Semi-simplicity

Semi-simple matrices

A square matrix or, equivalently, a linear operator T on a finite-dimensional vector space V is called semi-simple if every T-invariant subspace has a complementary T-invariant subspace. This is equivalent to the minimal polynomial of T being square-free.

For vector spaces over an algebraically closed field F, semi-simplicity of a matrix is equivalent to diagonalizability. This is because such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant hyperplane, which itself has an eigenvector, and thus by induction is diagonalizable. Conversely, diagonalizable operators are easily seen to be semi-simple, as invariant subspaces are direct sums of eigenspaces, and any eigenbasis for this subspace can be extended to an eigenbasis of the full space.

Semi-simple modules and rings

For a fixed ring R, a nontrivial R-module M is simple, if it has no submodules other than 0 and M. An R-module M is semi-simple if every R-submodule of M is an R-module direct summand of M (the trivial module 0 is semi-simple, but not simple). For an R-module M, M is semi-simple if and only if it is the direct sum of simple modules (the trivial module is the empty direct sum). Finally, R is called a semi-simple ring if it is semi-simple as an R-module. As it turns out, this is equivalent to requiring that any finitely generated R-module M is semi-simple.

Examples of semi-simple rings include fields and, more generally, finite direct products of fields. For a finite group G Maschke's theorem asserts that the group ring R[G] over some ring R is semi-simple if and only if R is semi-simple and |G| is invertible in R. Since the theory of modules of R[G] is the same as the representation theory of G on R-modules, this fact is an important dichotomy, which causes modular representation theory, i.e., the case when |G| does divide the characteristic of R to be more difficult than the case when |G| does not divide the characteristic, in particular if R is a field of characteristic zero. By the Artin–Wedderburn theorem, a unital Artinian ring R is semisimple if and only if it is (isomorphic to) ${\displaystyle M_{n_1}(D_1)\times M_{n_2}(D_2)\times \cdots \times M_{n_r}(D_r)}$, where each ${\displaystyle D_i}$ is a division ring and ${\displaystyle M_n(D)}$ is the ring of n-by-n matrices with entries in D.

An operator T is semi-simple in the sense above if and only if the subalgebra ${\displaystyle F[T]\subseteq {\mathrm{End}}_F(V)}$ generated by the powers (i.e., iterations) of T inside the ring of endomorphisms of V is semi-simple.

As indicated above, the theory of semi-simple rings is much more easy than the one of general rings. For example, any short exact sequence

${\displaystyle 0\to M^{\prime }\to M\to M^{″}\to 0}$

of modules over a semi-simple ring must split, i.e., ${\displaystyle M\cong M^{\prime }\oplus M^{″}}$. From the point of view of homological algebra, this means that there are no non-trivial extensions. The ring Z of integers is not semi-simple: Z is not the direct sum of nZ and Z/n.

Semi-simple categories

Many of the above notions of semi-simplicity are recovered by the concept of a semi-simple category C. Briefly, a category is a collection of objects and maps between such objects, the idea being that the maps between the objects preserve some structure inherent in these objects. For example, R-modules and R-linear maps between them form a category, for any ring R.

An abelian category C is called semi-simple if there is a collection of simple objects ${\displaystyle X_{\alpha }\in C}$, i.e., ones with no subobject other than the zero object 0 and ${\displaystyle X_{\alpha }}$ itself, such that any object X is the direct sum (i.e., coproduct or, equivalently, product) of finitely many simple objects. It follows from Schur's lemma that the endomorphism ring

${\displaystyle {\mathrm{End}}_C(X)={\mathrm{Hom}}_C(X,X)}$

in a semi-simple category is a product of matrix rings over division rings, i.e., semi-simple.

Moreover, a ring R is semi-simple if and only if the category of finitely generated R-modules is semisimple.

An example from Hodge theory is the category of polarizable pure Hodge structures, i.e., pure Hodge structures equipped with a suitable positive definite bilinear form. The presence of this so-called polarization causes the category of polarizable Hodge structures to be semi-simple. Another example from algebraic geometry is the category of pure motives of smooth projective varieties over a field k ${\displaystyle \mathrm{Mot}(k{)}_{\sim }}$ modulo an adequate equivalence relation ${\displaystyle \sim }$. As was conjectured by Grothendieck and shown by Jannsen, this category is semi-simple if and only if the equivalence relation is numerical equivalence. This fact is a conceptual cornerstone in the theory of motives.

Semisimple abelian categories also arise from a combination of a t-structure and a (suitably related) weight structure on a triangulated category.

Semi-simplicity in representation theory

One can ask whether the category of finite-dimensional representations of a group or a Lie algebra is semisimple, that is, whether every finite-dimensional representation decomposes as a direct sum of irreducible representations. The answer, in general, is no. For example, the representation of ${\displaystyle \mathbb{R}}$ given by

${\displaystyle \mathrm{\Pi }(x)=\left(\begin{array}{cc}1& x\\ 0& 1\end{array}\right)}$

is not a direct sum of irreducibles. (There is precisely one nontrivial invariant subspace, the span of the first basis element, ${\displaystyle e_1}$.) On the other hand, if ${\displaystyle G}$ is compact, then every finite-dimensional representation ${\displaystyle \mathrm{\Pi }}$ of ${\displaystyle G}$ admits an inner product with respect to which ${\displaystyle \mathrm{\Pi }}$ is unitary, showing that ${\displaystyle \mathrm{\Pi }}$ decomposes as a sum of irreducibles. Similarly, if ${\displaystyle \mathfrak{g}}$ is a complex semisimple Lie algebra, every finite-dimensional representation of ${\displaystyle \mathfrak{g}}$ is a sum of irreducibles. Weyl's original proof of this used the unitarian trick: Every such ${\displaystyle \mathfrak{g}}$ is the complexification of the Lie algebra of a simply connected compact Lie group ${\displaystyle K}$. Since ${\displaystyle K}$ is simply connected, there is a one-to-one correspondence between the finite-dimensional representations of ${\displaystyle K}$ and of ${\displaystyle \mathfrak{g}}$. Thus, the just-mentioned result about representations of compact groups applies. It is also possible to prove semisimplicity of representations of ${\displaystyle \mathfrak{g}}$ directly by algebraic means, as in Section 10.3 of Hall's book.

See also: Fusion category (which are semisimple).