Semi-simplicity

In abstract algebra, semisimplicity refers to a structural property of modules, rings, and algebras that allows them to decompose into direct sums of irreducible or simple components, facilitating the study of representations and symmetries.^[1] A module M over a ring R is semisimple if it is a finite direct sum of simple R-modules, where a simple module has no proper nonzero submodules.^[1] This decomposition ensures that semisimple modules are both Artinian and Noetherian, with submodules corresponding to sums of subsets of the simple summands.^[1] For rings and algebras, semisimplicity is defined equivalently as the ring being left-Artinian with trivial Jacobson radical, meaning every finitely generated module is semisimple.^[1] The Artin-Wedderburn theorem characterizes semisimple rings (or Artinian semisimple rings) as direct products of matrix rings over division rings, providing a complete structural description.^[1] This theorem is central to representation theory, where, for example, the group algebra \mathbb{C}[G] of a finite group G is semisimple by Maschke's theorem, decomposing into matrix blocks corresponding to irreducible representations.^[1] In linear algebra, a linear operator A on a finite-dimensional vector space V over a field F is semisimple if every A-invariant subspace admits an A-invariant complement, allowing a direct sum decomposition of V.^[2] Over algebraically closed fields, this is equivalent to A being diagonalizable, and in general, it holds if and only if the minimal polynomial of A is square-free (has no repeated roots).^[2] Semisimple operators thus generalize diagonalizable ones, with applications in spectral theory and Jordan canonical forms. The concept extends to Lie algebras over fields of characteristic zero, where a Lie algebra \mathfrak{g} is semisimple if its radical (the largest solvable ideal) is zero, or equivalently, if the Killing form is nondegenerate.^[3] Semisimple Lie algebras, such as \mathfrak{sl}_n(\mathbb{C}) or \mathfrak{so}_n(\mathbb{C}) for n > 2, form the building blocks for classifying simple Lie algebras via root systems and Dynkin diagrams.^[4]

Foundations in linear algebra

Introductory example with vector spaces

In the context of finite-dimensional vector spaces over a field, semi-simplicity of a linear operator can be introduced through its defining property of allowing a complete decomposition of the space into simpler building blocks. Specifically, a linear operator T: V \to V on a finite-dimensional vector space V over a field F is semisimple if V decomposes as a direct sum V = W_1 \oplus W_2 \oplus \cdots \oplus W_k of T-invariant irreducible subspaces, where each W_i is T-invariant (i.e., T(W_i) \subseteq W_i) and irreducible under T.^[2] This decomposition captures the essence of semi-simplicity by breaking down the action of T into independent, indecomposable components. A subspace W \subseteq V is irreducible with respect to T if it admits no proper nontrivial T-invariant subspaces, meaning the only T-invariant subspaces of W are \{0\} and W itself.^[2] For instance, one-dimensional subspaces are always irreducible, as they contain no proper subspaces other than the zero subspace. To illustrate semi-simplicity, consider the identity operator \mathrm{Id} on the two-dimensional vector space \mathbb{R}^2. Here, \mathbb{R}^2 decomposes as a direct sum of any two one-dimensional eigenspaces, such as \langle (1,0) \rangle \oplus \langle (0,1) \rangle, both of which are invariant under \mathrm{Id} and irreducible.^[2] In contrast, not all operators exhibit this behavior. A nilpotent operator, such as the one represented by the Jordan block matrix \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} on \mathbb{R}^2, fails to be semisimple. This operator has a one-dimensional invariant subspace W = \langle (1,0) \rangle, but no complementary one-dimensional invariant subspace exists, as any complement intersects W nontrivially or is not invariant.^[2] Thus, \mathbb{R}^2 cannot decompose into a direct sum of irreducible invariant subspaces under this operator. A fundamental property equivalent to semi-simplicity is that every T-invariant subspace admits a complementary T-invariant subspace, ensuring the space splits flexibly without "stuck" components.^[2] The matrix representations of such semisimple operators inherit this structure.^[2]

Semi-simple endomorphisms and matrices

In linear algebra, an endomorphism T: V \to V on a finite-dimensional vector space V over a field F is called semi-simple if its minimal polynomial factors into distinct linear factors over an algebraic closure of F.^[2] Over an algebraic closure of F, V decomposes as a direct sum of the eigenspaces of T, V = \bigoplus_{\lambda} \ker(T - \lambda I), where the sum runs over the distinct eigenvalues \lambda of T.^[2] This condition ensures that T acts by scalar multiplication on each eigenspace, capturing the intuitive notion of "simplicity" extended to multiple irreducible components. A key criterion for semi-simplicity is that T is semi-simple if and only if it is diagonalizable over an algebraically closed field containing F.^[2] Over such a field, the characteristic polynomial of T splits completely, and the absence of repeated roots in the minimal polynomial guarantees the existence of a basis of eigenvectors. In matrix terms, a matrix A \in M_n(F) representing T with respect to some basis is semi-simple if it is similar to a diagonal matrix over an algebraic closure of F, i.e., there exists an invertible matrix P such that A = P D P^{-1} where D is diagonal.^[2] This diagonalizability connects directly to the Jordan canonical form: a matrix is semi-simple if and only if its Jordan form consists solely of 1-by-1 blocks (i.e., no nontrivial Jordan blocks larger than size 1).^[2] A representative example is the rotation matrix in \mathbb{R}^2 by an angle \theta \notin \{0, \pi\}, given by

R = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.

