Glossary

11.2 Weights and root systems

2 min read•july 25, 2024

decomposition breaks down representations into eigenspaces of the Cartan subalgebra. This reveals the structure of representations and helps identify common eigenvectors, which is crucial for understanding Lie algebras.

Root systems emerge from the adjoint representation and provide a framework for classifying semisimple Lie algebras. The root space decomposition expresses Lie algebras in terms of these roots, offering insights into their structure and properties.

Weight Space Decomposition and Root Systems

Weight space decomposition of representations

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  • Weight space decomposition breaks down representation into eigenspaces of Cartan subalgebra V=⨁λ∈h∗VλV = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda revealing structure
  • Weight spaces VλV_\lambda comprise vectors vv satisfying h.v=λ(h)vh.v = \lambda(h)v for all h∈hh \in \mathfrak{h}, where λ\lambda denotes weight
  • Computation involves diagonalizing Cartan subalgebra action and identifying common eigenvectors (Jordan canonical form)
  • Multiplicity of weight measured by dimension of corresponding weight space informs representation structure (highest weight representations)

Roots and root space decomposition

  • Roots emerge as non-zero weights in adjoint representation, elements α∈h∗\alpha \in \mathfrak{h}^* with gα≠0\mathfrak{g}_\alpha \neq 0
  • Root space decomposition expresses Lie algebra as g=h⊕⨁α∈Φgα\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, Φ\Phi denoting root set
  • Root spaces gα\mathfrak{g}_\alpha contain elements xx satisfying [h,x]=α(h)x[h,x] = \alpha(h)x for all h∈hh \in \mathfrak{h}
  • Semisimple Lie algebras feature one-dimensional root spaces, property crucial for classification
  • Root spaces exhibit multiplication property [gα,gβ]⊆gα+β[\mathfrak{g}_\alpha, \mathfrak{g}_\beta] \subseteq \mathfrak{g}_{\alpha+\beta}, key for understanding Lie algebra structure

Relationships and Classification of Root Systems

Relationship of weights vs roots

  • Roots function as special weights arising in adjoint representation (sl(2,C))
  • Root lattice forms sublattice within larger , crucial for representation theory
  • Highest weight theory characterizes irreducible representations by unique highest weight (Verma modules)
  • Weyl character formula expresses characters using roots, connecting weights and roots
  • Root strings organize weights of representation along root directions, revealing structure (sl(3,C))

Classification of root systems

  • Irreducible root systems classified as AnA_n, BnB_n, CnC_n, DnD_n, E6E_6, E7E_7, E8E_8, F4F_4, G2G_2, each corresponding to simple Lie algebra
  • Root systems exhibit key properties finite, span space, closed under reflection, satisfy integrality condition
  • Weyl group generated by root reflections acts on weights and roots, crucial for representation theory
  • Dynkin diagrams visually encode structure, particularly angles between (ADE classification)
  • Root systems determine semisimple Lie algebra structure, classify irreducible representations, provide framework for study
  • Cartan matrix encodes simple root information, uniquely determines Lie algebra up to isomorphism

Key Terms to Review (15)

