nLab simple object

Contents

Contents

Definition

Definition

An object XX in a category CC with a terminal object 11 is simple if there are precisely two quotient objects of XX, namely 11 and XX.

Remark

If CC is abelian, then the terminal object is a zero object and we may use subobjects in place of quotient objects in the definition, and this is more common; the result is the same.

Remark

The terminal object itself is not simple, as it has only one quotient object. It is too simple to be simple.

Remark

In constructive mathematics, we want to phrase the definition as: a quotient object of XX is XX if and only if it is not 11.

Definition

An object which is a direct sum of simple objects is called a semisimple object.

Properties

In an abelian category

Proposition

(Schur's lemma)

In an abelian category CC, every morphism between simple objects is either a zero morphism or an isomorphism, and the endomorphism algebra of any simple object is a division ring.

If CC is also enriched in finite-dimensional vector spaces over an algebraically closed field kk, then hom(X,Y)hom(X, Y) has dimension 00 or 11 for any pair of simple objects XX and YY. In this case the endomorphism algebra of any simple object is kk.

Proof

Suppose XX and YY are simple objects in an abelian category CC. If f:XYf \colon X \to Y is any morphism then the kernel of ff must be either 00 or XX, while its cokernel must be 00 or YY. If the kernel and cokernel are both 00, ff is an isomorphism; otherwise f=0f = 0. It follows that every element of the endomorphism ring End(X)End(X) of a simple object XX is zero or invertible, so End(X)End(X) is a division ring.

Next suppose CC is enriched over finite-dimensional vector spaces over an algebraically closed field kk. In this case End(X)End(X) is a finite-dimensional division algebra over kk, but any such algebra is isomorphic to kk. Post-composing with an isomorphism f:XYf: X \to Y gives a vector space isomorphism End(X)hom(X,Y)End(X) \to hom(X,Y), so if such an isomorphism ff exists then hom(X,Y)hom(X,Y) is one-dimensional. If no such isomorphism exists all the morphisms from XX to YY are zero.

Remark

If an abelian category is enriched over finite-dimensional vector spaces over a field kk that is not algebraically closed, the endomorphism algebra of a simple object can be a division algebra other than kk. For example consider k=k = \mathbb{R}. In the category of real representations of the Lie group SO(2), the usual action of rotations on the plane gives a simple object (that is, irreducible representation) XX with End(X)End(X) \cong \mathbb{C}. In the category of real representations of Sp(1))?, the action of this group by right multiplication on the quaternions gives a simple object XX with End(X)End(X) \cong \mathbb{H}.

Examples

Last revised on December 13, 2023 at 06:45:13. See the history of this page for a list of all contributions to it.