lie algebras and their representations

1. Lie Algebras and their Representations Sam Espahbodi Advisor: James Wells Michigan Physics REU

2. What are Lie Algebras? Let�s start with an algebra, which is a vector space that is given additional structure Take a vector space �g� and define on it a bilinear map A Lie algebra is an algebra with the additional requirements that The Jacobi identity

3. More on Lie Algebras Instead of writing f(_,_), we write [_,_] so that we have, for example, We call the map the Lie bracket This looks like a commutator, but it is not the same thing However, given an algebra with an associative product operation (remember it must be bilinear), we can define a Lie algebra on the same vector space by defining the Lie bracket as the familiar commutator The Jacobi identity is then trivially satisfied Not all Lie algebras can be defined this way, using a so called enveloping algebra, however

4. How do Lie Algebras Arise? Lie algebras arise in physics from Lie groups A Lie group is a group that is also a differential manifold; the Lie algebra is the tangent space of this manifold at the identity An example is SO(3), the set of rotations of Euclidean space, given mathematically by 3x3 orthogonal matrices Elements of the Lie algebra are said to generate elements of the Lie group, or they are referred to as �infinitesimal rotations� What this really means is that for elements of the group �close� to the identity (specifically their exists a neighborhood in which this is true), there is a map from the Lie algebra to a piece of the Lie group (around 0) that is a homeomorphism If we are dealing with matrix Lie groups such as SO(3), this mapping is the matrix exponential

5. How do Lie Algebras Arise? Lets look closely at the beginning of this series Thus to �first order� elements close to the identity transformation of the Lie group are given by elements of the Lie algebra plus the identity These are referred to as infinitesimal transformations by physicists, with X often being called an �infinitesimal matrix� The reason why no �infinitesimal map� can reverse orientation is only elements that are connected to the identity can be mapped to using the exponential mapping SO(3) on the other hand is entirely connected, no maps reverse orientation

6. Representations of Lie Algebras Most Lie algebras we work with can be defined as matrices with certain constraints Elements of these algebras are therefore maps from vector spaces onto themselves (automorphisms of vector spaces) For example, SO(3) maps the familiar Euclidean space with vector addition onto itself But Lie algebras and Lie groups can be used to describe symmetries For example, the unit sphere in our Euclidean space looks the same even after we rotate it, or in other words act on it with any element of SO(3) What if we want to see if something that doesn�t live in Euclidean space has the same symmetry? We must associate with each element of SO(3) a map which acts over a space the thing we are interested in does live in, and this map is called a representation

7. More on Representations We want our representations to preserve the structure of the group, so we demand that the commutator must be respected by the mapping Thus we define a representation of a Lie algebra g as a map such that

8. Back to Lie Algebras: A Review of su(2) The Lie group SU(2) is the set of 2x2 unitary matrices with determinant 1 su(2), its Lie Algebra, is the set of 2x2 skew Hermitian matrices with zero trace, but we always work with its complexification, , the set of complex matrices with zero trace (given one constraint equation, this should be three dimensional) We choose the familiar basis We can easily find the commutation relations

9. Representations of su(2) Any representation of su(2) must abide by the commutator relations given, so that we have , etc. Now, suppose we have an eigenvector of Let�s see what we can do with So we have another eigenvector of X and Y are the raising and lower operators we use when describing spin in quantum mechanics

10. Representations of su(2) Now, just as in QM, there can�t be infinitely many distinct eigenvalues in a finite dimensional space, so we must have some state that lowers to zero, and some state that is raised to zero The algebra works out so that all the eigenvalues must be integers For every non-negative integer m there is a m+1 dimensional representation of su(2) with eigenvalues m,m-2,m-4,�,2-m,-m, which we get by saying the highest eigenvalue is m and then acting on that eigenvector with the lowering operator Y

11. Representations of larger groups: su(3) The representation theory of su(3) is built on the representation theory of su(2) Using the following basis for su(3)

12. Representations of larger groups: su(3) The important things to notice are that the two boxes contain subgroups isomorphic to su(2) Also and commute so we can diagnolize them both in any representation

13. Representations of su(3) Now, given a simultaneous eigenvalue of H1 and H2 We call the ordered pair (m1,m2) a weight It turns out we can lower or raise these weights like in su(2) to get new eigenvectors The amounts that we can raise and lower by are actually the weights of the adjoint representation, which is a representation over the vector space that the group is defined on For su(3) there are six roots, each corresponding to a one of X�s or Y�s we made the basis out of

14. Representations of su(3) For example, the root corresponding to X1 is (2,-1) so if we have Then we will get Of course sometimes we will get Every (irreducible) representation of su(3) is specified by two numbers, (m1,m2), that correspond to a highest weight. The eigenvector corresponding to this weight will always be raised to zero

15. Simple Lie Algebras Simple Lie algebras (Lie algebras with no non-trivial ideals) can be decomposed as a direct product of three subgroups (h is an ideal in g if ) The first is called the Cartan subalgebra Set of elements that commute with each other (or have zero Lie bracket amongst themselves) This was the span of H1 and H2 in the su(3) case The other two thirds is the sum of the weight spaces (span of the eigenvectors of weights) which can be broken into positive and negative parts by introducing an ordering on the weights

16. What else can we do with representations? We can form more representations! Given two representations one can take their direct sum or tensor product to give representations over larger vector spaces Irreducible representations are those which are not a direct sum of two other representations All the representations we found for su(2) and su(3) were irreducible For most algebras we work with all representations decompose as a direct sum of irreducible representations In particular, we can decompose tensor products of representations as a direct sum of irreducible representations This is what Clebsch-Gordon coefficients are, the coefficients of the decomposition of the tensor product of two representations of su(2)