additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
An object in a category with a terminal object is simple if there are precisely two quotient objects of , namely and .
If 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.
The terminal object itself is not simple, as it has only one quotient object. It is too simple to be simple.
In constructive mathematics, we want to phrase the definition as follows: A quotient object of is if and only if it is not .
An object which is a direct sum of simple objects is called a semisimple object.
In an abelian category , 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 is also enriched in finite-dimensional vector spaces over an algebraically closed field , then has dimension or for any pair of simple objects and . In this case the endomorphism algebra of any simple object is .
Suppose and are simple objects in an abelian category . If is any morphism then the kernel of must be either or , while its cokernel must be or . If the kernel and cokernel are both , is an isomorphism; otherwise . It follows that every element of the endomorphism ring of a simple object is zero or invertible, so is a division ring.
Next suppose is enriched over finite-dimensional vector spaces over an algebraically closed field . In this case is a finite-dimensional division algebra over , but any such algebra is isomorphic to . Post-composing with an isomorphism gives a vector space isomorphism , so if such an isomorphism exists then is one-dimensional. If no such isomorphism exists all the morphisms from to are zero.
If an abelian category is enriched over finite-dimensional vector spaces over a field that is not algebraically closed, the endomorphism algebra of a simple object can be a division algebra other than . For example consider . 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) with . In the category of real representations of Sp(1))?, the action of this group by right multiplication on the quaternions gives a simple object with .
In a top bounded partial order its simple objects are its coatoms.
In the category Vect of vector spaces over some field , the simple objects are precisely the lines: the -dimensional vector spaces, i.e. 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 a group and 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 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 that is not accepted as a simple Lie algebra is the -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.