Glossary

11.3 Highest weight theory

2 min read•july 25, 2024

Highest weight theory is a powerful tool in representation theory, focusing on special vectors in Lie algebra representations. It helps classify and construct irreducible representations, making complex algebraic structures more manageable.

The theory introduces highest weight vectors, unique irreducible representations, and Verma modules. These concepts are crucial for understanding finite-dimensional representations and their characters, connecting abstract algebra to practical applications in physics and beyond.

Highest Weight Theory in Representation Theory

Concept of highest weight vectors

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  • vv in representation VV of Lie algebra g\mathfrak{g} annihilated by all positive root vectors and eigenvector of elements
  • Highest weight λ\lambda linear functional on Cartan subalgebra corresponds to eigenvalues of highest weight vector
  • Highest weight representation generated by highest weight vector characterized by highest weight λ\lambda (simple roots, Dynkin diagrams)

Uniqueness of irreducible representations

  • states each highest weight λ\lambda has unique irreducible highest weight representation L(λ)L(\lambda)
  • Proof constructs U(g)U(\mathfrak{g}), defines IλI_\lambda annihilating highest weight vector
  • Shows L(λ)≅U(g)/IλL(\lambda) \cong U(\mathfrak{g})/I_\lambda and demonstrates for any other irreducible highest weight representation with weight λ\lambda

Construction of Verma modules

  • M(λ)M(\lambda) induced representation from one-dimensional representation of Borel subalgebra
  • Universal property any highest weight module of weight λ\lambda quotient of M(λ)M(\lambda)
  • Infinite-dimensional for non-trivial Lie algebras contains unique
  • Quotient by maximal submodule yields L(λ)L(\lambda)
  • application provides basis for Verma modules using PBW basis of U(n−)U(\mathfrak{n}^-) (monomials, ordered basis)

Highest weight vs semisimple representations

  • Finite-dimensional irreducible representations are highest weight representations
  • correspond to finite-dimensional irreducible representations
  • expresses character of irreducible finite-dimensional representations using highest weight
  • Category O\mathcal{O} includes Verma modules and finite-dimensional representations studied using highest weight theory
  • Tensor product of highest weight representations decomposed using (Young tableaux, weight diagrams)
  • Highest weight representations of Lie algebras correspond to representations of associated Lie group (compact Lie groups, maximal torus)

Key Terms to Review (28)

