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Definition & Example

the 5 Finite Cartan types of size 8

Additional information

Finite root systems

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. $\Phi \cap \mathbb{R}\alpha = \{\alpha, -\alpha\}$,
  2. $s_\alpha \Phi = \Phi$ for all $\alpha \in \Phi$ and
  3. $\langle\beta,\alpha\rangle := 2\frac{(\beta,\alpha)}{(\alpha,\alpha)} \in \mathbb{Z}$ 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 [Hum72].

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 [Hum72].

Irreducible crystallographic root systems are classified by Dynkin diagrams and named by finite Cartan types, see here.

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 [Hum92]. 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) and $H_4$ (symmetry group of the 120-cell and of the 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;
  2. $[x,x] = 0$ for all $x\in \mathfrak{g}$;
  3. $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ 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 [Hum72].

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 [ASS06]

Cluster algebras of finite type

Cluster algebras were introduced by Fomin and Zelevinsky in the early 2000s in [FZ02] and they obtained the classification of cluster algebras of finite type in [FZ03].


Sage examples

Technical information for database usage

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