Classical Lie groups

 Lecture topic

A Lie group $G$ is both a group and a manifold, with a smooth map $G\times G\to G$, given by $(g,h)\mapsto gh^{-1}$. The Lie algebra $\mathfrak g$ of $G$ is the tangent space $T_eG$ of $G$ at the identity.

We distinguish between real and complex Lie groups by saying that the base manifold is either real or complex analytic, respectively.

Example: Here are some examples of classical Lie groups and their dimension:
\[
\begin{array}{r c r c l}
\text{general linear group} & n^2 & GL(n) & = & \{n\times n\ \text{matrices with non-zero determinant}\} \\
\text{special linear group} & n^2-1 & SL(n) & = & \{M\in GL(n)\ :\ \det(M)=1\} \\
\text{orthogonal group} & n(n-1)/2 & O(n) & = & \{M\in GL(n)\ :\ MM^t = I\} \\
\text{special orthogonal group} & n(n-1)/2 & SO(n) & = & \{M\in O(n)\ :\ \det(M)=1\} \\
\text{unitary group} & n^2 & U(n) & = & \{M\in GL(n,\textbf{C})\ :\ MM^* = I\} \\
\text{special unitary group} & n^2-1 & SU(n) & = & \{M\in U(n)\ :\ \det(M)=1\} \\
\text{symplectic group} & n(2n+1) & Sp(n) & = & \{n\times n\ \text{matrices}:\ \omega(Mx,My)=\omega(x,y)\}
\end{array}
\]
For the symplectic group, the skew-symmetric bilinear form $\omega$ is defined as
\[
\omega(x,y) = \sum_{i=1}^n x_iy_{i+n} - y_ix_{i+n} = \begin{pmatrix} 0 & -I \\ I & 0 \end{pmatrix} x\cdot y,
\]
where $\cdot$ is the regular dot product (a symmetric bilinear form). Also note that the unitary group is a real Lie group - real because there is no holomorphic map $G\times G\to G$ as would be necessary, so we view the entries of a matrix in $U(n)$ in terms of its real and imaginary parts. Hence the dimension indicated above is real dimension.

References: Kirillov Jr (An introduction to Lie groups and Lie algebras, Chapter 2)

Higgs fields of principal bundles

The goal here is to understand the setting of Higgs fields on Riemannian manifolds, in the manner of Hitchin. First we consider general topological spaces $X$ and groups $G$.

Definition: Let $X$ be a topological space and $G$ a group. A principal bundle (or principal $G$-bundle) $P$ over $X$ is a fiber bundle $\pi:P\to X$ together with a continuous, free, and transitive right action $P\times G\to P$ that preserves the fibers. That is, if $p\in \pi^{-1}(x)$, then $pg\in \pi^{-1}(x)$ for all $g\in G$ and $x\in X$.

Now suppose we have a principal bundle $\pi:P\to X$, a representation $\rho$ of $G$, and another space $Y$ on which $G$ acts on the left. Define an equivalence relation $(p,y)\sim (p',y')$ on $P\times Y$ iff there is some $g\in G$ for which $p'=pg$ and $y'=\rho(g^{-1})y$. This is an equivalence relation. We will be interested in the adjoint representation (induced by conjugation).

Proposition: The projection map $\pi':P\times_\rho Y := (P\times Y)/\sim\ \to X$, where $\pi'([p,y]) = \pi(p)$, defines a vector bundle over $X$, called the associated bundle of $P$.

Recall a Lie group $G$ is a group that is also a topological space, in the sense that there is a continuous map $G\times G\to G$, given by $(g,h)\mapsto gh^{-1}$. The Lie algebra $\mathfrak g$ of the Lie group $G$ is the tangent space $T_eG$ of $G$ at the identity $e$. We will be interested in principal $G$-bundles $P\to \R^2$ and associated bundles $P\times_\ad \mathfrak g\to \R^2$, where $\ad$ is the adjoint representation of $G$.

Next, recall we had the space $\mathcal A^k_M$ of $k$-differential forms on $M$ (see post "Smooth projective varieties as Kähler manifiolds," 2016-06-16), defined in terms of wedge products of elements in the cotangent bundle $(TM)^* = T^*M$ of $M$. Now we generalize this to get differential forms over arbitrary vector bundles.

Definition: Let $E\to M$ be a vector bundle. Let
\begin{align*}
\mathcal A^k_M(E) & := \Gamma(E\otimes \textstyle\bigwedge^k T^*M) = \Gamma(E)\otimes_{\mathcal A^0_M}\mathcal A^k_M, \\
\mathcal A^{p,q}_M(E) & := \Gamma(E\otimes \textstyle\bigwedge^p (T^{1,0}M)^*\otimes \bigwedge^q (T^{0,1}M)^*) = \Gamma(E)\otimes_{\mathcal A^0_M}\mathcal A^{p,q}_M
\end{align*}
be the spaces of $k$- and $(p,q)$-differential forms, respectively, over $M$ with values in $E$.

Equality above follows by functoriality. Now we are close to understanding where exactly the Higgs field lives, in Hitchin's context.

Definition: Given a function $f:\C\to \C$, the conjugate of $f$ is $\bar f$, defined by $\bar f(z) = \overline{f(\bar z)}$.

Hitchin denotes this as $f^*$, but we will stick to $\bar f$. Finally, let $P$ be a $G$-principal bundle over $\R^2$ and $P\times_\ad \mathfrak g$ the associated bundle of $P$. Given $f\in \mathcal A^0_{\R^2}( (P\times_\ad \mathfrak g)\otimes \C)$, set
\begin{align*}
\theta & = \textstyle \frac12 f(dx+i\ dy) \in \mathcal A^{1,0}_{\R^2}((P\times_\ad\mathfrak g)\otimes \C) ,\\
\theta^* & = \textstyle \frac12 \bar f(dx-i\ dy) \in \mathcal A^{0,1}_{\R^2}((P\times_\ad\mathfrak g)\otimes \C),
\end{align*}
called a Higgs field over $\R^2$ and (presumably) a dual (or conjugate) Higgs field over $\R^2$. Note this agrees with the definition in a previous post ("Connections, curvature, and Higgs bundles," 2016-07-25).

References: Hitchin (Self-duality equations on a Riemann surface), Wikipedia (article on associated bundles, article on vector-valued differential forms)