69 articles
Highest-weight theory
Highest-weight theory is the branch of representation theory classifying representations of semisimple Lie algebras and Lie groups by a single linear functional, the highest weight, attached to each irreducible representation.
History of Kac–Moody algebra theory
The history of Kac–Moody algebra theory traces the infinite-dimensional Lie algebras defined independently by Victor Kac and Robert Moody in 1967–68, through Macdonald's identities and physics applications.
Kac–Moody algebra
A Kac–Moody algebra is a usually infinite-dimensional Lie algebra defined by generators and relations from a generalized Cartan matrix, studied independently by Victor Kac and Robert Moody.
Kac–Moody flag variety
A Kac–Moody flag variety is the homogeneous space of flags attached to a Kac–Moody group, built from a generalized Cartan matrix; in finite type it is the familiar projective variety G/P.
Killing form
The Killing form is a symmetric bilinear form on a Lie algebra, defined as a trace in the adjoint representation, used to test solvability and semisimplicity.
Knizhnik–Zamolodchikov equations
The Knizhnik–Zamolodchikov equations, or KZ equations, are a holonomic system of linear partial differential equations written down by Vladimir Knizhnik and Alexander Zamolodchikov in 1984 for conformal blocks of the Wess–Zumino–Witten model.
Kostant partition function
The Kostant partition function is a representation theory function counting the ways a weight can be written as a sum of positive roots, introduced by Bertram Kostant in 1958.
Langlands dual group
The Langlands dual group of a reductive group G, introduced by Robert Langlands in the late 1960s, is the reductive group obtained by exchanging characters with cocharacters and roots with coroots.
Lie algebra
A Lie algebra is a vector space with a bilinear, antisymmetric Lie bracket satisfying the Jacobi identity, recording the infinitesimal behavior of continuous symmetry groups, named after Sophus Lie.
Lie algebra extension
A Lie algebra extension is a Lie algebra fitting a short exact sequence, with splitting, central extensions, and classification by Lie algebra cohomology, founded by Chevalley and Eilenberg in 1948.
Lie algebra representation
A Lie algebra representation is a way of realizing a Lie algebra as linear operators on a vector space, via a bracket-preserving homomorphism into endomorphisms.
Lie group
A Lie group is a set carrying compatible structures of a group and a smooth manifold, named for Sophus Lie and central to continuous symmetry in mathematics and physics.
Lie superalgebra
A Lie superalgebra is a ℤ₂-graded algebra with even and odd parts whose bracket obeys graded symmetry and Jacobi identities, generalizing a Lie algebra and underlying supersymmetry in physics.
Loop algebra
A loop algebra is an infinite-dimensional Lie algebra formed from a Lie algebra by extending its scalars to Laurent polynomials, and it underlies affine Kac-Moody algebras.
Module over a restricted Lie algebra
A module over a restricted Lie algebra is a representation compatible with the pth power map, organized around reduced enveloping algebras and baby Verma modules in prime characteristic.
Nilpotent Lie algebra
A nilpotent Lie algebra is a Lie algebra in which all iterated brackets of sufficient length equal zero, so it is nearly abelian and underlies nilpotent Lie groups.
Poincaré algebra
The Poincaré algebra is the ten-dimensional Lie algebra of the Poincaré group, the symmetry group of flat spacetime, named for Henri Poincaré, who studied the group in 1905.
Representation theory of Kac–Moody and affine algebras
The representation theory of Kac–Moody and affine algebras studies how infinite-dimensional Lie algebras act on vector spaces; for affine algebras, integrable representations carry a level, and the Weyl–Kac character formula yields number-theoretic identities.
Representation theory of semisimple Lie algebras
The representation theory of semisimple Lie algebras, or Cartan–Weyl theory, classifies how semisimple Lie algebras act linearly on vector spaces, with every irreducible determined by its highest weight.
Representation theory of SL2(R)
The representation theory of SL2(R), the non-compact simple Lie group of 2 × 2 real matrices with determinant 1, studies its unitary representations, classified by Bargmann in 1947 into principal, complementary, and discrete series.
Representation theory of the Lorentz group
The representation theory of the Lorentz group studies how Lorentz transformations act linearly on vector spaces, dividing into non-unitary finite-dimensional representations for classical fields and unitary representations for particles.
Root system
A root system is a finite set of vectors in a Euclidean space closed under perpendicular reflections, encoding the symmetry of semisimple Lie algebras and classified by Dynkin diagrams.
Special linear Lie algebra
The special linear Lie algebra slₙ consists of n×n matrices with trace zero under the commutator bracket, and is the Lie algebra of the special linear group of determinant-1 matrices.
Special unitary group
The special unitary group SU(n), also called the unitary unimodular group, is the group of n × n unitary matrices with determinant 1, of dimension n² − 1.
Theorem of the highest weight
The theorem of the highest weight, or theorem on highest weights, is a classification theorem in representation theory that classifies finite-dimensional irreducible representations of a semisimple Lie algebra by dominant integral weights.
Unitary group
The unitary group U(n) is the group of n × n complex matrices that preserve the Hermitian inner product, with dimension n² as a compact Lie group.
Universal enveloping algebra
The universal enveloping algebra U(g) of a Lie algebra g is the associative algebra in which the Lie bracket becomes a commutator, so Lie algebra representations and modules of U(g) coincide.
Verma module
A Verma module is an infinite-dimensional highest-weight module for a complex semisimple Lie algebra, induced from a Borel subalgebra; Daya-Nand Verma introduced it in 1966.
Vertex operator algebra
A vertex operator algebra, also called a vertex algebra, is an algebraic structure axiomatized by Richard Borcherds in 1986 that encodes the operator products of two-dimensional conformal field theory.
Virasoro algebra
The Virasoro algebra is an infinite-dimensional Lie algebra, the universal central extension of the Witt algebra of vector fields, and the algebraic home of two-dimensional conformal symmetry in conformal field theory and string theory.