Contents

category theory

# Contents

## Definition

###### Definition

An object $X$ in a category $C$ with a terminal object $1$ is simple if there are precisely two quotient objects of $X$, namely $1$ and $X$.

###### Remark

If $C$ 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 follows: A quotient object of $X$ is $X$ if and only if it is not $1$.

###### 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 $C$, 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 $C$ is also enriched in finite-dimensional vector spaces over an algebraically closed field $k$, then $hom(X, Y)$ has dimension $0$ or $1$ for any pair of simple objects $X$ and $Y$. In this case the endomorphism algebra of any simple object is $k$.

###### Proof

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

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

###### Remark

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

## Examples

• In a top bounded partial order its simple objects are its coatoms.

• In the category Vect of vector spaces over some field $k$, the simple objects are precisely the lines: the $1$-dimensional vector spaces, i.e. $k$ itself, up to isomorphism.

• A simple group is a simple object in Grp. (Here it is important to use quotient objects instead of subobjects.)

• For $G$ a group and $Rep(G)$ its category of representations, the simple objects are the irreducible representations.

• A simple ring is a simple object in Ring. Equivalently, it is a ring $R$ that is simple in its category of bimodules.

• A simple Lie algebra is a simple object in LieAlg that (by conventional fiat) is not abelian. As an abelian Lie algebra is simply a vector space, the only simple object of $Lie Alg$ that is not accepted as a simple Lie algebra is the $1$-dimensional Lie algebra.

Last revised on February 24, 2024 at 04:28:18. See the history of this page for a list of all contributions to it.