Lie Algebras and Finite-Dimensional Representations¶

Job: structure of complex semisimple Lie algebras via roots and the Cartan–Weyl classification of finite-dimensional irreps — with $\mathfrak{sl}_2$ as the universal local engine.

Core¶

Lie algebras¶

A Lie algebra over $k$ (here $k=\mathbb{C}$) is a vector space $\mathfrak{g}$ with bilinear bracket $[-,-]$ skew-symmetric satisfying Jacobi: $$ [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0. $$ Morphism: linear map preserving brackets. Representation: $\rho:\mathfrak{g}\to\mathfrak{gl}(V)=\operatorname{End}(V)$ Lie homomorphism, i.e. $\rho([x,y])=[\rho(x),\rho(y)]$. Equiv. $U(\mathfrak{g})$-module via universal enveloping algebra.

Ideal / type	Definition
Ideal $\mathfrak{i}$	$[\mathfrak{g},\mathfrak{i}]\subset\mathfrak{i}$
Solvable	derived series $\mathfrak{g}^{(k)}=0$
Nilpotent	lower central series $=0$
Simple	$[\mathfrak{g},\mathfrak{g}]\neq 0$, no nontrivial ideals
Semisimple	$0$ radical (max solvable ideal); equiv. $\bigoplus$ simple

Killing form $B(x,y)=\operatorname{tr}(\operatorname{ad}x\circ\operatorname{ad}y)$. Cartan’s criterion: $\mathfrak{g}$ semisimple $\Leftrightarrow B$ nondegenerate (char $0$).

$\mathfrak{sl}_2$ — the atomic theory¶

$\mathfrak{sl}_2=\mathfrak{sl}_2(\mathbb{C})=\{A\in M_2: \operatorname{tr} A=0\}$ with basis $$ h=\begin{pmatrix}1&0\0&-1\end{pmatrix},\quad e=\begin{pmatrix}0&1\0&0\end{pmatrix},\quad f=\begin{pmatrix}0&0\1&0\end{pmatrix}, $$ relations $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$.

Theorem (finite-dim irreps of $\mathfrak{sl}_2$). For each integer $n\ge 0$ there is a unique irrep $V(n)$ of dim $n+1$ with weights $\{-n,-n+2,\ldots,n\}$ for $h$ (eigenvalues), highest weight vector $v$ with $h\cdot v=nv$, $e\cdot v=0$, and $f^k v$ spanning. Every f.d. rep is completely reducible (semisimplicity of $\mathfrak{sl}_2$ / Weyl).

Object	Formula
Casimir $C$	$\frac{1}{2}h^2+ef+fe$; acts by $\frac{n(n+2)}{2}$ on $V(n)$
Tensor	$V(m)\otimes V(n)\cong\bigoplus_{k} V(

Every root $\mathfrak{sl}_2$-triple in a semisimple $\mathfrak{g}$ reproduces this locally on weight spaces.

Cartan subalgebras and roots¶

$\mathfrak{h}\subset\mathfrak{g}$ Cartan: nilpotent, self-normalizing (equiv. maximal toral in semisimple char $0$). For $\mathfrak{g}$ semisimple f.d.: $$ \mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}\alpha,\qquad \mathfrak{g}\alpha={x:[h,x]=\alpha(h)x\ \forall h\in\mathfrak{h}}, $$ $\Phi\subset\mathfrak{h}^*\setminus\{0\}$ finite root system, each $\dim\mathfrak{g}_\alpha=1$, and for $\alpha\in\Phi$ there is $h_\alpha\in\mathfrak{h}$ and $\mathfrak{sl}_2$-triple $(e_\alpha,h_\alpha,f_\alpha)$.

Root system axioms (abstract): $\Phi$ finite spanning $E=\mathbb{R}\Phi$, $\alpha\in\Phi\Rightarrow -\alpha\in\Phi$, reflections $s_\alpha(\beta)=\beta-\langle\beta,\alpha^\vee\rangle\alpha$ preserve $\Phi$, $\langle\beta,\alpha^\vee\rangle\in\mathbb{Z}$.

Data	Role
Simple roots $\Delta$	basis; positive roots $\Phi^+$
Weyl group $W$	$\langle s_\alpha\rangle$; finite reflection group
Cartan matrix	$A_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle$
Dynkin diagram	encodes $A$ up to iso

Classification (Cartan–Killing). Connected Dynkin diagrams: $A_n,B_n,C_n,D_n,E_6,E_7,E_8,F_4,G_2$ — biject with simple complex Lie algebras.

