Lie Algebras and their Representations

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Lie Algebras and their Representations

• Sam Espahbodi
• Advisor James Wells
• Michigan Physics REU

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What are Lie Algebras?

• Lets 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

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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

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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

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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

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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 doesnt
  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

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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

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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

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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
• Lets 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

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Representations of su(2)

• Now, just as in QM, there cant 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 m1
  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

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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)

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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

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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 Xs or Ys we made the basis out of

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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

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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

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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)