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Weight is a concept arising often in representation theory of Lie groups and Lie algebras, a branch of mathematics. In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...
In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
The motivation is that, given a set S of complex matrices, each of which is diagonalizable and any two of which commute, it is always possible to diagonalize all the elements of S simultaneously. In basis-free terms, for any set of mutually commuting semisimple operators on a finite-dimensional complex vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. The "generalized eigenvalue" of such an eigenvector is called weight. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...
Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ...
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...
In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...
In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
Definition of a weight
Weight of a representation of a Lie algebra Let be a Lie algebra, a maximal commutative Lie subalgebra consisting of semi-simple elements (sometimes called Cartan subalgebra) and let V be a representation of (sometimes called -module). A weight is any linear map . A weight space of weight λ is defined by In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...
In mathematics, a Cartan subalgebra is a certain kind of subalgebra of a Lie algebra. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
Nonzero elements of this weight space are called weight vectors.
It is well known that if is semisimple and the representation V is finite dimensional, it decomposes as a direct sum of its weight spaces:
Weight of a representation of a Lie group Let G be a Lie group, H a maximal commutative Lie subgroup. Let V be a representation of G (sometimes called G-module). A homomorphism from H into the multiplicative group of complex numbers is called character. A weight is usually defined to be the differential of a character . In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...
In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G. According to Cartans theorem, this is equivalent to H being a closed subset in the topological structure of G. Then the Lie algebra h of H is...
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...
In its simplest form, multiplication is a quick way of adding identical numbers. ...
This picture illustrates how the hours in a clock form a group. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
A weight space of weight λ is defined by
where exp(λ) is the character so that λ = d(exp(λ)) (sometimes, exp(λ)(h) is denoted by hλ).
Elements of this weight space are called weight vectors.
We say that λ is a weight of the representation V, if the weight space Vλ is nonzero.
It is well known that if G is semisimple and the representation V is finite dimensional, it decomposes as a direct sum of its weight spaces:
Clearly, if λ is a weight of the representation V of G, it is also a weight of V as a representation of .
Properties of weights Suppose that for the Lie algebra and the Cartan subalgebra , a set of positive roots Φ + is chosen. This is equivalent to the choice of a set of simple roots. We will assume that the Lie algebra resp. the Lie group in question are semisimple. See also Simple Lie group. ...
In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...
Ordering on the space of weights Let be the real subspace of (if it is complex) generated by the roots of .
There are two concepts how to define an ordering of . Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...
The first one is the partial ordering
- if and only if λ − μ is a sum of positive roots with nonnegative integral coefficients.
The second concept is a total ordering given by an element and - if and only if . Usually, f is chosen so, that β(f) > 0 for each positive root β.
Fundamental weight The fundamental weights are defined by the property that they form a basis of dual to the set of simple coroots .
Integral weight A weight is integral (or -integral), if for each coroot Hγ such that γ is a positive root. Equivalently, λ is integral, if it is an integral combination of the fundamental weights. The set of all -integral weights is a lattice in called weight lattice for , denoted by . Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. ...
A weights λ of the Lie group G is called integral (or G-integral), if for each such that . For G semisimple, the set of all G-integral weights is a sublattice . If G is further simply connected, then . If G is not simply connected, then the lattice P(G) is smaller than and their quotient is isomorphic to the fundamental group of G. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
Dominant weight A weight λ is dominant, if for each coroot Hγ such that γ is a positive root. Equivalently, λ is dominant, if it is a non-negative linear combination of the fundamental weights. Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. ...
The set of all dominant weights is sometimes called the fundamental Weyl chamber.
Sometimes, the term dominant weight is used to denote a dominant (in the above sense) and integral weight.
Highest weight A weight λ of a representation V is called highest weight, if no other weight of V is larger than λ (in the total ordering). Sometimes, it is assumed that a highest weight is a weight, such that all other weights of V are strictly smaller than λ in the partial ordering given above. The term highest weight denotes often the highest weight of a highest weight module.
Similarly, we define the lowest weight.
See also In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ...
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ...
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i. ...
In mathematics, a Cartan subalgebra is a certain kind of subalgebra of a Lie algebra. ...
In mathematics, a fundamental representation is a representation of a mathematical structure, such as a group, that satisfies the following condition: All other irreducible representations of the group can be found in the tensor products of the fundamental representation with many copies of itself. ...
References - Fulton W., Harris J., Representation theory: A first course, Springer, 1991
- Goodmann R., Wallach N. R., Representations and Invariants of the Classical Groups, Cambridge University Press, Cambridge 1998.
- Humphreys J., Introduction to Lie Algebras and Representation Theory, Springer Verlag, 1980.
- Knapp A. W., Lie Groups Beyond an introduction, Second Edition, (2002)
- Roggenkamp K., Stefanescu M., Algebra - Representation Theory, Springer, 2002.
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