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 Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...

\lambda

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

\lambda

Definition

* A finite-dimensional

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

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: \mathfr ...

\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 additio ...

) can be decomposed into

irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

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 multiplic ...

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

, where

M_\lambda

denotes the

highest weight module 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 ...

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