Matrix groups and Lie groups

## Matrices

with product matrix multiplication
is not a group
because, if $g=\left(\begin{array}{ccc}1& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$ then there does not exist ${g}^{-1}$ with $g{g}^{-1}={g}^{-1}g=1$.

The general linear group is The special linear group is is $SLnℂ =ker ϕ$ where $\varphi :{GL}_{n}\left(ℂ\right)\to {GL}_{1}\left(ℂ\right)$ is the homomorphism given by $ϕg=detg.$

## Orthogonal, Symplectic and Unitary groups

The orthogonal group is $On= g∈GLnℂ∣ g⋅1⋅gt=1 .$ The symplectic group is $Spn= g∈GLnℂ∣ g⋅J⋅gt=1 where J= 0 1 0 ⋱ 0 1 -1 0 ⋱ 0 -1 0$ The conjugation involution on $ℂ$ is $-:ℂ\to ℂ$ given by $\stackrel{‾}{x+yi}=x-yi.$ The unitary group is $Unℂ = g∈GLnℂ∣ g⋅1⋅ g ‾ t ,$ where $g ‾ t = g11 … g1n ⋮ ⋮ gn1 … gnn t ‾ = g11 … gn1 ⋮ ⋮ g1n … gnn ‾ = g ‾ 11 … g ‾ n1 ⋮ ⋮ g ‾ 1n … g ‾ nn .$

## Linear transformations

Let $V$ be a vector space. If ${b}_{1},\dots ,{b}_{n}$ is a basis of $V$ then $EndV → ∼ Mnℂ g ↦ g11 … g1n ⋮ ⋮ gn1 … gnn$ where $g{b}_{i}={g}_{1i} {b}_{1}+\dots +{g}_{ni} {b}_{n}.$ such that
1. If ${a}_{1},{a}_{2}\in ℂ$ and ${v}_{1},{v}_{2},{v}_{3}\in V$ then $⟨{a}_{1}{v}_{1}+{a}_{2}{v}_{2},{v}_{3}⟩={a}_{1}⟨{v}_{1},{v}_{3}⟩+{a}_{2}⟨{v}_{2},{v}_{3}⟩,$
2. If ${v}_{1},{v}_{2}\in V$ then $\left\{\begin{array}{l}⟨{v}_{2},{v}_{1}⟩=⟨{v}_{1},{v}_{2}⟩.\\ ⟨{v}_{2},{v}_{1}⟩=-⟨{v}_{1},{v}_{2}⟩.\\ ⟨{v}_{2},{v}_{1}⟩=\stackrel{‾}{⟨{v}_{1},{v}_{2}⟩}.& & \end{array}\right\$

## Linear groups

The general linear group is Let $⟨,⟩$ be a symmetric form on $V$. The orthogonal group is Let $⟨,⟩$ be a skew-symmetric form on $V$. The symplectic group is Let $⟨,⟩$ be a Hermitian form on $V$. The unitary group is

## Exponential maps

satisfies $ex ey= ex+y and dex dx x=0 =1.$ If ${e}^{x}$ is a polynomial then $ex= 1+x+ 12 x2+ 13! x3+⋯.$ So we have an exponential map $e: ℂ → ℂ∖ 0 r+iθ ↦ er eθ$ which is really a map $e: M1ℂ → GL1ℂ x ↦ ex$ which is a special case of $e: Mnℂ → GLnℂ x ↦ ex.$

Example: $e 1 0 0 2 = e 0 0 e2 and e 0 1 0 0 = 1 1 0 1 .$

## Lie groups

Conceptually, a Lie group is a group $G$ with an exponential map.

${M}_{n}\left(ℂ\right)$ is a vector space with basis $Eij∣ 1≤i≤j≤n where Eij= jth ith 0 ⋱ 1 0 0$

Example: $\left(\begin{array}{cc}0& 5\\ 2& 1\end{array}\right)=5{E}_{12}+2{E}_{21}+{E}_{22}.$

Let $G$ be a group of matrices (so $G\subseteq {GL}_{n}\left(ℂ\right)$). If it turns out that $𝔤=logG= x∈Mnℂ∣ ex∈G$ is a subspace of ${M}_{n}\left(ℂ\right)$, then $e:𝔤→G$ is an exponential map for $G$.

$𝔤$ is the Lie algebra of $G$.

