Weight (representation theory)

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Weight is a concept arising often in representation theory of Lie groups and Lie algebras, a branch of mathematics.

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.

1 Definition of a weight
- 1.1 Weight of a representation of a Lie algebra
- 1.2 Weight of a representation of a Lie group
2 Properties of weights
3 See also
4 References

[edit] Definition of a weight

[edit] Weight of a representation of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{h}$ a maximal commutative Lie subalgebra consisting of semi-simple elements (sometimes called Cartan subalgebra) and let $V$ be a representation of $\mathfrak{g}$ (sometimes called $\mathfrak{g}$ -module). A weight is any linear map $\lambda: \mathfrak{h}\rightarrow \mathbb{C}$ . A weight space $V_\lambda\subset V$ of weight $λ$ is defined by

$V_\lambda:=\{v\in V; \forall h\in \mathfrak{h}\quad h\cdot v=\lambda(h)v\}$

Nonzero elements of this weight space are called weight vectors.

It is well known that if $\mathfrak{g}$ is semisimple and the representation $V$ is finite dimensional, it decomposes as a direct sum of its weight spaces:

$V=\oplus_{\mu\in\mathfrak{h}^*} V_\mu$

[edit] 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 $\theta: H\rightarrow \mathbb{C}^\times$ from $H$ into the multiplicative group of complex numbers is called character. A weight is usually defined to be the differential of a character $d\theta\in\mathfrak{h}^*$ .

A weight space $V_\lambda\subset V$ of weight $λ$ is defined by

$V_\lambda:=\{v\in V; \forall h\in H\quad h\cdot v=\exp(\lambda)(h) v\}$

where $exp(λ)$ is the character so that $λ = d (exp(λ))$ (sometimes, $exp(λ)(h)$ is denoted by $h λ$ ).

Elements of this weight space are called weight vectors.

We say that $λ$ is a weight of the representation $V$ , if the weight space $V λ$ 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:

$V=\oplus_{\mu\in\mathfrak{h}^*} V_\mu$

Clearly, if $λ$ is a weight of the representation $V$ of $G$ , it is also a weight of $V$ as a representation of $\mathfrak{g}$ .

[edit] Properties of weights

Suppose that for the Lie algebra $\mathfrak{g}$ and the Cartan subalgebra $\mathfrak{h}$ , 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.

[edit] 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 the partial ordering

$\mu\leq\lambda$ 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 $f\in\mathfrak{h}_0$ and

$\mu\leq\lambda$ if and only if $\mu(f)\leq \lambda(f)$ . Usually,

f

is chosen so, that

β(f) > 0

for each positive root

β

[edit] Fundamental weight

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 $H_{\alpha_1}, \ldots, H_{\alpha_n}$ .

[edit] Integral weight

A weight $\lambda\in\mathfrak{h}^*$ is integral (or $\mathfrak{g}$ -integral), if $\lambda(H_\gamma)\in\Z$ for each coroot $H γ$ 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 $\mathfrak{g}$ -integral weights is a lattice in $\mathfrak{h}^*$ called weight lattice for $\mathfrak{g}$ , denoted by $P(\mathfrak{g})$ .

A weights $λ$ of the Lie group $G$ is called integral (or $G$ -integral), if for each $t\in\mathfrak{h}$ such that $\exp(t)=1\in G,\,\,\lambda(t)\in 2\pi i \mathbb{Z}$ . For $G$ semisimple, the set of all $G$ -integral weights is a sublattice $P(G)\subset P(\mathfrak{g})$ . If $G$ is further 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 $G$ .

[edit] Dominant weight

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

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.

[edit] 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.

[edit] See also

[edit] 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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Categories: Lie groups | Representation theory