Physics Notes by Jakob Schwichtenberg

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group_theory:groups:u1

U(1)

Lie Algebra, Covering Group, Fundamental group

For U(1) [the Lie algebra] is usually identified with the set of pure imaginary numbers $Im \mathbb{C} = \{ i \theta : \theta \in \mathbb{R} \}$ with the vector space structure it inherits from $\mathbb{C}$ (the tangent space to the circle at 1 is, of course, just a copy $\mathbb{R}$ but the isomorphic space $Im \mathbb{C}$ is more convenient because its elements can be “exponentiated” to give the elements $e^{i \theta}$ of $U(1)$.

Topology, Geometry and Gauge fields by G. Naber

You should be able now to think of other example of $\mathfrak{G}=U(1)$ and $\tilde{\mathfrak{G}}=(\mathbb{R},\,+)$ in these terms: they both have the same Lie algebra $\mathfrak{g}=(\mathbb{R},\,+)$ and $\tilde{\mathfrak{G}}=(\mathbb{R},\,+)$ is the universal covering group of $\mathfrak{G}=U(1)$. The fundamental group of course is $\mathbb{Z} = \cdots ,\,-2,\,-1,\,0,\,1,\,2,\,\cdots$, where the integer $n$ corresponds to a loop around the circle $U(1) = \{e^{i\,\theta}:\;\theta\in\mathbb{R}\}$ comprising $n$ bights in the anticlockwise direction.

The group G =U(1) of complex numbers of modulus 1, seen as the unit circle S 1 , is non simply connected: a path from the identity 1 to 1 may wind an arbitrary number of times around the circle and this (positive or negative) winding number characterizes the different homotopy classes: the homotopy group is π1(U(1)) = Z . The group Ge is nothing else than the additive group R and may be visualised as a helix above U(1). The quotient is R/Z ' U(1), which must be interpreted as the fact that a point of U(1), i.e. an angle, is a real number modulo an integer multiple of 2π. One may also say that π1(S 1 ) = Z. More generally one may convince oneself that for spheres, π1(S n ) is trivial (all loops are contractible) as soon as n > 1.