The isotypic component of

weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar q ...

\lambda

of a

Lie algebra module In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket ...

is the sum of all submodules which are isomorphic to the highest weight module with weight

\lambda

Definition

* A finite-dimensional module

V

of a

reductive Lie algebra In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: \mathfra ...

\mathfrak

(or of the corresponding

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

) can be decomposed into irreducible submodules :

V = \oplus_^N V_i

. * Each finite-dimensional irreducible representation of

\mathfrak

is uniquely identified (up to isomorphism) by its

highest weight 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 multipli ...

\forall i \in \ \exists \lambda \in P(\mathfrak) : V_i \simeq M_\lambda

, where

M_\lambda

denotes the highest weight module with highest weight

\lambda

. * In the decomposition of

V

, a certain isomorphism class might appear more than once, hence :

V \simeq \oplus_ ( \oplus_^ M_)

. This defines the isotypic component of weight

\lambda

of V:

\lambda(V) := \oplus_^ V_i \simeq \mathbb^ \otimes  M_

where

d_\lambda

is maximal.

References

* * {{Cite journal , doi = 10.1007/s00220-005-1330-9 , last = Heinzner , first = P. , author2=A. Huckleberry , author3=M. R Zirnbauer , title = Symmetry classes of disordered fermions , journal = Communications in Mathematical Physics , volume = 257 , issue = 3 , pages = 725–771 , year = 2005 , arxiv = math-ph/0411040 , bibcode = 2005CMaPh.257..725H Representation theory of Lie algebras

Definition

See also

References