Andrei Zelevinsky: Andrei Zelevinsky is a prominent mathematician known for his contributions to representation theory, particularly in the study of highest weight representations and their applications. His work has been instrumental in advancing the understanding of the structure and classification of representations of semisimple Lie algebras, connecting algebraic concepts to geometric and combinatorial perspectives.
Borel-Weil Theorem: The Borel-Weil Theorem establishes a deep connection between algebraic geometry and representation theory by describing the relationship between line bundles on projective varieties and representations of algebraic groups. Specifically, it provides a way to construct representations of a semisimple Lie group from the cohomology of line bundles over its flag varieties, which are important geometric objects in the study of these groups.
Cartan subalgebra: A Cartan subalgebra is a maximal abelian subalgebra of a Lie algebra consisting of semisimple elements, which is fundamental in understanding the structure and representation of the algebra. It plays a critical role in the classification of Lie algebras and helps to define weights and irreducible representations, linking directly to the theory of highest weights and the classification of irreducible representations.
Category O: Category O refers to a specific type of representation in the context of highest weight theory, primarily associated with the representation theory of semisimple Lie algebras. It consists of representations that have a highest weight and exhibit certain desirable properties, such as being finite-dimensional and integrable. This concept plays a crucial role in understanding the structure and classification of representations, particularly in relation to their weights and decompositions.
Characteristic function: In representation theory, a characteristic function is a tool used to represent a mathematical object, typically a group or an algebra, in terms of its action on a vector space. It plays a vital role in studying representations by encoding the information of how elements of the group interact with the space. This function can also help identify properties of representations, such as irreducibility and decomposability, which are essential for understanding highest weight theory.
Dominant integral weights: Dominant integral weights are specific types of weights in representation theory that are used to describe the representations of Lie algebras and algebraic groups. These weights are characterized by being non-negative linear combinations of simple roots, making them essential for identifying highest weight representations, which are vital in understanding the structure and classification of representations.
Dominant weight: A dominant weight is a specific type of weight associated with a representation of a semisimple Lie algebra or a reductive group, typically corresponding to the highest weight of a representation. This concept plays a crucial role in understanding the structure and classification of representations, especially in highest weight theory, where dominant weights help identify irreducible representations and their properties.
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.
Generic representation: Generic representation refers to a specific class of representations of a group that can be parameterized by weights, typically within the context of highest weight theory. It allows us to study the structure and behavior of representations that exhibit certain common features, making it easier to analyze and classify them within the larger framework of representation theory.
Highest weight vector: A highest weight vector is a special type of vector in the context of representation theory, particularly within the study of semisimple Lie algebras and their representations. It is defined as a non-zero vector in a representation space that is an eigenvector for all the elements of a Cartan subalgebra, with the property that its eigenvalues are the largest possible among all vectors in that representation, termed 'weights'. This concept is central to highest weight theory, which categorizes representations based on these highest weight vectors and their associated weights.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces, that respects the operations defined on those structures. This concept is crucial for understanding how different algebraic entities relate to each other, especially when exploring their properties and behaviors under transformations.
Induction: Induction is a method of reasoning that establishes the truth of a statement by proving it for a base case and then showing that if it holds for an arbitrary case, it also holds for the next case. This technique is especially useful in areas like representation theory, where it helps in constructing representations and understanding their properties, connecting foundational concepts with complex applications in group theory and algebra.
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.
Isomorphism: Isomorphism refers to a structural-preserving mapping between two algebraic structures, such as groups, that allows for the preservation of operations and relationships. This concept is vital in understanding how different mathematical systems can be equivalent in structure, enabling the classification of groups and representations based on their essential properties.
Left ideal: A left ideal is a subset of a ring that is closed under addition and absorbs multiplication from the left by elements of the ring. In other words, if 'I' is a left ideal of a ring 'R', then for any 'a' in 'I' and 'r' in 'R', the product 'ra' is also in 'I'. This concept plays a crucial role in representation theory, particularly in understanding modules and the structure of representations associated with Lie algebras.
Littlewood-Richardson Rule: The Littlewood-Richardson Rule provides a combinatorial method to compute the coefficients appearing in the expansion of the product of two Schur functions in terms of a basis of Schur functions. This rule is crucial for understanding how representations of certain groups can be decomposed, particularly in contexts involving tensor products, symmetric and alternating groups, and highest weight theory.
Maximal proper submodule: A maximal proper submodule is a submodule of a module that is not equal to the entire module and is not contained in any other proper submodule. It represents the largest possible proper submodule, providing a crucial structure in the study of modules and representation theory. Understanding maximal proper submodules helps identify important elements in the analysis of modules, particularly in relation to weight representations and highest weight theory.
PBW Theorem: The PBW Theorem, or Poincaré-Birkhoff-Witt Theorem, establishes a fundamental isomorphism between the universal enveloping algebra of a Lie algebra and a polynomial algebra generated by the elements of a basis of that Lie algebra. This theorem is essential for understanding the structure of representations and the construction of modules over Lie algebras, especially within the framework of highest weight theory.
Restriction: Restriction refers to the process of limiting a representation of a group to a smaller subgroup. This concept allows us to study how representations behave when we focus on just a part of the group, providing insight into the relationship between different representations and their induced counterparts.
Typical Representation: Typical representation refers to a specific type of representation of a group, often used in the context of highest weight theory, where representations are characterized by their highest weight vectors. This concept is crucial because it allows the classification and understanding of representations of semisimple Lie algebras and groups, revealing important structural information about their actions on various mathematical objects.
Unique irreducible representation: A unique irreducible representation is a type of representation of a group that cannot be decomposed into simpler representations, and is unique up to isomorphism. This concept is central to understanding the structure of representations in the context of Lie algebras and groups, particularly when analyzing how different representations relate to each other and how they can be classified using highest weights.
Uniqueness Theorem: The uniqueness theorem in representation theory states that, under certain conditions, a highest weight module for a semisimple Lie algebra is uniquely determined by its highest weight. This theorem is fundamental as it assures that for a given highest weight, there is essentially one way to construct the corresponding representation, simplifying the study of these modules and their interrelations.
Universal Enveloping Algebra: The universal enveloping algebra of a Lie algebra is a fundamental construction that allows one to represent elements of the Lie algebra as operators on a vector space, effectively bridging the gap between Lie theory and representation theory. This algebra is key in studying representations of Lie algebras, as it captures their structure in a way that facilitates the exploration of their modules. The universal enveloping algebra provides a way to express irreducible representations and highest weight theory, making it a cornerstone in the study of symmetries in mathematics and physics.
Verma module: A Verma module is a specific type of module constructed from a highest weight representation of a Lie algebra, defined using the universal enveloping algebra. It is generated by the action of the algebra on a highest weight vector, which makes it essential in understanding representations of semisimple Lie algebras. These modules are crucial for studying irreducible representations, as they provide a systematic way to build them and analyze their properties through highest weight theory.
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 character formula: The Weyl character formula is a mathematical expression that provides a way to compute the characters of irreducible representations of a semisimple Lie algebra in terms of its highest weights. This formula highlights the deep relationship between representation theory and geometry, specifically through the roots and weights of the Lie algebra. By utilizing this formula, one can systematically classify irreducible representations and understand their structure in terms of the highest weight theory.
Weyl's Dimension Formula: Weyl's Dimension Formula is a crucial result in representation theory that gives a way to calculate the dimensions of irreducible representations of a semisimple Lie algebra. The formula relates the highest weight of a representation to its dimension, reflecting the structure of the underlying algebra. Understanding this formula is essential for exploring the properties of highest weight representations and their applications in various areas, such as geometry and mathematical physics.
William J. Adams: William J. Adams is a prominent mathematician known for his significant contributions to representation theory, particularly in the area of highest weight theory. His work has helped establish a framework for understanding representations of Lie algebras and algebraic groups, where highest weights play a crucial role in classifying representations and their properties.
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