What is a root system in mathematics?

In mathematics, a root system is a specific arrangement of vectors within a Euclidean space. These vectors must satisfy certain geometric properties, making the concept fundamental in the study of Lie groups and Lie algebras, particularly in classifying and understanding semisimple Lie algebras.

What is the primary application of root systems in mathematics?

Root systems are fundamental to the theory of Lie groups and Lie algebras. They are particularly crucial for the classification and representation theory of semisimple Lie algebras, and their study extends to areas like algebraic groups.

Beyond Lie theory, in which other mathematical fields are root systems applied?

Root systems find applications in other areas of mathematics, including singularity theory and spectral graph theory. Their classification scheme, often represented by Dynkin diagrams, appears in contexts seemingly unrelated to their origins in Lie theory.

Can you describe the example of the A2 root system?

The A2 root system is exemplified by six vectors in a 2-dimensional Euclidean space (R^2). These vectors span the space, and for any root $\alpha$, the reflection across the hyperplane perpendicular to $\alpha$ maps every other root $\beta$ to $\beta + n\alpha$, where n is an integer. This geometric property is key to the definition of a root system.

What are the four defining conditions for a set of vectors to be considered a root system in a Euclidean space E?

A finite set of non-zero vectors $\Phi$ in a Euclidean space E forms a root system if it satisfies these conditions: 1. The roots $\Phi$ span the entire space E. 2. The only scalar multiples of a root $\alpha$ that are also in $\Phi$ are $\alpha$ itself and $-\alpha$. 3. For any root $\alpha$, the reflection of the set $\Phi$ across the hyperplane perpendicular to $\alpha$ leaves the set $\Phi$ unchanged. 4. For any two roots $\alpha$ and $\beta$, the projection of $\beta$ onto the line defined by $\alpha$ is an integer or half-integer multiple of $\alpha$.

What is the significance of the integrality condition (condition 4) in the definition of a root system?

The integrality condition states that for any two roots $\alpha$ and $\beta$, the quantity $2(\alpha, \beta) / (\alpha, \alpha)$ must be an integer. This condition ensures specific geometric relationships between roots, constraining the possible angles and lengths within the system.

How do crystallographic and reduced root systems differ in their definitions?

A root system is called 'crystallographic' if it satisfies the integrality condition (condition 4). Some definitions omit condition 2 (scalar multiples), and root systems that satisfy condition 2 are then called 'reduced'. This article assumes all root systems discussed are both reduced and crystallographic.

What is the rank of a root system?

The rank of a root system is defined as the dimension of the Euclidean space E in which the vectors are situated. This dimension corresponds to the number of linearly independent roots needed to span the space.

What does it mean for a root system to be irreducible?

A root system is considered irreducible if it cannot be decomposed into two or more smaller, independent root systems. This occurs when the set of roots cannot be partitioned into two non-empty subsets such that the vectors in one subset are orthogonal to the vectors in the other subset.

How are two root systems considered isomorphic?

Two root systems are isomorphic if there exists an invertible linear transformation between their respective Euclidean spaces that maps one set of roots onto the other while preserving the geometric relationships, specifically the inner product values between pairs of roots.

What is the root lattice associated with a root system?

The root lattice of a root system $\Phi$ is the set of all integer linear combinations of the roots in $\Phi$. This lattice is a discrete subgroup within the Euclidean space E, capturing the fundamental structure generated by the roots.

How does the Weyl group relate to the symmetry of a root system?

The Weyl group consists of isometries (transformations that preserve distances) of the Euclidean space that map the root system onto itself. While it captures many symmetries, the Weyl group might not encompass all symmetries of the root system itself, as seen in the A2 example where rotations are symmetries but not part of the Weyl group.

What is the relationship between semisimple Lie algebras and root systems?

Semisimple Lie algebras are intrinsically linked to root systems. A complex semisimple Lie algebra has a unique Cartan subalgebra, and the roots of the characteristic polynomial associated with elements of this subalgebra form a root system. This root system is crucial for understanding the algebra's structure and classifying it.

Who first introduced the concept of a root system and when?

The concept of a root system was first introduced by Wilhelm Killing around 1889 in German, using the term 'Wurzelsystem'. He developed this concept as part of his effort to classify simple Lie algebras over complex numbers.

What was Wilhelm Killing's initial contribution and subsequent correction regarding root systems?

Wilhelm Killing introduced root systems in his attempt to classify simple Lie algebras. Although he initially made an error in the classification, listing two exceptional rank 4 root systems instead of one (F4), Élie Cartan later corrected this by demonstrating the isomorphism between Killing's two systems.

