Root system

Root systems are used in mathematics as a tool for classification of finite reflection groups and the finite-dimensional semisimple complex Lie algebras.

• 7.1 Lie algebras 7.1.1 properties
• 7.1.2 example

Definitions

A subset of a vector space over a field of characteristic 0 is the root system, if it satisfies the following conditions:

• For is.
• The linear map with maps to.

A reduced root system is available, if in addition applies

It can be shown that the linear form of 3 is unique to each. You will be called the Kowurzel; the designation is justified that the coroots form a root system in the dual space. The figure is a reflection and of course also uniquely determined.

Are roots and two with, one can show that also applies, and is called an orthogonal relationship. Can you write the root system such as a union of two non-empty subsets that each root in orthogonal to each root in, it means the root system reducible. In this case, can be split into a direct sum, so that and root systems. However, is not reducible a non-empty root system, so it is called irreducible.

The dimension of the vector space is called the rank of the root system. A subset of a root system is based, if a base, and each element of can be represented as a linear combination of the integral elements of exclusively positive or exclusively negative coefficients.

Two root systems and are accurate then each isomorphic if there is a Vektorraumisomorphismus with.

Scalar product

One can define a scalar product with respect to which the pictures are reflections. In the reducible case, you can put together this scalar from the components. However, if is irreducible, this scalar product is clearly even up to a factor. You can still normalize this so that the shortest roots have length 1.

So you can basically assume that a root system in a (mostly) with its standard scalar product "alive". The integrality of and represents a significant constraint on the possible angle between two roots. It is clear from

That must be an integer. Again, this is only for the angle 0 °, 30 °, 45 °, 60 °, 90 °, 120 °, 135 °, 150 °, 180 ° of the case. Between two different roots of a base even just the angle 90 °, 120 °, 135 °, 150 ° are possible. All these angles occur in fact, see the examples of rank 2 is also clear that only a few values ​​are also possible for the length ratio of two roots in the same irreducible component.

Weyl group

The sub-group of the automorphism of which is generated from the amount of reflection is, Weyl group ( by Hermann Weyl ), and is generally referred to. With respect to the scalar product defined all elements of the Weyl group are orthogonal, which are reflections.

The group operates on faithful and therefore is always finite. Furthermore, acts transitively on the set of bases of.

In the case of the reflection planes decompose the space in each half-spaces in total in several open convex subsets, called Weylkammern. Also on these acts transitively.

Positive roots, Simple roots

After selecting a Weyl chamber, one can define the set of positive roots by

This defines an arrangement by

The positive or negative roots are therefore those with or. ( Note that this definition depends on the choice of the Weyl chamber. At any Weyl chamber gives an arrangement. )

A simple root is a positive root, which can not be decomposed as the sum of several positive roots.

The simple roots form a basis of. Any positive (negative ) root can be decomposed as a linear combination of simple roots with non-negative ( non-positive ) coefficients.

Examples

The empty set is the only root system of rank 0 and is also the only root system, which is neither reducible nor irreducible.

There are up to isomorphism only reduced root system of rank 1 It consists of two different roots of 0 and is denoted by. If one also considers non-reduced root systems, so the only other example of rank 1

All reduced root systems of rank 2, up to isomorphism, one of the following forms. is in each case a base of the root system.

In the first example, is the ratio of the lengths of any and, on the other hand clearly determined in the other cases by the geometric conditions.

Classification

Up to isomorphism, all information about reduced root system in its Cartan matrix

Included. One can represent this in the form of a Dynkin diagram. It relies on each element of a basis of a point and connects the points α and β by dashes, the number of

Is determined. These are more than one, you have to set additional points between both a relation symbol > or <, ie a, arrow 'in the direction of the shorter root. The connected components of the Dynkin diagram correspond exactly to the irreducible components of the root system. As a diagram of an irreducible root system can only occur:

The index gives the rank of each case, and thus the number of points in the graph. From the Dynkin diagrams can be read multiple identities. Thus, for example, or. Therefore, only maps and only from a standalone class. To the series belong to the root systems are also referred to as a classical root systems, and the remaining five as exceptional or exception root systems. All of these root systems occur for example on semisimple root system as complex Lie algebras.

Non- reduced root systems

For irreducible, non-reduced root systems, there are few options that can be thought of as the union of a ( with ) or as one in which short for any root whose double was added.

Other applications

Lie algebras

It is a finite-dimensional semisimple Lie algebra and a Cartan subalgebra. Then is called a root if

Is. Here, the means of the Killing form by

Defined linear map.

Be the set of roots, then one can show that

A root system is.

Properties

This root system has the following features:

Finite-dimensional semisimple complex Lie algebras are classified by their root systems, so by their Dynkin diagrams.

Example

It should be. The Killing form is a Cartan subalgebra is the algebra of diagonal matrices with trace 0, ie. We denote identical with the diagonal matrix with diagonal entry system and the other diagonal entries 0

The root system of is. The dual form is to

As a positive Weyl chamber can be

Select. The positive roots are then

The simple roots are

Reflection groups

A Coxeter group is abstractly defined as a group with presentation

With and for, and the convention case has infinite order, ie there is no relation to the shape.

Coxeter groups are an abstraction of the concept of mirroring group.

Every Coxeter group corresponding to an undirected Dynkin diagram. The points of the graph correspond to the generators. The corresponding points and are connected by edges.

Singularities

After Vladimir Arnold Elementary to disasters by Dynkin diagrams of type ADE can be classified:

• A0 - is a non- singular point.
• A1 - a local extremum, either a stable or an unstable minimum maximum.
• A2 - the folding, fold
• A3 - the tip, cusp
• A4 - the swallowtail, swallowtail
• A5 - the butterfly, butterfly
• Ak - an infinite sequence of shapes in a variable
• D4 - the elliptical umbilical disaster
• D4 - the hyperbolic umbilical disaster
• D5 - the parabolic umbilical disaster
• Dk - an infinite sequence of further disasters umbilischer
• E6 - the umbilical disaster
• E7
• E8