Cartan matrix

Intuition [Beginner]

The Cartan matrix is a square matrix that encodes the angles and relative lengths of the simple roots of a Lie algebra. Each entry tells you how two root directions interact.

Think of a crystal lattice: the simple roots are the basic directions that generate all the other directions through addition and subtraction. The Cartan matrix records the inner products between these generators.

For the Lie algebra $\mathfrak{sl}(3)$, there are two simple roots ${\alpha }_1$ and ${\alpha }_2$ at a $120^{\circ }$ angle. The Cartan matrix is $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$. The $2$'s on the diagonal are always there (a root has angle $0$ with itself). The $-1$'s off-diagonal encode the $120^{\circ }$ angle.

The Cartan matrix captures the entire root system. From it, you can reconstruct all roots, the Weyl group, and eventually the complete Lie algebra.

Visual [Beginner]

The root diagram of $\mathfrak{sl}(3)$: six roots arranged in a hexagonal pattern around the origin. The two simple roots ${\alpha }_1$ and ${\alpha }_2$ are highlighted as arrows from the origin, at $120^{\circ }$ to each other. The Cartan matrix entries are shown as labels on the arrows.

The Cartan matrix: the DNA of a semisimple Lie algebra, encoding the geometry of its root system.

Worked example [Beginner]

For $\mathfrak{sl}(4)$ (type $A_3$), there are three simple roots. Each pair of adjacent roots has angle $120^{\circ }$, and non-adjacent roots are perpendicular.

The Cartan matrix is:

$$A=\left(\begin{array}{ccc}2& -1& 0\\ -1& 2& -1\\ 0& -1& 2\end{array}\right)$$

The pattern: $2$ on the diagonal, $-1$ between adjacent roots, $0$ between non-adjacent roots. This tridiagonal pattern is the signature of type $A_n$.

For $\mathfrak{so}(5)$ (type $B_2$), there are two simple roots of different lengths at $135^{\circ }$. The Cartan matrix is $\left(\begin{array}{cc}2& -2\\ -1& 2\end{array}\right)$. The asymmetry ($-2$ vs $-1$) reflects the unequal root lengths.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Cartan matrix). Let $\Phi$ be a root system with base (set of simple roots) $\Delta =\{{\alpha }_1,\dots ,{\alpha }_{\ell }\}$. The Cartan matrix $A=(A_{ij})$ is the $\ell \times \ell$ integer matrix:

$$A_{ij}=\frac{2({\alpha }_i,{\alpha }_j)}{({\alpha }_i,{\alpha }_i)}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$$

where ${\alpha }^{\vee }=2\alpha /(\alpha ,\alpha )$ is the coroot and $(\cdot ,\cdot )$ is the Killing-form-induced inner product.

Properties:

1. $A_{ii}=2$ for all $i$.
2. $A_{ij}\le 0$ for $i\ne j$.
3. $A_{ij}=0$ if and only if $A_{ji}=0$.
4. $A_{ij}{A}_{ji}\in \{0,1,2,3\}$ for $i\ne j$.

The last property restricts the possible angles between simple roots: $90^{\circ }$, $120^{\circ }$, $135^{\circ }$, or $150^{\circ }$.

Key theorem with proof [Intermediate+]

Theorem (Cartan matrix determines the root system). The Cartan matrix determines the root system $\Phi$ up to isomorphism. Moreover, every root system is determined by its Cartan matrix together with the set of simple roots.

Proof. The Cartan matrix encodes the reflections $s_i({\alpha }_j)={\alpha }_j-A_{ij}{\alpha }_i$. Since the Weyl group $W$ is generated by the simple reflections $\{s_1,\dots ,s_{\ell }\}$ and each $s_i$ is determined by its action on the simple roots (given by $A_{ij}$), the Weyl group is determined by $A$.

The full root system is $\Phi =W\cdot \Delta$ (the orbit of the simple roots under the Weyl group). Since both $W$ and $\Delta$ are determined by $A$, so is $\Phi$.

Conversely, the root system determines $A$ by definition. So the correspondence is bijective. $\square$

Bridge. The Cartan matrix is the combinatorial fingerprint of the Lie algebra; the foundational reason classification works is that the severe constraints on the entries (integers, non-positive off-diagonal, product at most $3$) allow only finitely many configurations. This pattern appears again in 07.06.20 where Serre's theorem reconstructs the Lie algebra from the Cartan matrix alone. The bridge is that the Cartan matrix identifies the Lie algebra up to isomorphism — it generalises the Dynkin diagram by encoding the same information in matrix form.

Exercises [Intermediate+]

Advanced results [Master]

The classification theorem. Every finite-dimensional semisimple Lie algebra over $\mathbb{C}$ has a Cartan matrix that decomposes as a direct sum of indecomposable Cartan matrices of the following types:

• Classical: $A_n$ ($n\ge 1$), $B_n$ ($n\ge 2$), $C_n$ ($n\ge 3$), $D_n$ ($n\ge 4$)
• Exceptional: $E_6$, $E_7$, $E_8$, $F_4$, $G_2$

The proof proceeds by showing that the Cartan matrix is a generalised Cartan matrix of finite type, and then classifying all such matrices by their associated Dynkin diagrams.

Serre's theorem. The Cartan matrix determines not just the root system but the Lie algebra itself. Serre's theorem (1966) constructs a Lie algebra from generators $e_i,f_i,h_i$ ($1\le i\le \ell$) subject to:

• $[h_i,h_j]=0$
• $[h_i,e_j]=A_{ij}{e}_j$, $[h_i,f_j]=-A_{ij}{f}_j$
• $[e_i,f_j]={\delta }_{ij}{h}_i$
• $(\text{ad }{e}_i{)}^{1-A_{ij}}{e}_j=0$, $(\text{ad }{f}_i{)}^{1-A_{ij}}{f}_j=0$ for $i\ne j$

The resulting Lie algebra is semisimple with Cartan matrix $A$.

Kac-Moody algebras. Relaxing the condition $\det A>0$ to allow generalised Cartan matrices (still $A_{ii}=2$, $A_{ij}\le 0$ for $i\ne j$, but possibly singular or indefinite) leads to infinite-dimensional Kac-Moody algebras. The affine types (where $\det A=0$) correspond to extended Dynkin diagrams and have deep connections to modular forms and conformal field theory.

Synthesis. The Cartan matrix is the bridge between the continuous world of Lie algebras and the discrete world of combinatorics; the central insight is that the severe restrictions on root angles and lengths (only four possible angles, three possible length ratios) force the classification into finitely many types. This pattern appears again in the Coxeter classification of finite reflection groups and the Tits building associated to the Lie group. The bridge is that the Cartan matrix converts the geometric data of the root system into purely combinatorial data, identifying Lie theory with the combinatorics of Dynkin diagrams.

Full proof set [Master]

Proposition (Off-diagonal product constraint). If $A_{ij}{A}_{ji}=3$, then one of $A_{ij},A_{ji}$ equals $-1$ and the other equals $-3$. The corresponding simple roots have length ratio $\sqrt{3}:1$ and angle $150^{\circ }$.

Proof. Let ${\alpha }_i,{\alpha }_j$ be simple roots with angle $\theta$ between them. By the definition of the Cartan matrix: $A_{ij}=2|{\alpha }_j|\cos \theta /|{\alpha }_i|$ and $A_{ji}=2|{\alpha }_i|\cos \theta /|{\alpha }_j|$. So $A_{ij}{A}_{ji}=4{\cos }^2\theta$. If this equals $3$, then ${\cos }^2\theta =3/4$, giving $\cos \theta =-\sqrt{3}/2$ (the negative root since $A_{ij}<0$), so $\theta =150^{\circ }$. The length ratio is $|{\alpha }_i|/|{\alpha }_j|=\sqrt{A_{ji}/A_{ij}}$. $\square$

Connections [Master]

Root-space decomposition 07.06.18 provides the root system from which the Cartan matrix is extracted; the simple roots are a basis of the root system chosen so that all roots have non-negative coefficients.

Serre relations and Serre's theorem 07.06.20 reconstruct the Lie algebra from the Cartan matrix, proving that the matrix captures the complete algebraic structure.

Dynkin diagrams are the graphical encoding of the Cartan matrix, where each node represents a simple root and edges encode the off-diagonal entries.

The Weyl character formula 07.06.07 computes characters using the root system, which is determined by the Cartan matrix.

Bibliography [Master]

@phdthesis{cartan1894,
  author = {Cartan, {\'E}lie},
  title = {Sur la structure des groupes de transformations finis et continus},
  school = {Paris},
  year = {1894}
}

@article{killing1888,
  author = {Killing, Wilhelm},
  title = {Die Zusammensetzung der stetigen endlichen Transformationsgruppen},
  journal = {Math. Ann.},
  volume = {31, 33, 34, 36},
  years = {1888--1890}
}

@book{humphreys-lie,
  author = {Humphreys, James E.},
  title = {Introduction to Lie Algebras and Representation Theory},
  publisher = {Springer},
  year = {1972}
}

@book{fulton-harris,
  author = {Fulton, William and Harris, Joe},
  title = {Representation Theory: A First Course},
  publisher = {Springer},
  year = {1991}
}

Historical & philosophical context [Master]

Wilhelm Killing classified the simple Lie algebras in his monumental four-paper series (1888-1890) ^{[Killing 1888]}, discovering the exceptional types $E_8$, $F_4$, and what we now call $G_2$. Killing's work was imperfect (he missed the distinction between $B_n$ and $C_n$, and his proofs had gaps) but the classification was essentially correct.

Elie Cartan refined and corrected Killing's work in his 1894 thesis ^{[Cartan 1894]}, giving rigorous proofs and the Cartan matrix as a systematic tool. The Cartan matrix crystallises the classification into a finite list of integer matrices, making the "DNA" of each Lie algebra explicit.

The philosophical point is that the continuous symmetry groups are classified by discrete combinatorial data. The Cartan matrix is where geometry becomes combinatorics, and this bridge between the continuous and the discrete is one of the deepest themes in mathematics.