What are the possible angles between any two roots in a root system?

The angles between any two roots in a root system are restricted. Specifically, the cosine of the angle $\theta$ between two roots $\alpha$ and $\beta$ must satisfy $(2\cos(\theta))^2$ being an integer, leading to possible angles of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

What is a simple root in the context of a root system?

A simple root is a specific type of positive root within a chosen set of positive roots. It is defined as a positive root that cannot be expressed as the sum of two other positive roots. The set of simple roots forms a basis for the entire root system.

What is a base for a root system?

A base for a root system is a set of simple roots. This set serves as a basis for the vector space spanned by the roots, and any root can be uniquely expressed as a linear combination of these simple roots with integer coefficients.

What are the properties of a set of simple roots?

A set of simple roots $\Delta$ for a root system $\Phi$ forms a basis for the space E. Every root $\alpha$ in $\Phi$ can be written as an integer linear combination of roots in $\Delta$, and for any root $\alpha$, its coefficients in this combination are either all non-negative or all non-positive.

What is a coroot, and how is it related to a root?

The coroot $\alpha^{\vee}$ of a root $\alpha$ is defined as $2\alpha / (\alpha, \alpha)$, where $(\alpha, \alpha)$ is the squared norm of the root. Coroots are essential for defining the dual root system and play a role in the integrality conditions.

What is the dual root system?

The dual root system, denoted $\Phi^{\vee}$, is the set of all coroots corresponding to the roots in the original root system $\Phi$. Importantly, the dual root system is also a root system, and $\Phi$ is the dual root system of $\Phi^{\vee}$.

What is the weight lattice of a root system?

The weight lattice associated with a root system is the set of integral elements. An element $\lambda$ is integral if its inner product with every coroot is an integer. This lattice is significant in representation theory, as its elements correspond to the possible weights of finite-dimensional representations of semisimple Lie algebras.

How are root systems classified, and what are Dynkin diagrams?

Root systems are classified based on their Dynkin diagrams, which are graphical representations derived from the simple roots. An irreducible root system corresponds to a connected Dynkin diagram. There are four infinite families (A, B, C, D) and five exceptional cases (E6, E7, E8, F4, G2).

How is a Dynkin diagram constructed from a root system?

A Dynkin diagram is constructed using a set of simple roots. Vertices represent the simple roots, and edges connect them based on the angle between the roots: no edge for orthogonal roots, a single edge for 120 degrees, a double edge for 135 degrees, and a triple edge for 150 degrees. Arrows indicate the direction towards the shorter root in cases of unequal lengths.

What is the relationship between the Dynkin diagram of a root system and its dual root system?

The Dynkin diagram of a dual root system is obtained by reversing the directions of all arrows on the edges of the original root system's Dynkin diagram. This highlights the duality between certain root systems, such as B_n and C_n.

What are Weyl chambers?

Weyl chambers are the connected components of the space remaining after removing all hyperplanes perpendicular to the roots of a root system. They represent fundamental regions of symmetry under the action of the Weyl group.

What is the fundamental Weyl chamber?

The fundamental Weyl chamber is a specific Weyl chamber associated with a chosen set of simple roots. It is defined as the set of points in the Euclidean space that lie on the positive side of all hyperplanes corresponding to the simple roots.

What is the connection between the order of the Weyl group and the number of Weyl chambers?

A key theorem states that the Weyl group acts freely and transitively on the Weyl chambers. This means the order (number of elements) of the Weyl group is precisely equal to the number of Weyl chambers for that root system.

How do root systems classify simple Lie algebras?

Irreducible root systems serve as a classification scheme for simple complex Lie algebras, simply connected complex Lie groups, and simply connected compact Lie groups. Each irreducible root system corresponds to a unique type of these algebraic structures.

What are the four infinite families of irreducible root systems called?

The four infinite families of irreducible root systems are known as the classical root systems: A_n, B_n, C_n, and D_n. The subscript 'n' typically denotes the rank of the root system.

What are the five exceptional root systems?

The five exceptional root systems are G2, F4, E6, E7, and E8. These systems do not fit into the general families and possess unique structures and properties.

What does it mean for a root system to be 'simply laced'?

A root system is 'simply laced' if all its roots have the same length. This property is characteristic of the A, D, and E series of root systems. In contrast, systems like B, C, F, and G have roots of two different lengths.

How are the B_n and C_n root systems related?

The B_n and C_n root systems are dual to each other, meaning the root system of B_n is isomorphic to the coroot system of C_n, and vice versa. They are generally not isomorphic to each other, except for the special case of n=2.

