# Semi-simplicity

(Redirected from Complete reducibility)

In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial sub-objects. The precise definitions of these words depends on the context.

For example, if G is a finite group, then a nontrivial finite-dimensional representation V over a field is said to be simple if the only subrepresentations it contains are either {0} or V (these are also called irreducible representations). Then Maschke's theorem says that any finite-dimensional representation is a direct sum of simple representations (provided the characteristic does not divide the order of the group). So, in this case, every representation of a finite group is semi-simple. Especially in algebra and representation theory, "semi-simplicity" is also called complete reducibility. For example, Weyl's theorem on complete reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple.

A square matrix (in other words a linear operator ${\displaystyle T:V\to V}$ with V finite dimensional vector space) is said to be simple if its only invariant subspaces under T are {0} and V. If the field is algebraically closed (such as the complex numbers), then the only simple matrices are of size 1 by 1. A semi-simple matrix is one that is similar to a direct sum of simple matrices; if the field is algebraically closed, this is the same as being diagonalizable.

These notions of semi-simplicity can be unified using the language of semi-simple modules, and generalized to semi-simple categories.

## Introductory example of vector spaces

If one considers all vector spaces (over a field, such as the real numbers), the simple vector spaces are those that contain no proper subspaces. Therefore, the one-dimensional vector spaces are the simple ones. So it is a basic result of linear algebra that any finite-dimensional vector space is the direct sum of simple vector spaces; in other words, all finite-dimensional vector spaces are semi-simple.

## Semi-simple matrices

A 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.[1][2] 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.[1] 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 basis for this space can be extended to an eigenbasis.

## 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.[3]

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) ${\displaystyle M_{n}(D_{1})\times M_{n}(D_{2})\times \cdots \times M_{n}(D_{r})}$, 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 ${\displaystyle F[T]\subset \operatorname {End} _{F}(V)}$ 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

${\displaystyle 0\to M'\to M\to M''\to 0}$

of modules over a semi-simple ring must split, i.e., ${\displaystyle M\cong M'\oplus M''}$. 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[4] 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

${\displaystyle \operatorname {End} _{C}(X)=\operatorname {Hom} _{C}(X,X)}$

in a semi-simple category is a product of division algebras, 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.[5] Another example from algebraic geometry is the category of pure motives of smooth projective varieties over a field k ${\displaystyle \operatorname {Mot} (k)_{\sim }}$ 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.[6] 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.[7]

## Semi-simplicity in representation theory

One can ask whether the category of (say, 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

${\displaystyle \Pi (x)={\begin{pmatrix}1&x\\0&1\end{pmatrix}}}$

is not a direct sum of irreducibles.[8] (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 ${\displaystyle \Pi }$ of ${\displaystyle G}$ admits an inner product with respect to which ${\displaystyle \Pi }$ is unitary, showing that ${\displaystyle \Pi }$ decomposes as a sum of irreducibles.[9] Similarly, if ${\displaystyle {\mathfrak {g}}}$ is a complex semisimple Lie algebra, every finite-dimensional representation of ${\displaystyle {\mathfrak {g}}}$ is a sum of irreducibles.[10] 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 ${\displaystyle {\mathfrak {g}}}$.[11] 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.

## References

1. ^ a b Lam (2001), p. 39
2. ^ Hoffman, Kenneth; Kunze, Ray (1971). "Semi-Simple operators". Linear algebra (2nd ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc. MR 0276251.
3. ^ * Lam, Tsit-Yuen (2001). A first course in noncommutative rings. Graduate texts in mathematics. 131 (2 ed.). Springer. ISBN 0-387-95183-0.
4. ^ More generally, the same definition of semi-simplicity works for pseudo-abelian additive categories. See for example Yves André, Bruno Kahn: Nilpotence, radicaux et structures monoïdales. With an appendix by Peter O'Sullivan. Rend. Sem. Mat. Univ. Padova 108 (2002), 107–291. https://arxiv.org/abs/math/0203273.
5. ^ Peters, Chris A. M.; Steenbrink, Joseph H. M. Mixed Hodge structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 52. Springer-Verlag, Berlin, 2008. xiv+470 pp. ISBN 978-3-540-77015-2; see Corollary 2.12
6. ^ Uwe Jannsen: Motives, numerical equivalence, and semi-simplicity, Invent. math. 107, 447~452 (1992)
7. ^ Bondarko, Mikhail V. (2012), "Weight structures and `weights' on the hearts of t-structures", Homology Homotopy Appl., 14 (1): 239–261, doi:10.4310/HHA.2012.v14.n1.a12, Zbl 1251.18006
8. ^ Hall 2015 Example 4.25
9. ^ Hall 2015 Theorem 4.28
10. ^ Hall 2015 Theorem 10.9
11. ^ Hall 2015 Theorem 5.6
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer