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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 Operator s on a finite-dimensional complex Vector Space ''V'' there exists a basis of ''V'' consisting of simultaneous Eigenvector s of all elements of ''S''. The "generalized eigenvalue" of such an eigenvector is called ''weight''.


DEFINITION OF A WEIGHT



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} ightarrow \mathbb{C}. A ''weight space'' V_\lambda\subset V of weight \lambda is defined by

:V_\lambda:=\{v\in V; orall 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_\mu



Weight of a representation of a Lie group


Let G be a Lie Group , H a Maximal Commutative Lie Subgroup .
  • .


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

:V_\lambda:=\{v\in V; orall h\in H\quad h\cdot v=\exp(\lambda)(h) v\}
where \exp(\lambda) is the character so that \lambda=d(\exp(\lambda))
(sometimes, \exp(\lambda)(h) is denoted by h^\lambda).

Elements of this weight space are called ''weight vectors''.

We say that \lambda is a weight of the representation V, if the weight space V_\lambda 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_\mu


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


PROPERTIES OF WEIGHTS


Suppose that for the Lie algebra \mathfrak{g} and the Cartan subalgebra \mathfrak{h}, a set of Positive Root s \Phi^+ is chosen. This is equivalent to the choice of a set of Simple Root s. We will assume that the Lie algebra resp. the Lie group in question are Semisimple .


Ordering on the space of weights


  • be the real subspace of \mathfrak{h}^--- (if it is complex) generated by the roots of \mathfrak{g}.


  • .


The first one is the partial ordering

:\mu\leq\lambda if and only if \lambda-\mu is a sum of Positive Root s 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 \beta(f)>0 for each Positive Root \beta.


Fundamental weight


  • dual to the set of Simple Coroot s H_{\alpha_1}, \ldots, H_{\alpha_n}.



Integral weight


  • is ''integral'' (or \mathfrak{g}-integral), if \lambda(H_\gamma)\in\Z for each Coroot H_\gamma such that \gamma is a positive root. Equivalently, \lambda is integral, if it is an integral combination of the Fundamental Weight s.

  • called ''weight lattice'' for \mathfrak{g}, denoted by P(\mathfrak{g}).


A weights \lambda 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.


Dominant weight


A weight \lambda is ''dominant'', if \lambda(H_\gamma)\geq 0 for each Coroot H_\gamma such that \gamma is a positive root. Equivalently, \lambda is dominant, if it is a non-negative linear combination of the Fundamental Weight s.

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 \lambda of a representation V is called ''highest weight'', if no other weight of V is larger than \lambda (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 \lambda 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




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.