Can you describe the E8 root system's structure?

The E8 root system consists of 240 roots in 8-dimensional Euclidean space (R^8). It can be defined as the set of integer or half-integer coordinate vectors in R^8 whose squared norm is 2 and whose coordinate sum is an even integer. This structure is also known as the E8 lattice.

How is the F4 root system constructed?

The F4 root system exists in 4-dimensional Euclidean space (R^4) and comprises 48 roots. These roots can be defined based on vectors whose coordinates are integers or half-integers, with specific conditions on their sum and squared norm, and are related to the vertices of a 24-cell and its dual.

What is the G2 root system, and how many roots does it have?

The G2 root system is an exceptional root system with 12 roots, which geometrically form the vertices of a hexagram (a six-pointed star). It is related to the A2 root system and has a distinctive Dynkin diagram with a triple edge.

What is the root poset, and how is it structured?

The root poset is formed by the set of positive roots, ordered such that $\alpha \leq \beta$ if $\beta - \alpha$ is a non-negative linear combination of simple roots. This poset is graded by the sum of the coefficients of the simple roots in the linear combination, and its structure is visualized using a Hasse diagram.

What is the significance of the 'ADE classification' in relation to root systems?

The ADE classification relates specific types of mathematical structures, including root systems, to the simply laced Dynkin diagrams A, D, and E. This classification highlights deep connections between seemingly disparate areas of mathematics.

What is an affine root system?

An affine root system is an extension of the concept of a finite root system, typically involving translations by elements of a root lattice. They are closely related to affine Lie algebras and affine Weyl groups.

What is a Coxeter group in the context of root systems?

A Coxeter group is a group defined by generators and relations, often related to reflections. The Weyl group of a root system is a finite Coxeter group, and affine Weyl groups are infinite Coxeter groups associated with affine root systems.

How does the Cartan matrix relate to root systems?

The Cartan matrix is a matrix whose entries are derived from the scalar products of simple roots (specifically, the integers $\langle \alpha_i, \alpha_j \rangle$). It encodes the geometric structure of the root system and is fundamental in its classification via Dynkin diagrams.

What is a root datum?

A root datum is a pair of root systems ($\Phi$, $\Phi^{\vee}$) along with a root-preserving isomorphism between them and a Weyl group acting on them. It provides a more abstract and general framework for studying Lie theory structures.

What are weights in the context of Lie algebra representation theory?

In the representation theory of semisimple Lie algebras, weights are linear functions on the Cartan subalgebra that describe how the algebra acts on vectors in a representation space. The integral elements of a root system correspond to the highest weights of finite-dimensional representations.

What is the role of the adjoint representation in relation to root systems?

The roots of a semisimple Lie algebra are precisely the non-zero weights of its adjoint representation. This means the root system describes the structure of the Lie algebra itself when viewed as a representation on its own elements.

What is the 'triality' symmetry associated with the D4 root system?

Triality is a special symmetry of the D4 root system and its associated Lie group/algebra. It relates the three different ways the root system can be interpreted, corresponding to the three types of Dynkin diagrams associated with D4.

What is the E8 lattice?

The E8 lattice is a specific arrangement of points in 8-dimensional space that is closely related to the E8 root system. It is defined by integer or half-integer coordinates with specific constraints on their sum and squared norm, and it is known for its high degree of symmetry and density.

How are the A_n, B_n, C_n, and D_n root systems constructed explicitly?

These classical root systems are constructed in Euclidean spaces R^k (where k is the rank) using vectors with integer or specific coordinate patterns related to standard basis vectors. For example, A_n involves vectors in R^(n+1) with coordinates summing to zero, while B_n and C_n involve vectors in R^n with specific length constraints.

What is the relationship between the A2 root system and the triangular tiling?

The root lattice generated by the A2 root system corresponds to the vertex arrangement of the triangular tiling of the plane. This illustrates how abstract mathematical structures can relate to geometric patterns.

What is the connection between the A3 root system and the face-centered cubic lattice?

The root lattice generated by the A3 root system is equivalent to the face-centered cubic lattice, which is fundamental in crystallography and describes the packing of spheres in a highly symmetric manner.

What is the significance of the 'simply laced' property for root systems?

Simply laced root systems (A, D, E series) have all roots of equal length. This property simplifies certain aspects of their analysis and connects them to specific types of Lie algebras and geometric structures.

Root System Wiki: Geometric Foundations: The Essence of Root Systems

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Definitions

Core Concept

A root system is a finite set of non-zero vectors in a Euclidean space that satisfies specific geometric conditions. These configurations are fundamental building blocks in the theory of Lie groups and Lie algebras, particularly in their classification and the study of their representations.

