2.3 Root systems and weight spaces

Root systems and weight spaces form the backbone of Non-associative Algebra, providing a geometric framework for understanding Lie algebras. These concepts allow us to classify and analyze complex algebraic structures, breaking them down into manageable components.

Weight spaces decompose representations of Lie algebras, while root systems encode the structure of these algebras. Together, they offer powerful tools for studying symmetries in mathematics and physics, from particle interactions to string theory.

Definition of root systems

• Root systems form a fundamental concept in Non-associative Algebra, providing a geometric framework for understanding the structure of Lie algebras
• These systems consist of vectors in a Euclidean space that satisfy specific properties, allowing for the classification and analysis of complex algebraic structures

Properties of root systems

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• Finite set of non-zero vectors in a Euclidean space
• Closed under reflection about hyperplanes perpendicular to roots
• Only scalar multiples of a root in the system are integer multiples
• Reflection of one root by another yields an integer linear combination of roots

Axioms for root systems

• Span the entire vector space $V$
• If $\alpha$ is a root, then $-\alpha$ is also a root, but no other scalar multiples
• For any roots $\alpha$ and $\beta$, the reflection $s_\alpha(\beta)$ is also a root
• For any roots $\alpha$ and $\beta$, the quantity $\frac{2(\alpha, \beta)}{(\alpha, \alpha)}$ is an integer

Examples of root systems

• $A_n$ root system corresponds to $SL(n+1)$ Lie algebra
• $B_n$ root system relates to odd-dimensional orthogonal groups
• $D_n$ system associated with even-dimensional orthogonal groups
• $G_2$ exceptional root system connected to the octonions

Types of root systems

• Root systems play a crucial role in the classification of semisimple Lie algebras in Non-associative Algebra
• Understanding different types of root systems helps in analyzing the structure and properties of various algebraic entities

Reducible vs irreducible systems

• Reducible systems decompose into orthogonal subsystems
• Irreducible systems cannot be further decomposed
• Classification of irreducible root systems leads to the ADE classification
• Reducible systems form direct sums of irreducible components

Crystallographic root systems

• Arise from lattices in Euclidean space
• Correspond to semisimple Lie algebras
• Characterized by integer-valued Cartan matrices
• Include classical series ($A_n$, $B_n$, $C_n$, $D_n$) and exceptional types ($G_2$, $F_4$, $E_6$, $E_7$, $E_8$)

Non-crystallographic root systems

• Do not correspond to lattices in Euclidean space
• Include $H_3$ (icosahedral symmetry) and $H_4$ (4-dimensional analogue of icosahedral symmetry)
• Related to quasicrystals and Coxeter groups
• Have applications in mathematical physics and condensed matter theory

Weight spaces

• Weight spaces form an essential component in the study of representations of Lie algebras within Non-associative Algebra
• They provide a way to decompose representations into simpler, more manageable components

Definition of weight spaces

• Subspaces of a representation where Cartan subalgebra elements act by scalar multiplication
• Characterized by weight vectors $\lambda$ in the dual space of the Cartan subalgebra
• For a representation $V$ and weight $\lambda$, the weight space $V_\lambda$ defined as $\{v \in V | h.v = \lambda(h)v \text{ for all } h \text{ in the Cartan subalgebra}\}$
• Dimensions of weight spaces give rise to weight multiplicities

Relationship to root systems

• Roots can be viewed as weights of the adjoint representation
• Weight lattice contains the root lattice as a sublattice
• Fundamental weights form a basis for the weight lattice
• Weyl group acts on both root and weight lattices

Weight lattices

• Discrete subgroup of the dual space of the Cartan subalgebra
• Generated by fundamental weights
• Contains all possible weights of finite-dimensional representations
• Integral linear combinations of fundamental weights form dominant weights

Cartan matrix

• The Cartan matrix serves as a compact representation of the root system in Non-associative Algebra
• It encodes essential information about the structure and relationships between simple roots

Construction of Cartan matrix

• Square matrix $A$ with entries $A_{ij} = \frac{2(\alpha_i, \alpha_j)}{(\alpha_i, \alpha_i)}$
• $\alpha_i$ and $\alpha_j$ are simple roots of the root system
• Diagonal entries always equal 2
• Off-diagonal entries are non-positive integers

Properties of Cartan matrix

• Symmetrizable (can be made symmetric by multiplying rows by positive real numbers)
• Determinant is positive for finite-dimensional semisimple Lie algebras
• Eigenvalues are all positive for positive definite Cartan matrices
• Inverse of Cartan matrix relates simple roots to fundamental weights

Dynkin diagrams

• Graphical representation of Cartan matrices
• Nodes represent simple roots
• Edges indicate relationships between simple roots
• Single, double, or triple edges denote different angles between roots
• Arrows on edges indicate relative lengths of roots

Classification of root systems

• Classification of root systems forms a cornerstone in the study of semisimple Lie algebras within Non-associative Algebra
• It provides a complete categorization of all possible root systems, enabling a systematic approach to understanding complex algebraic structures

ADE classification

• Refers to simply-laced root systems
• Includes $A_n$, $D_n$, and $E_6$, $E_7$, $E_8$ series
• Corresponds to Dynkin diagrams with only single edges
• Related to various mathematical objects (Platonic solids, McKay correspondence)

BCFG classification

• Non-simply-laced root systems
• Includes $B_n$, $C_n$, $F_4$, and $G_2$ series
• Dynkin diagrams contain double or triple edges
• Associated with orthogonal and symplectic Lie algebras

