Why Are There No Triple Affine Hecke Algebras?

What role do Lie groups and/or Lie algebras play in physics?

• A group is a set together with an associative law of composition that contains the identity and inverses. I can understand the basic definitions of http://en.wikipedia.org/wiki/Lie_groups and http://en.wikipedia.org/wiki/Lie_algebras. Lie algebras are apparently used to study Lie groups. I'd like to know what role the structures play in physics. For instance, the three angular momentum commutators in QM form a representation of a Lie algebra. Describing their functionality is similar to describing why they are useful-- I'd be grateful for a discussion of either. Be as specific or general as you see fit.

• Answer:

  Lie algebras describe continuous symmetries in infinitesimal form--- so if you have a geometrical continuous group, the Lie algebra describes the transformations near the identity. If G is a transformation near the identity, G=I + A where A is infinitesimal, then A is part of the Lie algebra, when you think of it as a concrete object, as a matrix you compute things with. But the lie algebra abstracts out the notion of group multiplication and leaves only the abstract properties of the infinitesimal parts. The product of G and G'  is to first order (I+ A)(I+A') = 1 + A + A', and to second order there's an AA' term which depends on details of the parametrization (for example, you might have A^2/2 term if you use an exponential). The group commutator of G and G' is GG'G^{-1}G'^{-1}, and to first order, this gives zero first order term, and the second order infinitesimal term is then AA' - A'A independent of parametrization. This quantity defines the Lie Bracket, it's the commutator of the A's, which are called the infinitesimal generators. [A,A'] = AA' - A'A Sophus Lie abstracted out the properties of the bracket which guarantee that an abstract bracket can be interpreted as a commutator. These are three properties--- they only refer to the bracket itself, not to the enveloping algebra that the objects live in: [A'.A] = - [A,A'] [a A + b B , C ] = a [A,C] + b [B,C] [[A,B],C] = [B,[A,C]] - [A,[B,C]] Where capital letters are generators, and little letters are constants. you can check these properties are all true when the bracket is a commutator. The third relation is the Jacobi identity, and it has a word interpretation when you think of the quantity A as acting on other things by commutation, it says that acting [A,B] is acting A then acting B and subtracting acting B then acting A. This gives intuition for this thing. The idea of the bracket is that it is the natural way that a generator can act on another generator. There are more applications in physics than one can count, and it is too broad a question, but here are some highlights: == Canonical Transformations == Every vector field on a manifold defines an infinitesimal motion, by moving all the points along the vector field. This is a subgroup of the group of all differential maps from the manifold to itself, if the vector field is differentiable it is an infinitesimal diffeomorphism (it is always invertible, because it is infinitesimally close to the identity). So you can define the commutator of these transformations, and this defines the Lie bracket of two vector fields: {U,V}j=Vi∂iUj−Ui∂iVj{U,V}j=Vi∂iUj−Ui∂iVj\{U,V\}^j = V^i \partial_i U^j - U^i \partial_i V^j This quantity doesn't depend on any connection, since it is an abstract commutator of diffeomorphisms. You can also explicitly see that the connection cancels out, so you can replace the derivative with a covariant derivative at will. For Hamiltonian manifolds (symplectic manifolds), that is, for classical phase space with position and momentum, every function on the phase space defines a vector field by taking the gradient and using the symplectic form. This is really just Hamilton's equations: dq = H_p dt dp = - H_q dt These equations tell you how to move a point in phase space infinitesimally given a scalar function H on the phase space. Such a transformation is an infinitesimal canonical transformation. So you can define the commutator of two different scalar functions H,G: first move p's and q's forward in time by dt as if H is the hamiltonian, then move them by dt as if G is the Hamiltonian, then move back using H for dt, then move back by dt using G, and divide the resulting motion by dt^2. This is the Lie bracket of the vector fields that tell you how p and q move in response to a Hamiltonian which is H or G. This Lie bracket is called the Poisson Bracket. == Quantum Mechanics == In quantum mechanics, the canonical transformations turn into unitary maps. The unitary maps have infinitesimal generators which are the Hermitian operators (actually anti-Hermitian, but it's conventional to multiply by i). So in quantum mechanics, the Poisson bracket is reinterpreted as an actual commutator of matrices. This is Dirac's transformation theory. In