Serre relations and Dynkin data¶

Given a Cartan matrix $A=(A_{ij})_{1\le i,j\le\ell}$ of finite type, the corresponding semisimple Lie algebra is presented by generators $e_i,f_i,h_i$ ($i=1,\ldots,\ell$) and the Serre relations:

Relation	Formula
Cartan	$[h_i,h_j]=0$, $[h_i,e_j]=A_{ji}e_j$, $[h_i,f_j]=-A_{ji}f_j$
$\mathfrak{sl}_2$	$[e_i,f_j]=\delta_{ij}h_i$
Serre (nilpotency)	$(\operatorname{ad} e_i)^{1-A_{ji}}e_j=0$, $(\operatorname{ad} f_i)^{1-A_{ji}}f_j=0$ for $i\neq j$

(Chevalley–Serre theorem: this presents the split semisimple Lie algebra with Cartan matrix $A$.) The integers $1-A_{ji}\in\{1,2,3,4\}$ match bond multiplicity in the Dynkin diagram.

Dynkin diagrams (connected, finite type) — one table.

Type	Diagram (bonds)	$\dim\mathfrak{g}$	Rank $\ell$	Classical realization
$A_n$ ($n\ge 1$)	$\circ—\circ—\cdots—\circ$ ($n$ nodes)	$n(n+2)$	$n$	$\mathfrak{sl}_{n+1}$
$B_n$ ($n\ge 2$)	$\circ—\cdots—\circ\Rightarrow\circ$ (double to short)	$n(2n+1)$	$n$	$\mathfrak{so}_{2n+1}$
$C_n$ ($n\ge 3$)	$\circ—\cdots—\circ\Leftarrow\circ$ (double to long)	$n(2n+1)$	$n$	$\mathfrak{sp}_{2n}$
$D_n$ ($n\ge 4$)	$\circ—\cdots—\circ$ with fork at end	$n(2n-1)$	$n$	$\mathfrak{so}_{2n}$
$G_2$	$\circ\equiv\hspace{-0.4em}\equiv\hspace{-0.4em}\equiv\circ$ (triple)	$14$	$2$	exceptional
$F_4$	$\circ—\circ\Rightarrow\circ—\circ$	$52$	$4$	exceptional
$E_6$	branched $E$ chain, arm length $2$	$78$	$6$	exceptional
$E_7$	branched $E$ chain, arm length $3$	$133$	$7$	exceptional
$E_8$	branched $E$ chain, arm length $4$	$248$	$8$	exceptional

Single bonds: $A_{ij}A_{ji}=1$; double: $2$; triple: $3$. Arrow (when drawn) points toward the shorter root. Cartan matrices are symmetrizable; $B_n$ vs $C_n$ swap long/short.

Weights and finite-dimensional modules¶

For $\mathfrak{h}$-diagonalizable action on $V$: weight spaces $V_\lambda=\{v:h\cdot v=\lambda(h)v\}$, $\lambda\in\mathfrak{h}^*$. Highest weight $\lambda$: $V_\lambda\neq 0$ and $\mathfrak{g}_\alpha\cdot V_\lambda=0$ all $\alpha\in\Phi^+$.

Theorem (Cartan–Weyl). Finite-dimensional irreducible $\mathfrak{g}$-modules are classified by dominant integral weights $\lambda\in\Lambda^+$: $$ \Lambda={\lambda\in\mathfrak{h}^*:\langle\lambda,\alpha^\vee\rangle\in\mathbb{Z}\ \forall\alpha\in\Phi},\qquad \Lambda^+={\lambda:\langle\lambda,\alpha^\vee\rangle\ge 0\ \forall\alpha\in\Delta}. $$ Unique irrep $L(\lambda)$ with highest weight $\lambda$; $\dim L(\lambda)$ by Weyl dimension formula; character by Weyl character formula.

Complete reducibility (Weyl). Every f.d. representation of a semisimple $\mathfrak{g}$ over char $0$ is completely reducible (Casimir / unitary trick via compact form).

PBW. $U(\mathfrak{g})$ has associated graded $\operatorname{Sym}(\mathfrak{g})$; provides Verma modules $M(\lambda)=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}k_\lambda$ with simple heads $L(\lambda)$.

See it¶

$\mathfrak{sl}_n$ fundamentals. $\mathfrak{h}=$ traceless diagonals; roots $\varepsilon_i-\varepsilon_j$; fundamental weights $\varpi_k$ correspond to $\bigwedge^k\mathbb{C}^n$. Irrep of highest weight $\varpi_1$ is the standard $n$-dimensional representation. Young diagrams label dominant weights for $\mathfrak{sl}_n$ and give explicit constructions via Schur functors — dictionary between symmetric group reps (via Schur–Weyl) and $\operatorname{GL}_n$ / $\mathfrak{sl}_n$ reps.

