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Definition & Example
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- An **irreducible finite Cartan type** is one of the following:
- $A_n$ for $n \geq 1$;
- $B_n$ for $n \geq 2$;
- $C_n$ for $n \geq 3$;
- $D_n$ for $n \geq 4$;
- $E_n$ for $n = 6,7,8$;
- $F_4$;
- $G_2$;
- The **rank** of an irreducible finite Cartan type is its index. It denotes the dimension of the corresponding irreducible representation of its Weyl group.
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Additional information
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Finite root systems
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Let $V$ be an Euclidean vector space of finite dimension, endowed with an inner product $(\cdot, \cdot)$, and let $\alpha\in V$ with $\alpha \neq 0$. The reflection $s_\alpha:V\rightarrow V$ orthogonal to $\alpha$ is defined as
$$s_\alpha(\lambda) = \lambda - 2\frac{(\lambda,\alpha)}{(\alpha,\alpha)}\alpha \text{ for }\lambda \in V.$$
A **(crystallographic) root system** in $V$ is a finite set $\Phi\subset V\setminus \{0\}$ that spans $V$ and satisfies
1. $\Phi \cap \mathbb{R}\alpha = \{\alpha, -\alpha\}$,
1. $s_\alpha \Phi = \Phi$ for all $\alpha \in \Phi$ and
1. $\langle\beta,\alpha\rangle := 2\frac{(\beta,\alpha)}{(\alpha,\alpha)} \in \mathbb{Z}$ for all $\alpha,\beta\in \Phi$.
The elements of $\Phi$ are called **roots**. The root system $\Phi$ is called **reducible**, if there is a disjoint union $\Phi = \Phi_1 \sqcup \Phi_2$ in nonempty sets such that $(\alpha,\beta) = 0$ for all $\alpha \in \Phi_1$ and $\beta \in \Phi_2$. Otherwise it is called **irreducible**.
Associated to a root system is its **Weyl group**
$$W_\Phi = \langle s_\alpha \mid \alpha \in \Phi \rangle \subset \operatorname{GL}(V)$$
that is generated by the reflections perpendicular to the roots of $\Phi$. The Weyl group is completely determined by the root system.
Let $\Phi$ be a root system. A subset $\Delta \subset \Phi$ is called a **simple system** if it is a basis of $V$ and every root in $\Phi$ is a linear combination of elements of $\Delta$ with integral coefficients all of the same sign (or zero).
Associated to a simple system $\Delta$ is the **positive system** $\Phi^+$ containing all roots that are linear combinations of $\Delta$ with only nonnegative coefficients.
Elements of $\Delta$ are called **simple roots** and elements of $\Phi^+$ are called **positive roots**. For every root system, simple systems exist, see <>.
Let $\Delta = \{\alpha_1,\dots,\alpha_n\}$ be a simple system of $\Phi$. The **Dynkin diagram** of $\Phi$ is defined as a graph on $n$ vertices where each pair $(i,j)$ is connected by $\langle\alpha,\beta\rangle\langle\beta,\alpha\rangle$ edges. Whenever $i$ and $j$ are connected by more than one edge, add an arrow indicating which root is of greater length. The Dynkin diagram only depends on $\Phi$ and not on the choice of $\Delta$, see <>.
Irreducible crystallographic root systems are classified by Dynkin diagrams and named by finite Cartan types, [see here.](https://en.wikipedia.org/wiki/Root_system#Classification_of_root_systems_by_Dynkin_diagrams)
### Noncrystallographic finite root systems
Dropping condition (3) in the definition extends the class of root systems by the non-crystallographic cases.
This gives a classification of finite Coxeter groups, see <>.
In addition to the classification of the cystallographic irreducible root systems (and their associated Weyl groups), there are finite Coxeter groups of types $I_2(m)$ (symmetry group of a regular $m$-gon) and types $H_3$ (symmetry group of the [regular icosahedron and dodecahedron](https://en.wikipedia.org/wiki/Icosahedral_symmetry)) and $H_4$ (symmetry group of the [120-cell](https://en.wikipedia.org/wiki/120-cell) and of the [600-cell](https://en.wikipedia.org/wiki/600-cell)).
