Lie groups and representation theory
Structure of finite-dimensional semisimple Lie algebras
Let $\mathbb{F}$ be a field. A vector space $\mathfrak{g}$ over $\mathbb{F}$ is called a Lie algebra if there exists an $\mathbb{F}$-bilinear map $\mathfrak{g\times g\to g}$ $(x,y)\mapsto [x,y]$ (called the commutator/Lie bracket) such that
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$[x,x]=0$ (alternating)
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$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ (Jacobi identity)
for all $x,y,z\in\mathfrak{g}.$ Lie algebras are a class of non-associative algebras - the Jacobi identity controls how non-associative the Lie bracket is.
Note: if $\text{char}(\mathbb F)\neq2$ then $[x,x]=0\;\forall x\in\mathfrak g$ is equivalent to $[x,y]=-[y,x]\;\forall x,y\in\mathfrak g.$ This is because
$$[x,y]=[x,y]+0=[x,y]-[y+x,y+x]=[x,y]-[y,x]-[y,y]-[x,x]-[x,y]=-[y,x].$$ On the other hand, $[x,y]=-[y,x]\implies[x,x]=-[x,x]\implies 2[x,x]=0\implies\text{char}(\mathbb F)=2$ or $[x,x]=0.$
Definition. A vector subspace $s$ of $\mathfrak g$ is called a
- subalgebra if $[x,y]\in s$ for all $x,y\in s,$ and
- ideal if $x\in\mathfrak g,\; y\in s\implies [x,y]\in s.$
A Lie algebra $\mathfrak g$ is called
- abelian if $[x,y]=0$ for all $x,y\in s,$
- simple if $\{0\}$ and $\mathfrak g$ are the only ideals of $\mathfrak g,$ and
- semisimple if $\mathfrak g$ is a direct sum of its simple ideals.
Example.
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Any vector space $V$ is an abelian Lie algebra.
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$\mathbb R^3$ is a Lie algebra with $[x,y]=\vec{x}\times\vec{y}.$
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Any associative algebra $A$ can be made into a Lie algebra by defining $[a,b]:=ab-ba.$
In particular, $\text{M}_n(\mathbb C)$ is a Lie algebra, denoted by $\mathfrak{gl}_n(\mathbb C).$ (general linear algebra)
Let $V$ be a finite dimensional vector space over a field $\mathbb F.$ Then the general linear algebra $\mathfrak{gl}(V)$ is $\mathcal{L}(V)$ (space of linear operators on $V$) endowed with the Lie bracket $$[\phi,\psi]=(\phi\circ\psi)-(\psi\circ\phi)$$ for all $\phi,\psi\in\mathcal{L}(V).$
More examples.
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$\mathfrak{sl}_n(\mathbb C)=\{A\in\mathfrak{gl}_n(\mathbb C):\mathrm{tr}(A)=0\}$ is an ideal of $\mathfrak{gl}_n(\mathbb C).$ (special linear algebra)
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The space of all diagonal matrices $\delta_n(\mathbb C),$ upper triangular matrices $t_n(\mathbb C)$ and strictly upper triangular matrices $\Pi_n(\mathbb C)$ are all Lie subalgebras of $\mathfrak{gl}_n(\mathbb C).$
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$\Pi_n(\mathbb C)$ is an ideal in $t_n(\mathbb C).$
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$\mathfrak{sl}_n(\mathbb C)$ is a simple Lie algebra for all $n\in\mathbb N.$
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If $L_1,L_2$ are two ideals of $\mathfrak g$ then $L_1+L_2$ and $[L_1,L_2]$ are both ideals of $\mathfrak g.$
$$L_1+L_2=\{x+y\mid x\in L_1, y\in L_2\},\quad[L_1,L_2]=\{[x,y]\mid x\in L_1, y\in L_2\}.$$
Definition. Given two Lie algebras $\mathfrak g$ and $\mathfrak g',$ a linear map $f:\mathfrak{g}\to\mathfrak{g'}$ is called a Lie algebra homomorphism if $$f\left([x,y]\right)=\left[f(x),f(y)\right].$$
Example.
- Given $x\in\mathfrak g,$ we can define the adjoint Lie algebra homomorphism $\text{ad}_x:\mathfrak{g}\to\mathfrak{g}$ by $$\text{ad}_x(y)=[x,y]$$ for all $y\in\mathfrak g.$
- The Killing form $\kappa$ is defined by $\kappa:\mathfrak{g\times g}\to\mathbb C$ such that $$\kappa(x,y)=\mathrm{tr}\left(\text{ad}_x\circ\text{ad}_y\right).$$
Definition. A representation $\phi$ of a Lie algebra $\mathfrak g$ is a Lie algebra homomorphism $$\phi:\mathfrak{g}\to\mathfrak{gl}(V).$$
The affine Lie algebras
Theorem. Root generated subalgebras are in bijective correspondence with closed subroot systems.
Proof. A proof of the above theorem appears in the paper [1].
References