Defining Axioms

Let E be a finite-dimensional Euclidean vector space. A root system $\Phi$ is a finite set of non-zero vectors (roots) satisfying:

The roots span the entire space E.
If $\alpha \in \Phi$ , then $-\alpha$ is also in $\Phi$ .
Reflection across the hyperplane perpendicular to any root $\alpha$ maps $\Phi$ to itself.
The projection of root $\beta$ onto the line through $\alpha$ is an integer or half-integer multiple of $\alpha$ .

A root system is called crystallographic if it satisfies the integrality condition (axiom 4). This document assumes crystallographic root systems.

Rank and Isomorphism

The rank of a root system is the dimension of the Euclidean space E it spans. Two root systems are considered isomorphic if there's an invertible linear transformation between their spaces that preserves the set of roots and the geometric relationships (inner products) between them. This classification is crucial for understanding the underlying symmetries.

Examples

Rank One: A₁

The simplest root system, in 1-dimensional space, consists of just two vectors: $\{\alpha, -\alpha\}$ . This system is denoted as A₁.

Rank Two: Geometric Possibilities

In two dimensions, there are four distinct irreducible root systems, characterized by the angles and length ratios between pairs of roots:

A₁ × A₁ (or D₂): Two pairs of opposite roots, orthogonal to each other.
A₂: Six roots forming a hexagonal pattern, with angles of 60° or 120° between adjacent roots.
B₂ (or C₂): Four roots forming a square, with two roots longer than the other two. Angles are 90° and 135°.
G₂: Six roots with angles of 30° or 150°, and a length ratio of $\sqrt{3}$ .

These examples illustrate the geometric constraints imposed by the root system axioms.

Dynkin Diagrams

The relationships between simple roots (a specific subset of positive roots) can be visually represented by Dynkin diagrams. These diagrams encode the angles and relative lengths of the simple roots, providing a compact way to classify root systems. Connected Dynkin diagrams correspond to irreducible root systems.

Dynkin diagrams use vertices for simple roots and edges to represent their relationships:

No edge: Orthogonal roots (90° angle).
Single edge: Angle of 120° (equal length roots).
Double edge (directed): Angle of 135°, length ratio $\sqrt{2}$ .
Triple edge (directed): Angle of 150°, length ratio $\sqrt{3}$ .

Arrows point from longer roots to shorter roots.

Algebraic Connections

Lie Algebras and Roots

Root systems are intrinsically linked to the structure of semisimple Lie algebras. For a complex semisimple Lie algebra $\mathfrak{g}$ and its Cartan subalgebra $\mathfrak{h}$ , the roots are non-zero linear functionals on $\mathfrak{h}$ that arise from the adjoint action of $\mathfrak{h}$ on $\mathfrak{g}$ .

Killing Form

The inner product used in the definition of root systems is often derived from the Killing form of the Lie algebra. This bilinear form provides the necessary geometric structure. The set of roots, when considered within the dual space of the Cartan subalgebra, naturally forms a root system under this structure.

Foundational Tool

The root system is a critical invariant for classifying semisimple Lie algebras and understanding their representation theory. It dictates the structure and properties of these algebraic objects.

Weyl Group

Symmetry Group

The Weyl group of a root system $\Phi$ is the finite group generated by reflections across the hyperplanes perpendicular to each root. It represents the symmetries of the root system.

Action on Roots

The Weyl group acts on the set of roots, permuting them. This action is crucial for understanding the structure and classification. For example, the Weyl group of the A₂ root system is the symmetry group of an equilateral triangle.

Dual Relationship

The Weyl group is also fundamental in relating a root system to its dual root system (formed by the coroots). The action of the Weyl group is consistent between the root system and its dual.

Historical Development

Killing's Contribution

The concept of root systems was introduced by Wilhelm Killing around 1889 in his ambitious attempt to classify all simple Lie algebras over the complex numbers. His initial work laid the groundwork for this fundamental area of mathematics.

Refinements and Corrections

Killing's initial classification contained errors, notably regarding the exceptional root systems. These were later corrected and refined by Élie Cartan, solidifying the classification scheme that is still used today. The concept itself proved remarkably robust and widely applicable.

Dynkin's Diagrams

The development of Dynkin diagrams by Eugene Dynkin provided a powerful combinatorial tool for representing and classifying root systems, simplifying the study of their complex structures and interrelationships.

