)-module of finite rank. One notable consequence is the finite dimensionality of irreducible L-modules, which markedly contrasts with the classical situation. For algebraically closed fields the irreducible L-modules may thus again be subdivided according to their characters on O(L). However, this time different irreducibles may give rise to the same character. In fact, the irreducible modules belonging to the same character are modules over a finite dimensional associative algebra, the so-called reduced enveloping algebra. One thus obtains an algebraic family of finite dimensional algebras that is parametrized by the maximal spectrum of O(L).
Btw. as an unrelated side note, is there an opportunity to involve the Killing form, Casimir operator, heighest weight, Clebsch-Gordan coefficients and Young tableux anywhere? Let's see what is in general available:
gap>A:=FullMatrixAlgebra(GF(5),2);;
gap>L:=LieAlgebra(A);;
gap>L=FullMatrixLieAlgebra(GF(5),2);
true
gap>Dimension(L);
4
gap>Size(L);
625
gap>d:=DirectSumDecomposition(L);;
gap>d[2]=sl25;
true
gap>LieNilRadical(L);;
gap>LieDerivedSeries(L);
gap>LieLowerCentralSeries(L);
gap>IsLieAbelian(sl25);
false
gap>IsLieNilpotent(sl25);
false
gap>IsLieSolvable(sl25);
false
gap>levi:=LeviMalcevDecomposition(sl25);
[ <Lie algebra of dimension 3 over GF(5)>,
<Lie algebra of dimension 0 over GF(5)> ]
gap>CartanSubalgebra(sl25);
<Lie algebra of dimension 1 over GF(5)>
gap>R:=RootSystem(sl25);
<root system of rank 1>
gap>PositiveRoots(R);
[[ Z(5) ]];
gap>PositiveRootVectors(R);
[ LieObject( [[ 0*Z(5), 0*Z(5) ], [Z(5)^0, 0*Z(5) ]] ) ]
gap>NegativeRoots(R);
[[ Z(5)^3 ]]
gap>SimpleSystem(R);
[[ Z(5) ]]
gap>C:=CartanMatrix(R);
[[ 2 ]]
gap>BilinearFormMat(R);
[[ Z(5)^3 ]]
gap>W:=WeylGroup(R);
Group([[[ -1 ]]])
gap>DisplayCompositionSeries(W);
G
| Z(2)
1 (size 1)
gap>IsRestrictedLieAlgebra(sl25);
true
gap>B:=Basis(sl25);;
gap>SCT:=StructureConstantsTable(B);;
gap>l:=LieAlgebraByStructureConstants(GF(5),SCT);;
gap>ll:=IsomorphismSCAlgebra(B);;
gap>AdjointBasis(B);;
gap>AdjointMatrix(B, B[1]);;
gap>KillingMatrix(B);
[[ Z(5)^3, 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^2 ],
[0*Z(5), Z(5)^2, 0*Z(5) ]]
gap>BB:=Basis(L);;
gap>IsNilpotentElement(L, BB[2]);
true
gap>sl2:=FindSl2(L, BB[2]);;
gap>sl2=sl25;
true
gap>IsPerfect(sl25);
true
gap>sl25=DerivedSubgroup(sl25);
true
And even some more obscure stuff like infinity algebras and modules with their cohomology are feasable:
gap>u:=UniversalEnvelopingAlgebra(sl25,B);; Dimension(u);
infinity
gap>V:=AdjointModule(sl25);;
gap>M:=FaithfulModule(sl25);;
gap>cs:=CochainSpace(V,2);
<vector space of dimension 9 over GF(5)>
gap>Size(cs);
1953125
gap>Size(CochainSpace(V,3));
125
gap>Cocycles(V,2);
<vector space of dimension 6 over GF(5)>
gap>Coboundaries(V,2);
<vector space over GF(5), with 9 generators>
Finally let's play around a little bit with normal forms:
gap>JordanDecomposition(SL25);
[[[ Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^3 ]],
[[ 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5) ]]]
Well, and now we finally have the covers of the holonomy groups, we realize that's actually the same as doing an embedding into GL (as well called linearization). For perfect groups (identical to the derived group, G = G') this is trivial, quite easy going. Perhaps it would be a Good Thing (TM) to construct matrix representations (looking at the Brauer character table to see what dimensions are possible over a to-be-determined GF field) explicitly, for the original group as well as for the cover, and then we should proceed with that logarithm stuff. Anyway we shouldn't forget about our main goal: to find s.th. useful for dealing with the strategy games.
Finally the theory of buildings, BN-pairs and Bruhat-Tits (see [Gar97]) should be mentioned as well.