Algebraic character
An algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is anal
Gell-Mann matrices
The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.They spa
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
Pauli–Lubanski pseudovector
In physics, the Pauli–Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang
Polarization (Lie algebra)
In representation theory, polarization is the maximal totally isotropic subspace of a certain skew-symmetric bilinear form on a Lie algebra. The notion of polarization plays an important role in const
Representation of a Lie superalgebra
In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z2-graded vector space V, such that if A and B are any two pure eleme
Category O
In the representation theory of semisimple Lie algebras, Category O (or category ) is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms o
Lie algebra representation
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector
Representation up to homotopy
A Representation up to homotopy has several meanings. One of the earliest appeared in the `physical' context of constrained Hamiltonian systems. The essential idea is lifting a non-representation on a
Kostant partition function
In representation theory, a branch of mathematics, the Kostant partition function, introduced by Bertram Kostant , of a root system is the number of ways one can represent a vector (weight) as a non-n
Harish-Chandra isomorphism
In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra,is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center Z(U(g)) o
Affine representation
In mathematics, an affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). S
Clebsch–Gordan coefficients for SU(3)
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decompo
Affine action
Let be the Weyl group of a semisimple Lie algebra (associate to fixed choice of a Cartan subalgebra ). Assume that a set of simple roots in is chosen. The affine action (also called the dot action) of
Dynkin index
In mathematics, the Dynkin index of a representation with highest weight of a compact simple Lie algebra that has a highest weight is defined by evaluated in the representation . Here are the matrices
Engel's theorem
In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra is a nilpotent Lie algebra if and only if for each , the adjoint map given by , is a nil
Generalized Verma module
In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in
Universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal env
Isotypic component
The isotypic component of weight of a Lie algebra module is the sum of all submodules which are isomorphic to the highest weight module with weight .
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 representa
Kazhdan–Lusztig polynomial
In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial is a member of a family of integral polynomials introduced by David Kazhdan and George Lusztig. They are indexed by pai
Jordan map
In theoretical physics, the Jordan map, often also called the Jordan–Schwinger map is a map from matrices Mij to bilinear expressions of quantum oscillators which expedites computation of representati