Dirac's version of quantum mechanics, the reason that commutation is how operators act on operators becomes obvious--- this is the Heisenberg equation of motion, it is just saying that when you make an infinitesimal unitrary transformaton, operators transform as the commutator. == Symmetries == For symmetries, there is an abstract group that you are representing, like abstract rotations. The Lie algebra structure of the rotations turns into the Poisson bracket structure of the rotation generators, and in quantum mechanics, to the commutator of the generators. The Lie algebra of the operators in quantum mechanics, or generator functions on phase space in classical mechanics coincides with the Lie algebra of the symmetry group. This is the most obvious application. == ADE classification == One of the great classifications of classical mathematics is the classification of the compact Lie groups into infinite families, and a few sporadic exceptions. The three families are all variations on rotations. There are rotations of real valued vectors in n dimensions which preserve length. These make the group SO(n). The case n even and the case n odd are distinguished because of the details of the Dirac algebra in the two cases. Then there are the rotations of n complex numbers, which preserve complex multiplication and complex length. These form a subgroup of SO(2n), since a complex number can be thought of as made up of a real and imaginary part. This special subgroup is the part which commutes with complex multiplication by scalars. This is the group SU(n). Then there are rotations of n quaternions. These are a subgroup of SU(2n) which preserve the quaternionic structure, and this is called Sp(n) (actually, it is most often called Sp(2n), but I think it's better to call it Sp(n), in analogy with SU(n), and I think a lot of modern people agree). These are the three infinite families. Then there are the exceptional groups too: G2, F4, E6, E7, E8, which can be thought of as arising from octonions and Jordan algebras, but they crap out after a finite number of examples, because the octonions are not associative. Each of these are useful in physics: SO(n) and it's relativistic analog SO(1,n) define symmetries of space time. SU(n) defines fundamental quantum mechanical phase space symmetries. Sp(n) defines symmetries in cases where there is a quaternionic structure. All of the infinite families, and the special groups, arise in string theory. == Particles == In the 1960s, Gell-Mann made an industry from classifying the symmetries of the low-lying hadrons, the particles called elementary then, using SU(3). This gave relations between particle masses which are ultimately a consequence of the quark model--- the SU(3) is quantum mechanical rotations of the three flavors of quarks into each other. The main buzzword here is "current algebra", and what it means is that you use the symmetry of the theory to make predictions about the spectrum and about scattering which don't rely on the field theory which is right. The same Lie groups then became useful in formulating the fundamental theory, when it was realized by Nambu that the quarks carry color, and the color group is coincidentally also SU(3). The modern particle physics theories use the representations of SU(2),SU(3), SU(5) and SO(10). The last two because the standard model fits inside SU(5), and therefore inside SO(10) in the most obvious way (by thinking of the 5 complex numbers as 10 real numbers). The SO(10) extends to E6 and this embeds into E8 in an interesting way in string theory E6 x SU(3) sits inside E8 in a similar way to how the standard model sits inside SU(5). Since E8 emerges from heterotic strings, this is the easiest path from strings to the standard model. == Kac Moody Algebras == In addition to the finite dimensional Lie algebras, the ones arising from compact groups, or complexifications of these, as appropriate for Minkowski space, there are also infinite dimensional Lie algebras that arise in 2 dimensional conformal theories. The Kac-Moody algebras are the symmetries of certain models of 2 dimensional physics, where there is a natural group acting on the space of configurations. The simplest example is to imagine a sheet of rigid rotators at every point in 2d (one dimension of space, one of time), with a spring making the nearby rotators want to be oriented in the same direction, and the generators are the motions which rotate the rotators at each point separately. This defines a Wess-Zumion-Witten model, or a chiral model in Polyakov's terminology. This is not really a symmetry, because the action relates orientations at nearby points, but the algebra of these transformations is useful for constructing the conformal symmetry by the Sugawara construction. I have given buzzwords, because writing more about it takes effort and time, and I think other people do a very good job on all these topics, better than what I can do right now.