Classification

The Five Families

All irreducible root systems fall into five families, classified by their Dynkin diagrams:

Classical Types: A_n, B_n, C_n, D_n
Exceptional Types: E₆, E₇, E₈, F₄, G₂

The subscript denotes the rank of the root system.

Duality

Root systems and their duals (coroots) are closely related. The classical types B_n and C_n are dual to each other but not isomorphic (except for n=2). The classical types A_n and D_n, along with all exceptional types, are self-dual.

Irreducibility

A root system is irreducible if it cannot be decomposed into a direct sum of smaller root systems. This property corresponds directly to the connectivity of its Dynkin diagram. Non-irreducible systems are simply combinations of irreducible ones.

Dynkin Diagrams

Visualizing Structure

Dynkin diagrams provide a graphical representation of the relationships between the simple roots of a root system. Each vertex represents a simple root, and the edges (with potential arrows and multiple lines) encode the angles and relative lengths between them.

The construction rules are based on the angles between simple roots:

Orthogonal roots (90°): No edge.
120° angle: Single edge.
135° angle: Double edge, directed towards the shorter root.
150° angle: Triple edge, directed towards the shorter root.

The length ratio of longer to shorter root determines the edge multiplicity (1, $\sqrt{2}$ , or $\sqrt{3}$ ).

Classification Power

The classification of possible connected Dynkin diagrams directly leads to the classification of all irreducible root systems. Each distinct connected diagram corresponds to a unique irreducible root system, making this a powerful combinatorial approach.

Lattices

Root Lattice

The root lattice associated with a root system $\Phi$ is the lattice (discrete subgroup) generated by the roots themselves. It captures the integer linear combinations of roots.

Weight Lattice

An integral element is a vector whose inner product with every coroot is an integer. The set of all integral elements forms the weight lattice. This lattice is crucial in representation theory, as it contains the weights of finite-dimensional representations of semisimple Lie algebras.

Distinction

While roots are always integral elements, the root lattice and the weight lattice do not always coincide. The weight lattice can be larger, containing integral elements that are not integer linear combinations of roots, reflecting a richer structure.

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References

See various parts of Chapters III, IV, and V of Humphreys 1972, culminating in Section 19 in Chapter V

A full list of references for this article are available at the Root system Wikipedia page

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This page was generated by an Artificial Intelligence and is intended for informational and educational purposes only. The content is derived from publicly available data and aims to provide a structured overview of root systems in mathematics.

This is not a substitute for formal mathematical study. While efforts have been made to ensure accuracy and clarity, the abstract nature of the subject requires rigorous study through textbooks and academic resources. Always consult authoritative mathematical literature for definitive understanding and application.

The creators of this page are not responsible for any errors or omissions, or for any actions taken based on the information provided herein.

Geometric Foundations

💡 Dive in with Flashcard Learning! 💡

Definitions ℹ️

📐 Core Concept

⚖️ Defining Axioms

📊 Rank and Isomorphism

Examples 📊

📐 Rank One: A1

📐 Rank Two: Geometric Possibilities

🔗 Dynkin Diagrams

Algebraic Connections 🧮

🔗 Lie Algebras and Roots

⚖️ Killing Form

⭐ Foundational Tool

Weyl Group ⚙️

👑 Symmetry Group

🔄 Action on Roots

↔️ Dual Relationship

Historical Development ⏳

📜 Killing's Contribution

✅ Refinements and Corrections

💡 Dynkin's Diagrams

Classification 🏷️

🔢 The Five Families

↔️ Duality

🧩 Irreducibility

Dynkin Diagrams 🔗

🖼️ Visualizing Structure

✅ Classification Power

Lattices 🌐

➕ Root Lattice

⭐ Weight Lattice

≠ Distinction

Teacher's Corner 🧑‍🏫

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📜 References

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📜 Important Notice

Dive in with Flashcard Learning!

Definitions

Core Concept

Defining Axioms

Rank and Isomorphism

Examples

Rank One: A₁

Rank Two: Geometric Possibilities

Dynkin Diagrams

Algebraic Connections

Lie Algebras and Roots

Killing Form

Foundational Tool

Weyl Group

Symmetry Group

Action on Roots

Dual Relationship

Historical Development

Killing's Contribution

Refinements and Corrections

Dynkin's Diagrams

Classification

The Five Families

Duality

Irreducibility

Dynkin Diagrams

Visualizing Structure

Classification Power

Lattices

Root Lattice

Weight Lattice

Distinction

Teacher's Corner

True or False?

Test Your Knowledge!

Gamer's Corner

Discover other topics to study!

References

Feedback & Support

Disclaimer

Important Notice