This matrix is not diagonalizable over \mathbb{R} since its minimal polynomial x^2 - 2\cos\theta \, x + 1 is irreducible, but over \mathbb{C}, it factors as (x - e^{i\theta})(x - e^{-i\theta}) with distinct roots, making R semi-simple.^[5]

Generalizations to modules and rings

Semi-simple modules

In ring theory, a left module M over a ring R (with unity) is semisimple if it admits a decomposition as a direct sum of simple submodules.^[6] Equivalently, M is semisimple if every submodule of M is a direct summand.^[7] A module S is simple if it is nonzero and possesses no proper nontrivial submodules.^[1] For instance, over the ring \mathbb{Z} of integers, the cyclic module \mathbb{Z}/p\mathbb{Z} for a prime p is simple, as its only submodules are the zero module and itself.^[8] Semisimple modules exhibit several key properties arising from their decomposition. Submodules and quotient modules of a semisimple module are themselves semisimple.^[1] Moreover, since every submodule is a direct summand, any short exact sequence $0 \to K \to M \to Q \to 0 with semisimple middle term M splits.^[1] A concrete example occurs in the category of vector spaces over a field k, where every vector space is semisimple as a k-module, decomposing as a direct sum of copies of the simple module k.^[6] The socle of a module M, denoted \mathrm{Soc}(M), is the sum (direct in the semisimple case) of all simple submodules of M; for a semisimple module, this coincides with M itself.^[8] The radical of M, denoted \mathrm{rad}(M), is the intersection of all maximal submodules of M; in semisimple modules of finite length, \mathrm{rad}(M) = 0. Over an Artinian ring, semisimple modules are precisely those of finite length whose radical vanishes, ensuring a decomposition into finitely many simple summands.^[1]

Semi-simple rings and Artinian structure

A ring R is defined as left semi-simple if every left R-module is semi-simple.^[1] Equivalently, R is left semi-simple if the left regular module {}_R R decomposes as a direct sum of simple left R-modules.^[9] For rings with unity, left semi-simplicity implies right semi-simplicity under the Artinian condition, making the notion symmetric in such cases.^[10] Semi-simple rings are precisely the Artinian rings whose Jacobson radical vanishes, that is, J(R) = 0.^[1] The Jacobson radical J(R) is the intersection of all maximal left ideals of R, and its vanishing ensures that every left ideal is a direct summand, facilitating the semi-simple decomposition of modules.^[9] This zero radical condition distinguishes semi-simple rings from more general Artinian rings, where J(R) may be non-zero and consist of nilpotent elements.^[1] The structure of semi-simple rings is fully characterized by the Wedderburn-Artin theorem, which states that every semi-simple Artinian ring R is isomorphic to a finite direct product of full matrix rings over division rings:

R \cong \prod_{i=1}^k M_{n_i}(D_i),

where each n_i is a positive integer and each D_i is a division ring.^[11] This decomposition arises because semi-simple rings decompose into simple Artinian components, and each simple Artinian ring is itself a matrix ring over its endomorphism division ring of a simple module.^[10] The theorem relies on the descending chain condition on left ideals and the absence of non-trivial two-sided ideals in the simple components.^[11] A concrete example is the ring M_n(\mathbb{C}) of n \times n matrices over the complex numbers \mathbb{C}, which is a simple semi-simple ring since \mathbb{C} is a division ring (in fact, a field).^[10] Here, the unique simple left module is the column space \mathbb{C}^n, and the ring acts faithfully as endomorphisms.^[9] This notion of semi-simplicity for rings is unrelated to the semi-simplicity of Lie algebras, which refers to Lie algebras with no non-trivial solvable ideals.^[10]

