- Schur's lemma
In
mathematics , Schur's lemma is an elementary but extremely useful statement inrepresentation theory of groups andalgebra s. In the group case it says that that if "M" and "N" are two finite-dimensionalirreducible representation s of a group "G" and "φ" is linear map from "M" to "N" that commutes with the action of the group, then either "φ" is invertible, or "φ" = 0. An important special case occurs when "M" = "N" and "φ" is a self-map. The lemma is named afterIssai Schur who used it to proveSchur orthogonality relations and develop the basics of therepresentation theory of finite groups . Schur's lemma admits generalisations toLie group s andLie algebra s, the most common of which is due toJacques Dixmier .Formulation in the language of modules
If "M" and "N" are two
simple module s over a ring "R", then anyhomomorphism "f": "M" → "N" of "R"-modules is either invertible or zero. In particular, theendomorphism ring of a simple module is adivision ring .The condition that "f" is a module homomorphism means that
: for all in and in
The group version is a special case of the module version, since any representation of a group "G" can equivalently be viewed as a module over the
group ring of "G".Schur's lemma is frequently applied in the following particular case. Suppose that "R" is an
algebra over the field C ofcomplex numbers and "M" = "N" is a finite-dimensional module over "R". Then Schur's lemma says that any endomorphism of the module "M" is either given by a multiplication by a non-zero scalar, or is identically zero. This can be expressed by saying that theendomorphism ring of the module "M" is C, that is, "as small as possible". More generally, this results holds for algebras over anyalgebraically closed field and for simple modules that are at most countably-dimensional. When the field is not algebraically closed, the case where the endomorphism ring is as small as possible is of particular interest: A simple module over "k"-algebra is said to be absolutely simple if its endomorphism ring is isomorphic to "k". This is in general stronger than being irreducible over the field "k", and implies the module is irreducible even over the algebraic closure of "k".Matrix form
Let "G" be a complex
matrix group . This means that "G" is a set of square matrices of a given order "n" with complex entries and "G" is closed undermatrix multiplication and inversion. Further, suppose that "G" is "irreducible": there is nosubspace "V" other than 0 and the whole space which is invariant under the action of "G". In other words,: if for all in , then either or
Schur's lemma, in the special case of a single representation, says the following. If "A" is a complex matrix of order "n" that
commute s with all matrices from "G" then "A" is ascalar matrix .Generalization to non-simple modules
The one module version of Schur's lemma admits generalizations involving modules "M" that are not necessarily simple. They express relations between the module-theoretic properties of "M" and the properties of the
endomorphism ring of "M".A module is said to be strongly indecomposable if its endomorphism ring is a
local ring . For the important class of modules of finite length, the following properties are equivalent harv|Lam|2001|loc=§19:
* A module "M" is indecomposable;
* "M" is strongly indecomposable;
* Every endomorphism of "M" is either nilpotent or invertible.In general, Schur's lemma cannot be reversed: there exist modules that are not simple, yet their endomorphism algebra is a
division ring . Such modules are necessarily indecomposable, and so cannot exist over semi-simple rings such as the complex group ring of a finite group. However, even over the ring ofinteger s, the module ofrational number s has an endomorphism ring that is a division ring, specifically the field of rational numbers. Even for group rings, there are examples when the characteristic of the field divides the order of the group: theJacobson radical of theprojective cover of the one-dimensional representation of thealternating group on five points over the field with three elements has the field with three elements as its endomorphism ring.References
*David S. Dummit, Richard M. Foote. "Abstract Algebra." 2nd ed., pg. 337.
*Citation | last1=Lam | first1=Tsit-Yuen | title=A First Course in Noncommutative Rings | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-95325-0 | year=2001
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