Exceptional root systems

• $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$
• Do not belong to infinite series of classical root systems
• Have unique properties and symmetries
• Connected to exceptional Lie groups and algebras

Applications of root systems

• Root systems find extensive applications in various areas of mathematics and physics, extending the reach of Non-associative Algebra
• They provide a powerful framework for analyzing symmetries and structures in diverse fields

Lie algebras

• Root systems classify semisimple Lie algebras
• Determine the structure constants of Lie algebras
• Enable the study of representations and character theory
• Facilitate the analysis of subalgebras and ideals

Representation theory

• Weight spaces decompose representations
• Highest weight theory builds representations from dominant weights
• Character formulas express dimensions of weight spaces
• Branching rules describe decomposition of representations under subalgebras

Quantum groups

• Deformations of universal enveloping algebras of Lie algebras
• Root systems determine the structure of quantum groups
• Crystal bases provide combinatorial models for representations
• Applications in statistical mechanics and integrable systems

Weyl group

• The Weyl group plays a crucial role in the study of root systems and Lie algebras within Non-associative Algebra
• It captures the symmetries of the root system and provides insights into the structure of representations

Definition of Weyl group

• Finite group generated by reflections with respect to simple roots
• Isomorphic to the quotient of the Lie group by its maximal torus
• Order of the Weyl group related to the dimensions of associated Lie algebras
• Preserves the inner product structure of the root system

Action on root systems

• Permutes the roots of the system
• Preserves the set of positive roots up to sign changes
• Generates all roots from simple roots
• Orbits of Weyl group action correspond to conjugacy classes in Lie groups

Reflection groups

• Weyl groups are examples of finite reflection groups
• Classified by Coxeter-Dynkin diagrams
• Include symmetry groups of regular polytopes
• Study of reflection groups extends beyond Lie theory to geometric group theory

Root space decomposition

• Root space decomposition provides a powerful tool for analyzing the structure of Lie algebras in Non-associative Algebra
• It breaks down complex algebraic structures into simpler, more manageable components

Root space basis

• Decomposes Lie algebra $\mathfrak{g}$ as direct sum $\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha$
• $\mathfrak{h}$ denotes the Cartan subalgebra
• $\mathfrak{g}_\alpha$ represents the root space corresponding to root $\alpha$
• Dimension of $\mathfrak{g}_\alpha$ equals the multiplicity of the root $\alpha$

Cartan subalgebra

• Maximal abelian subalgebra of $\mathfrak{g}$
• Consists of simultaneously diagonalizable elements
• Dimension equals the rank of the Lie algebra
• Generators of Cartan subalgebra form a basis for weight space

Nilpotent subalgebras

• Positive and negative root spaces form nilpotent subalgebras
• $\mathfrak{n}^+ = \bigoplus_{\alpha > 0} \mathfrak{g}_\alpha$ and $\mathfrak{n}^- = \bigoplus_{\alpha < 0} \mathfrak{g}_\alpha$
• Borel subalgebra $\mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}^+$
• Triangular decomposition $\mathfrak{g} = \mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$

Highest weight theory

• Highest weight theory forms a cornerstone in the representation theory of Lie algebras within Non-associative Algebra
• It provides a systematic way to construct and classify irreducible representations

Dominant weights

• Weights $\lambda$ satisfying $(\lambda, \alpha_i) \geq 0$ for all simple roots $\alpha_i$
• Correspond to highest weights of irreducible finite-dimensional representations
• Form a cone in the weight lattice
• Weyl character formula expresses characters of irreducible representations in terms of dominant weights

Fundamental weights

• Dual basis to the simple coroots
• Generate the weight lattice
• Every dominant weight expressed as non-negative integer combination of fundamental weights
• Fundamental representations correspond to fundamental weights

Character formulas

• Weyl character formula expresses characters of irreducible representations
• Freudenthal formula computes weight multiplicities recursively
• Kostant multiplicity formula gives combinatorial expression for weight multiplicities
• Littelmann path model provides geometric interpretation of characters and crystal bases

Root systems in physics

• Root systems and their associated algebraic structures find numerous applications in theoretical physics, showcasing the relevance of Non-associative Algebra in understanding fundamental physical phenomena
• They provide a mathematical framework for describing symmetries and interactions in various physical theories

Gauge theories

• Root systems classify gauge groups in particle physics
• $SU(3)$ (corresponding to $A_2$ root system) describes strong interactions in quantum chromodynamics
• Electroweak theory based on $SU(2) \times U(1)$ (related to $A_1$ and $U(1)$ factors)
• Grand Unified Theories utilize larger gauge groups ($SU(5)$, $SO(10)$, $E_6$) corresponding to more complex root systems

String theory

• Exceptional Lie groups ($E_8 \times E_8$ and $SO(32)$) appear in heterotic string theory
• Compactifications of string theory related to ADE classifications
• Calabi-Yau manifolds and their singularities connected to root systems
• Dualities in string theory often described using Lie algebraic techniques

Conformal field theory

• Kac-Moody algebras (infinite-dimensional generalizations of semisimple Lie algebras) play crucial role in conformal field theory
• Virasoro algebra related to diffeomorphisms of the circle
• WZW models based on Lie groups described by root systems
• Fusion rules in rational conformal field theories connected to representation theory of affine Lie algebras