Weight (representation theory)

From Wikipedia, the free encyclopedia

You have new messages (last change).

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.

Contents

[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 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:

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 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 \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 Hγ 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.
Retrieved from "http://en.wikipedia.org../../../w/e/i/Weight_%28representation_theory%29.html"