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.

Motivation and general concept

Weights

Given a set "S" of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to diagonalize all of the elements of "S" simultaneously. Equivalently, for any set "S" of mutually commuting semisimple linear transformations of a finite-dimensional vector space "V" there exists a basis of "V" consisting of simultaneous eigenvectors of all elements of "S". The set "S" spans a linear subspace "U" of End("V") and for each eigenvector "v" ∈ "V", there is linear functional on "U" which associates to each element of "U" its eigenvalue on the eigenvector "v". This "generalized eigenvalue" is a prototype for the notion of a weight.

The notion is closely related to the idea of a multiplicative character in group theory, which is a homomorphism "χ" from a group "G" to the multiplicative group of a field F. Thus "χ": "G" → F^× satisfies "χ"("e") = 1 (where "e" is the identity element of "G") and:$chi(gh)\; =\; chi(g)chi(h)$ for all "g", "h" in "G".If "G" acts on a vector space "V" over F and there is a simultaneous eigenspace for every element of "G", then the eigenvalue of each element determines a multiplicative character on "G".

This notion of multiplicative character can be extended to any algebra "A" over F, by replacing "χ": "G" → F^× by a linear map "χ": "A" → F with:$chi(ab)\; =\; chi(a)chi(b)$ for all "a", "b" in "A".If an algebra "A" acts on a vector space "V" over F and there is a simultaneous eigenspace, then there is a corresponding algebra homomorphism from "A" to F assigning to each element of "A" its eigenvalue.

If "A" is a Lie algebra, then the commutativity of the field and the anticommutativity of the Lie bracket imply that "χ"( [a,b] )=0. A weight on a Lie algebra g over a field F is a linear map "λ": g → F with "λ"( [x,y] )=0 for all "x", "y" in g. Any weight on a Lie algebra g vanishes on the derived algebra [g,g] and hence descends to a weight on the abelian Lie algebra g/ [g,g] . Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuting linear transformations.

If "G" is a Lie group or an algebraic group, then a multiplicative character "θ": "G" → F^× induces a weight "χ" = d"θ": g → F on its Lie algebra by differentiation. (For Lie groups, this is differentiation at the identity element of "G", and the algebraic group case is an abstraction using the notion of a derivation.)

Weight space of a representation

Let "V" be a representation of a Lie algebra g over a field F and let "λ" be a weight of g. Then the "weight space" of "V" with weight "λ": g → F is the subspace:$V\_lambda:=\{vin\; V:\; forall\; xiin\; mathfrak\{g\},quad\; xicdot\; v=lambda(xi)v\}.$A "weight of the representation" "V" is a a weight "λ" such that the corresponding weight space is nonzero. Nonzero elements of the weight space are called "weight vectors".

If "V" is the direct sum of its weight spaces:then it is called a called a "weight module".

Similarly, we can define a weight space "V"_"λ" for any representation of a Lie group or an associative algebra.

emisimple Lie algebras

Let g be a Lie algebra, h a maximal commutative Lie subalgebra consisting of semi-simple elements (sometimes called Cartan subalgebra) and let "V" be a finite dimensional representation of g. If g is semisimple, then [g,g] = g and so all weights on g are trivial. However, "V" is, by restriction, a representation of h, and it is well known that "V" is a weight module for h, i.e., equal to the direct sum of its weight spaces. By an abuse of language, the weights of "V" as a representation of h are often called weights of "V" as a representation of g.

Similar definitions apply to a Lie group "G", a maximal commutative Lie subgroup "H" and any representation "V" of "G". Clearly, if "λ" is a weight of the representation "V" of "G", it is also a weight of "V" as a representation of the Lie algebra g of "G".

If "V" is the adjoint representation of g, its weights are called roots, the weight spaces are called root spaces, and weight vectors are sometimes called root vectors.

We now assume that g is semisimple, with a chosen Cartan subalgebra h and corresponding root system. Let us suppose also that a choice of positive roots $Phi^+$ has been fixed. This is equivalent to the choice of a set of simple roots.

Ordering on the space of weights

Let $mathfrak\{h\}\_0^*$ be the real subspace of $mathfrak\{h\}^*$ (if it is complex) generated by the roots of $mathfrak\{g\}$.

There are two concepts how to define an ordering of $mathfrak\{h\}\_0^*$.

The first one is:"μ" ≤ "λ" if and only if "λ" − "μ" is nonnegative linear combination of simple roots.

The second concept is given by an element $finmathfrak\{h\}\_0$ and:"μ" ≤ "λ" if and only if "μ"("f") ≤ "λ"("f").Usually, "f" is chosen so, that "β"("f") > 0 for each positive root "β".

Integral weight

A weight $lambdainmathfrak\{h\}^*$ is "integral" (or $mathfrak\{g\}$-integral), if for each coroot $H\_gamma$ such that $gamma$ is a positive root.

The fundamental weights $varpi\_1,ldots,varpi\_n$ are defined by the property that they form a basis of $mathfrak\{h\}^*$ dual to the set of simple coroots $H\_\{alpha\_1\},\; ldots,\; H\_\{alpha\_n\}$.

Hence "λ" is integral if it is an integral combination of the fundamental weights. The set of all $mathfrak\{g\}$-integral weights is a lattice in $mathfrak\{h\}^*$ called "weight lattice" for $mathfrak\{g\}$, denoted by $P(mathfrak\{g\})$.

A weight "λ" of the Lie group "G" is called integral, if for each $tinmathfrak\{h\}$ such that $exp(t)=1in\; G,,,lambda(t)in\; 2pi\; i\; mathbb\{Z\}$. For "G" semisimple, the set of all "G"-integral weights is a sublattice $P(G)subset\; P(mathfrak\{g\})$. If "G" is simply connected, then $P(G)=P(mathfrak\{g\})$. If "G" is not simply connected, then the lattice $P(G)$ is smaller than $P(mathfrak\{g\})$ and their quotient is isomorphic to the fundamental group of "G".

Dominant weight

A weight "λ" is "dominant", if $lambda(H\_gamma)geq\; 0$ for each coroot $H\_gamma$ such that "γ" is a positive root. Equivalently, "λ" is dominant, if it is a non-negative linear combination of the fundamental weights.

The convex hull of the 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". 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".

The space of all possible weights is a vector space. Let's fix a total ordering of this vector space such that a nonnegative linear combination of positive vectors with at least one nonzero coefficient is another positive vector.

Then, a representation is said to have "highest weight" "λ" if "λ" is a weight and all its other weights are less than "λ".

Similarly, it is said to have "lowest weight" "λ" if "λ" is a weight and all its other weights are greater than it.

A weight vector $v\_lambda\; in\; V$ of weight "λ" is called a "highest weight vector", or "vector of highest weight", if all other weights of "V" are smaller than "λ".

Highest weight module

A representation "V" of $mathfrak\{g\}$ is called "highest weight module", if it is generated by a weight vector $vin\; V$ that is annihilated by the action of all positive root spaces in $mathfrak\{g\}$.

Note that this is something more special than a $mathfrak\{g\}$-module with a highest weight.

Similarly we can define a highest weight module for representation of a Lie group or an associative algebra.

Verma module

For each weight $lambdainmathfrak\{h\}^*$, there exists a unique (up to isomorphism) simple highest weight $mathfrak\{g\}$-module with highest weight "λ", which is denoted "L"("λ").

It can be shown that each highest weight module with highest weight "λ" is a quotient of the Verma module "M"("λ"). This is just a restatement of "universality property" in the definition of a Verma module.

A highest weight modules is a weight module. The weight spaces in a highest weight module are always finite dimensional.

References

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