Bernhard Riemann: Bernhard Riemann was a German mathematician renowned for his contributions to analysis, differential geometry, and number theory. His work laid the foundation for many concepts in modern mathematics, particularly in understanding complex functions and the geometry of surfaces. Riemann’s theories have had significant implications in various fields, influencing the study of weights and root systems, as they relate to symmetry and mathematical structures.
Coxeter diagram: A Coxeter diagram is a graphical representation of a Coxeter group, which is a type of group defined by reflections in Euclidean space. The diagram consists of nodes representing generators of the group and edges indicating the relations between these generators, with specific labels on the edges that provide information about the angles between the corresponding reflections. This visualization helps in understanding the underlying geometric structures and symmetries associated with root systems and weights.
Dynkin Diagram: A Dynkin diagram is a graphical representation that encodes the structure of a root system in a finite-dimensional Lie algebra. Each node of the diagram represents a simple root, while the edges signify the angles between these roots, providing crucial insight into the symmetry and relationships within the root system. This diagram is essential for classifying Lie algebras and understanding their representations, as it directly connects to weights and the representation theory associated with these algebraic structures.
Finite-dimensional representation: A finite-dimensional representation is a way of expressing a group or algebra as linear transformations on a finite-dimensional vector space. This concept is crucial for understanding how algebraic structures can act on spaces and provides insight into their properties, such as reducibility and irreducibility. It connects to various areas like the interaction between different representations through functors, the decomposition of tensor products, and classifications of Lie algebras and their representations.
Irreducible Representation: An irreducible representation is a linear representation of a group that cannot be decomposed into smaller, non-trivial representations. This concept is crucial in understanding how groups act on vector spaces, as irreducible representations form the building blocks from which all representations can be constructed, similar to prime numbers in arithmetic.
Kac-moody algebras: Kac-Moody algebras are a class of infinite-dimensional Lie algebras that generalize finite-dimensional semisimple Lie algebras. They are defined using a generalized Cartan matrix and play a crucial role in various areas of mathematics and theoretical physics, especially in the study of representations, root systems, and quantum groups.
Nicolas Bourbaki: Nicolas Bourbaki is a collective pseudonym used by a group of primarily French mathematicians, founded in the 1930s, who aimed to reformulate mathematics on a more rigorous and abstract basis. They are particularly known for their work in algebra, topology, and representation theory, focusing on the axiomatic approach and the structure of mathematical concepts. Their contributions have significantly influenced modern mathematics and how mathematical disciplines are taught and understood.
Root multiplicity: Root multiplicity refers to the number of times a particular root appears in the context of a root system, which is fundamental in the study of Lie algebras and representation theory. It reflects the geometric and algebraic significance of the roots and their corresponding weights, affecting the structure and representation of algebraic groups. Understanding root multiplicity helps in analyzing how these roots interact within the weight space and contributes to classifying representations based on their symmetries.
Root system: A root system is a mathematical structure that arises in the context of Lie algebras and reflects symmetries in algebraic and geometric contexts. It consists of a finite set of vectors, called roots, in a Euclidean space, which satisfy specific properties regarding their relationships and can be used to study the representation theory of algebraic groups. This concept plays a crucial role in understanding the weights associated with representations and helps classify irreducible representations through their connections to these roots.
Simple Roots: Simple roots are the basis elements of a root system that cannot be expressed as a linear combination of other roots. These roots play a crucial role in defining the structure of a root system and are used to generate all the other roots in the system through their integer combinations. Understanding simple roots is key to analyzing the representation theory of Lie algebras, as they help in classifying representations and understanding symmetries.
Unitary Representation: A unitary representation is a way of representing a group as a set of linear transformations on a complex inner product space, such that the group elements preserve the inner product. This concept is significant because it provides a framework for studying groups through their actions on vector spaces, connecting directly to matrix representations, tensor products, and the structure of weights and root systems in representation theory.
Weight lattice: A weight lattice is a mathematical structure that organizes the weights of a representation in relation to the root system of a Lie algebra. It provides a way to understand how weights are distributed and how they relate to each other, which is crucial for analyzing the representations of semisimple Lie algebras. The weight lattice plays an essential role in classification by linking weights to irreducible representations and their dimensions.
Weight multiplicity: Weight multiplicity refers to the number of times a particular weight appears in the decomposition of a representation of a Lie algebra. It is a crucial concept in understanding how representations can be categorized based on their weights, especially in relation to root systems. This term highlights the relationship between weights and the structure of representations, playing a significant role in characterizing the behavior of these representations under group actions.
Weight space: Weight space is a mathematical concept used in representation theory that describes the set of all weights associated with a representation of a Lie algebra or a Lie group. Each weight corresponds to an eigenvalue of a Cartan subalgebra acting on a vector in the representation, and weight spaces are critical for understanding the structure of representations, particularly when analyzing highest weights and their roles in classification.
Weyl's Character Formula: Weyl's Character Formula provides a powerful tool for calculating the characters of representations of compact Lie groups using weights and root systems. The formula relates the character of a representation to its weights and their multiplicities, giving a deep connection between representation theory and the underlying structure of Lie algebras and root systems. It plays a crucial role in understanding how representations behave under group actions.
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