Lecture

Singular Vectors in Liouville CFT: Representation Theory Insights

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Description

This lecture discusses singular vectors in the context of Liouville Conformal Field Theory (CFT). The instructor begins by recalling concepts from previous lectures, particularly focusing on the representation theory associated with singular vectors. They explain the significance of singular vectors as axiomatic elements in physics, particularly in the study of representations of the Viasso algebra. The lecture delves into the mathematical framework, including the Hilbert space structure and the Hamiltonian involved in the analysis. The instructor highlights the probabilistic approach to Liouville theory, emphasizing the role of Gaussian fields and their properties. They also discuss the implications of the catch table on the linear independence of singular vectors and the conditions under which they vanish. The lecture concludes with applications of these concepts to Riemann surfaces and the classification of modules, providing a comprehensive overview of the theoretical underpinnings of singular vectors in Liouville CFT.

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Verma module

Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients of them can be used to construct finite-dimensional representations with highest weight , where is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds.

Weight (representation theory)

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).

Hilbert space

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Linear independence

In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

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