## One parameter subgroups

Let $G$ be a Lie group. A one parameter subgroup of $G$ is a map $γ:ℂ→G such that γt1+t2 =γt1 γt2.$

Example: Let $x\in {M}_{n}\left(ℂ\right)$. Then $γx: ℂ → GLnℂ t ↦ etx$ is a one-parameter subgroup of ${GL}_{n}\left(ℂ\right)$.

Example: Let $x\in 𝔤$. Then $γx: ℂ → G t ↦ etx$ is a one-parameter subgroup of $G$.

A one parameter subgroup of ${GL}_{n}\left(ℂ\right)$ is a homomorphism $ℂ\to {GL}_{n}\left(ℂ\right).$

## ${SL}_{2}$ embeddings

$SL2ℂ = g∈GL2∣ detg=1 .$ The group ${SL}_{2}\left(ℂ\right)$ is generated by For each $1\le i, $ϕij: SL2ℂ → GL2ℂ 1 t 0 1 ↦ 1 0 ⋱ t 0 1 1 0 s 1 ↦ 1 0 s ⋱ 0 1$ is an embedding of $S{L}_{2}\left(ℂ\right)$ in $G{L}_{n}\left(ℂ\right)$. A maximal torus in $G{L}_{n}\left(ℂ\right)$ is an embedding of the group A Borel subgroup in $G{L}_{n}\left(ℂ\right)$ is an embedding of the group

## Weyl groups

Let $G$ be a Lie group. Let $T$ be a maximul torus in $G$.

The normalizer of $T$ in $G$ is the

largest subgroup $N$ of $G$ such that $T$ is normal in $N$.
The Weyl group of $G$ is $W=N/T.$

Example: The normalizer of $T$ in ${GL}_{n}\left(ℂ\right)$ is The map that changes nonzero entries to $1$ is a homomorphism $ϕ:N→Sn with ker ϕ=T.$

Example:

So the symmetric group ${S}_{n}$ is the Weyl group of ${GL}_{n}\left(ℂ\right)$.

## Generators and relations for Weyl groups

Let

${S}_{n}$ is generated by ${s}_{1},{s}_{2},\dots ,{s}_{n-1}.$

 Proof. Stretch $w$. $=$ $=$ $s2s3s2s1s2.$ $\square$

The symmetric group is given by generators ${s}_{1},\dots ,{s}_{n-1}$ and relations

## Elementary matrices

$Eij∣ 1≤i,j≤n$ is a basis of ${M}_{n}\left(ℂ\right)$ and $xij: ℂ → GLnℂ t ↦ etEij$ is a one parameter subgroup. $xijt = 1+tEij + 12 t2 Eij2 + 13! t3 Eij3 +⋯ =1+tEij+0+0+⋯ = jth ith 1 ⋱ t 0 1 .$ The elementary matrices in ${GL}_{n}\left(ℂ\right)$ are If $g\in {GL}_{n}\left(ℂ\right)$ then So elementary matrices in ${GL}_{n}\left(ℂ\right)$ are row and column operations.

## Generators and relations for Lie groups

${GL}_{n}\left(ℂ\right)$ is generated by

 Proof. Let $g\in {GL}_{n}\left(ℂ\right)$. Finding ${g}^{-1}$ by row reduction is equivalent to finding $xiljl tl ⋯ xi1j1 t1 g=1.$ Since ${x}_{ij}{\left(t\right)}^{-1}={x}_{ij}\left(-t\right),$ $g= xi1j1 -t1 ⋯ xiljl -tl is a product of elementary matrices.$ $\square$

The relations are $xijt1 xijt2 = xij t1+t2 , hit1 hit2 = hit1t2,$ $xijt xkls = xkls xijt, xijt xjls = xjls xijt xilst,$ $xijt xji -t-1 xijt = sij hit hj-t-1,$ $hxijt = xij hithj-1, for h= h1 0 ⋱ 0 hn ∈T,$ where ${s}_{ij}=\begin{array}{c}\begin{array}{ccccccccccc}& & & {i}^{th}& & & & & {j}^{th}& & \end{array}\\ \left(\begin{array}{ccc}1& & \\ & \ddots & \\ & & 1\\ & & & 0& & & & 1\\ & & & & 1\\ & & & & & \ddots \\ & & & & & & 1\\ & & & 1& & & & 0\\ & & & & & & & & 1\\ & & & & & & & & & \ddots \\ & & & & & & & & & & 1\end{array}\right).\end{array}$

## References

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)