Minuscule weights and exceptional groups. For $E_6,E_7$, minuscule representations appear in algebraic geometry (Severi varieties) and Freudenthal’s magic square — root data control projective geometry of homogeneous spaces $G/P$.

Check¶

Why are the only f.d. irreps of the Heisenberg algebra (or nontrivial nilpotent) non-faithful / why does complete reducibility fail for solvable $\mathfrak{g}$?

Lie’s theorem: over alg. closed char $0$, f.d. irreps of solvable $\mathfrak{g}$ are $1$-dimensional. Nonabelian nilpotent algebras cannot act faithfully irreducibly in finite dimension. Unipotent Jordan-type actions give nonsplit extensions (cf. $\mathfrak{b}\subset\mathfrak{sl}_2$ Borel acting on $V(n)$ with flags).

Sketch why $\langle\lambda,\alpha^\vee\rangle\in\mathbb{Z}_{\ge 0}$ is necessary for $L(\lambda)$ to be finite-dimensional.

Restrict to $\mathfrak{sl}_2^{(\alpha)}$: the $h_\alpha$-string through highest weight must match $V(n)$ with $n=\langle\lambda,\alpha^\vee\rangle\ge 0$ integer; otherwise the $\mathfrak{sl}_2$-module generated is infinite-dimensional (Verma for $\mathfrak{sl}_2$).

Pitfalls¶

Real forms: classification above is over $\mathbb{C}$; $\mathfrak{su}(n)$ vs $\mathfrak{sl}_n(\mathbb{R})$ have different unitary/representation theories.
Kac–Moody / infinite-dim: same root language, but f.d. reps rare; integrable highest weight modules replace $L(\lambda)$.
Weights vs roots: roots are weights of the adjoint rep; not every weight is a root.
Cartan subalgebras are conjugate (char $0$ f.d. semisimple) — root system independent of choice up to $W$.

Refs¶

Humphreys, Introduction to Lie Algebras and Representation Theory
Fulton–Harris, Representation Theory
Hall, Lie Groups, Lie Algebras, and Representations
Bourbaki, Lie Groups and Lie Algebras (Ch. 4–6, root systems)

Next¶

Foundations · Lie groups · RT index · Group theory · Homological algebra · Algebraic geometry

Data	Role
Simple roots \(\Delta\)	basis; positive roots \(\Phi^+\)
Weyl group \(W\)	\(\langle s_\alpha\rangle\); finite reflection group
Cartan matrix	\(A_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle\)
Dynkin diagram	encodes \(A\) up to iso

Type	Diagram (bonds)	\(\dim\mathfrak{g}\)	Rank \(\ell\)	Classical realization
\(A_n\) (\(n\ge 1\))	\(\circ—\circ—\cdots—\circ\) (\(n\) nodes)	\(n(n+2)\)	\(n\)	\(\mathfrak{sl}_{n+1}\)
\(B_n\) (\(n\ge 2\))	\(\circ—\cdots—\circ\Rightarrow\circ\) (double to short)	\(n(2n+1)\)	\(n\)	\(\mathfrak{so}_{2n+1}\)
\(C_n\) (\(n\ge 3\))	\(\circ—\cdots—\circ\Leftarrow\circ\) (double to long)	\(n(2n+1)\)	\(n\)	\(\mathfrak{sp}_{2n}\)
\(D_n\) (\(n\ge 4\))	\(\circ—\cdots—\circ\) with fork at end	\(n(2n-1)\)	\(n\)	\(\mathfrak{so}_{2n}\)
\(G_2\)	\(\circ\equiv\hspace{-0.4em}\equiv\hspace{-0.4em}\equiv\circ\) (triple)	\(14\)	\(2\)	exceptional
\(F_4\)	\(\circ—\circ\Rightarrow\circ—\circ\)	\(52\)	\(4\)	exceptional
\(E_6\)	branched \(E\) chain, arm length \(2\)	\(78\)	\(6\)	exceptional
\(E_7\)	branched \(E\) chain, arm length \(3\)	\(133\)	\(7\)	exceptional
\(E_8\)	branched \(E\) chain, arm length \(4\)	\(248\)	\(8\)	exceptional

Relation	Formula
Cartan	\([h_i,h_j]=0\), \([h_i,e_j]=A_{ji}e_j\), \([h_i,f_j]=-A_{ji}f_j\)
\(\mathfrak{sl}_2\)	\([e_i,f_j]=\delta_{ij}h_i\)
Serre (nilpotency)	\((\operatorname{ad} e_i)^{1-A_{ji}}e_j=0\), \((\operatorname{ad} f_i)^{1-A_{ji}}f_j=0\) for \(i\neq j\)