General remarks

Section 4.5 General remarks

The presentation of Lie groups and Lie algebras in this course is necessarily unsystematic, and does not do justice to this beautiful area of mathematics. But let us conclude this section with a few general remarks about Lie groups and Lie algebras.

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Remark 4.5.1.

To every Lie group is associated a unique Lie algebra. The Lie algebra is uniquely determined as being the tangent space of the Lie group at the origin, or, in the language of matrix Lie groups, as being the algebra of the infinitesimal generators of the group.

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Remark 4.5.2.

Two Lie groups that have isomorphic Lie algebras are not necessarily isomorphic. Different Lie groups may share the same Lie algebras, such as $SO(3)$ and $SU(2)\text{.}$ However, Lie groups that have the same Lie algebras are locally isomorphic near the identity. Globally, they are generally related by surjective group homomorphisms, i.e. covering maps. For instance, $SU(2)$ covers $SO(3)$ twice, as we have seen. The “largest” of the groups associated to a Lie algebra is called the universal covering group. It is simply connected (i.e. it consists of one piece and has no “holes” in it, that is, its fundamental group is trivial). In the case of $SO(3)\text{,}$ its universal covering is $SU(2)\text{,}$ which is simply connected (since it is diffeomorphic to $S^3\text{,}$ see Example 4.1.10).

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Remark 4.5.3.

While different Lie groups may share the same Lie algebra, there is nonetheless a uniqueness result that is crucial. Lie's third theorem states that every finite-dimensional real Lie algebra is the Lie algebra of some simply connected Lie group. Moreover, if two Lie groups are simply connected and have isomorphic Lie algebras, then the groups themselves are isomorphic. What this means is that to every finite-dimensional real Lie algebra one can associate a unique Lie group that is simply connected. $SU(2)$ plays this role for the Lie algebras $\mathfrak{so}(3) \cong \mathfrak{su}(2)\text{.}$

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Our goal for the remaining of this class is to study representation theory of Lie groups such as

$SO(3)$ and

$SU(2)\text{.}$ More precisely, from physics we are particularly interested in representation theory of

$SO(3)\text{,}$ which corresponds to non-relativistic particles.

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Representation theory of Lie algebras is generally easier than representation theory of Lie groups, because Lie algebras are linear. We have already seen that given a representation of a Lie algebra, one can reconstruct a representation of a Lie group using exponentiation. So the goal is to study representation theory of Lie groups by studying representation theory of Lie algebras. However, one has to be careful, since different Lie groups may share the same Lie algebra. If we start with a representation of a Lie algebra, and define a representation via exponentiation, how do we know what Lie group we are talking about?

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The main result here is that every representation of a simply connected Lie group comes from a representation of its corresponding Lie algebra. So by starting with representations of the Lie algebra, what we are constructing is all representations of the unique simply connected Lie group associated to the algebra (the universal covering). The simply connected property is crucial.

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For instance, representation theory of the Lie algebra

$\mathfrak{su}(2) \cong \mathfrak{so}(3)$ constructs all representations of the universal covering

$SU(2)\text{.}$ However, some of those will not be honest representations of the non-simply connected

$SO(3) \cong SU(2)/\mathbb{Z}_2\text{.}$ As we will see, they are so-called “projective”, or “spin”, representations. Those are mappings that preserve group multiplication but only up to a constant. So from the point of view of

$SO(3)\text{,}$ representation theory of the Lie algebra

$\mathfrak{so}(3)$ constructs not only the honest representations of

$SO(3)\text{,}$ but also some spin representations.

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Fortunately, in quantum mechanics it is sufficient to preserve only group operation up to a constant (since wave-functions are only defined up to phase), so we actually do care about spin representations, and should include them. Indeed, as we will see, the odd-dimensional representations (corresponding in physics to particles with integer spin) coming from

$\mathfrak{so}(3)$ are honest representations of

$SO(3)\text{,}$ while the even-dimensional ones (corresponding in physics to particles with half-integer spin) are spin representations of

$SO(3)$ (they are however true representations of the simply connected

$SU(2)$ ).

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Let us now move on to representation theory!