Other objects classified by finite Cartan types
-----------------------------------------------
### Semisimple Lie Algebras
Let $F$ be an algebraically closed field of characteristic zero.
A **Lie algebra** is a vector space $\mathfrak{g}$ over $F$ together with an operation $\mathfrak{g}\times \mathfrak{g}\rightarrow \mathfrak{g}, (x,y) \mapsto [x,y]$, called **Lie bracket**, such that the following is satisfied:
1. The bracket operation is bilinear;
1. $[x,x] = 0$ for all $x\in \mathfrak{g}$;
1. $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ for all $x,y,z\in \mathfrak{g}$.
A subspace $I$ of $\mathfrak{g}$ is called an **ideal** of $\mathfrak{g}$ if $[x,y]\in I$ for all $x\in \mathfrak{g}, y\in I$. A special ideal of $\mathfrak{g}$ is the **derived algebra** $[\mathfrak{g},\mathfrak{g}]$ consisting of all possible brackets of elements from $\mathfrak{g}$. A Lie algebra is called **simple** if $[\mathfrak{g},\mathfrak{g}]\neq 0$ and $0$ and $\mathfrak{g}$ are its only ideals.
A Lie algebra $\mathfrak{g}$ is called **abelian** if the bracket vanishes on $\mathfrak{g}$, i.e. $[\mathfrak{g},\mathfrak{g}]=0$.
It is called **solvable** if the **derived series**
$$\mathfrak{g} \supset [\mathfrak{g},\mathfrak{g}] \supset [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]] \supset [[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]] \supset \dots$$
becomes zero eventually.
If $\mathfrak{g}$ is of finite dimension, then it has a unique maximal solvable ideal, called its **radical** $\operatorname{Rad} \mathfrak{g}$.
A **semisimple Lie algebra** is a Lie algebra $\mathfrak{g}$ such that $\operatorname{Rad} \mathfrak{g} = 0$. This is equivalent to a decomposition
$$\mathfrak{g}=\mathfrak{g}_1 \oplus \dots \oplus \mathfrak{g}_p$$
into simple Lie algebras $\mathfrak{g}_1,\dots,\mathfrak{g}_p$.
Given a semisimple Lie algebra L, there is a canonical way to construct a root system $\Phi$ associated to $\mathfrak{g}$ that completely determines the structure of $\mathfrak{g}$. Furthermore, a simple Lie algebras is associated to an irreducible root system. Thus, semisimple Lie algebras are completely classified by the finite Cartan types. Details on this can be found in <>.
### Quiver Representations
A **quiver** a directed graph $Q=(V,E)$ with possibly multiple edges and loops. For a ring $R$, a **representation of $Q$ over $R$** is an assignment of a $R$-module $R_v$ to each vertex $v\in V$ and a linear map $f_{v,w}:R_v\rightarrow R_w$ to each edge $(v,w) \in E$. A representation of $Q$ is **indecomposable** if it is not a sum of smaller nontrivial representations of $Q$.
Let $F$ be an algebraically closed field. *Gabriel's Theorem* states that a quiver $Q$ has only finitely many non-isomorphic representations of finite dimension if and only if the underlying undirected graph $\bar Q$ is of type $A$, $D$ or $E$. For a thorough introduction into quiver representations see <>
### Cluster algebras of finite type
**Cluster algebras** were introduced by Fomin and Zelevinsky in the early 2000s in <> and they obtained the classification of cluster algebras of finite type in <>.
References
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<>
Sage examples
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{{{#!sagecell
for ct in [/'A',4],['B',4],['C',4],['D',4],['F',4 'A',4],['B',4],['C',4],['D',4],['F',4]:
ct = CartanType(ct)
print ct, ct.coxeter_number()
}}}
Technical information for database usage
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- The database contains all irreducible finite Cartan types up to rank $8$.