Categorical framework

Semi-simple abelian categories

In an abelian category \mathcal{A}, a simple object is a nonzero object whose only subobjects are $0and itself. An objectX \in \mathcal{A}is semisimple if it is isomorphic to a [direct sum](/page/Direct_sum) of simple objects, and the category\mathcal{A}$ is semisimple if every object is semisimple. Moreover, in any abelian category, every morphism between simple objects is either the zero morphism or an isomorphism.^[12]^[13] A key property of semisimple abelian categories is that every short exact sequence splits. This follows because every subobject of a semisimple object is itself semisimple and hence a direct summand. Consequently, every object in a semisimple abelian category is both projective and injective, ensuring that the category has enough projectives and enough injectives. Additionally, every idempotent morphism splits, which implies that every object admits a direct sum decomposition corresponding to the image and kernel of the idempotent.^[13]^[14] Classic examples of semisimple abelian categories include the category of finite-dimensional vector spaces over a field k, where every object decomposes into a direct sum of one-dimensional simple modules. Another example is the category of finite-dimensional complex representations of a finite group, which is semisimple due to the complete reducibility of representations.^[12]^[13] Semisimple abelian categories generalize the structure of module categories over semisimple rings. Specifically, if R is a semisimple Artinian ring, then the category of left R-modules, \mathrm{Mod}\text{-}R, is semisimple, with simple modules corresponding to the simple components in the Artin-Wedderburn decomposition of R. Conversely, the module category over a semisimple ring provides a prototypical instance of semisimple abelian categories.^[12]^[13]

Krull-Schmidt theorem in semi-simple categories

In semi-simple abelian categories, indecomposable objects are defined as nonzero objects that cannot be expressed as a direct sum of two nonzero subobjects.^[15] These coincide with the simple objects in such categories, as every object decomposes into a direct sum of simples, and simples are precisely the indecomposables.^[12] The Krull-Schmidt theorem asserts that in a semi-simple abelian category where objects have finite length, any two direct sum decompositions of an object into indecomposables are unique up to permutation of the summands and isomorphism of the components.^[15] This uniqueness holds more generally in Krull-Schmidt categories, which are additive categories where every object admits a decomposition into a finite direct sum of indecomposables with local endomorphism rings, but in the semi-simple case, the finite length condition ensures the existence of such decompositions.^[12] A sketch of the proof relies on the endomorphism rings of indecomposable objects being local rings, meaning the non-units form an ideal (in the semi-simple context, these are division rings with trivial Jacobson radical). This locality implies that any idempotent endomorphism is either zero or an identity, preventing non-trivial splittings and ensuring that isomorphic decompositions must match up to reordering.^[15] For example, consider the category of finite-dimensional vector spaces over an algebraically closed field, which is semi-simple. A semisimple endomorphism (one whose minimal polynomial has distinct roots) admits a unique diagonalization up to permutation of eigenvalues, corresponding to the unique multiplicities of the one-dimensional eigenspaces (simple subobjects) in its decomposition.^[16] The theorem's key application is that it makes the isomorphism classes and multiplicities in a decomposition well-defined, independent of the choice of summands, which is essential for classifying objects up to isomorphism in such categories.^[12] This requires additional conditions like the category being artinian (every descending chain of subobjects stabilizes) or Hom-spaces between objects being finite-dimensional to guarantee finite-length decompositions.^[15]

Applications in representation theory

Semi-simple representations

In representation theory, a representation \rho: G \to \mathrm{GL}(V) of a group G on a finite-dimensional vector space V over a field k is called semisimple (or completely reducible) if V decomposes as a direct sum of irreducible subrepresentations.^[17] Similarly, for an algebra A, a representation \rho: A \to \mathrm{End}(V) is semisimple if V is a direct sum of simple A-modules.^[17] This decomposition implies that every invariant subspace has an invariant complement, ensuring the representation splits completely into irreducibles.^[17] Equivalently, a representation is semisimple if and only if its endomorphism algebra \mathrm{End}_G(V) (or \mathrm{End}_A(V)) is a semisimple algebra, meaning it is a direct product of matrix algebras over division rings.^[17] This equivalence highlights the structural similarity between the representation space and the algebra acting on it. For finite groups over fields of characteristic zero, semi-simplicity holds for all finite-dimensional representations by Maschke's criterion.^[17] In character theory for finite groups over \mathbb{C}, a representation is semisimple if and only if its character \chi is a \mathbb{Z}_{\geq 0}-linear combination of irreducible characters.^[17] The multiplicity of an irreducible representation with character \psi in a semisimple representation with character \chi is given by the inner product

\langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)},

which counts the dimension of the space of G-equivariant intertwiners \mathrm{Hom}_G(V_\chi, V_\psi).^[17] This formula allows explicit computation of isotypic components, where the dimension of the component isomorphic to a given irreducible is the multiplicity times the irreducible's dimension. A concrete example is the regular representation of the symmetric group S_3, which acts on \mathbb{C}[S_3] by left multiplication and has dimension 6. It decomposes as the direct sum of the 1-dimensional trivial representation (multiplicity 1), the 1-dimensional sign representation (multiplicity 1), and the 2-dimensional irreducible representation (multiplicity 2).^[17] The character table of S_3 confirms this via inner products: for the 2-dimensional irreducible \psi, \langle \chi_{\mathrm{reg}}, \psi \rangle = 2.^[17] Over non-splitting fields like \mathbb{R}, representations such as those of \mathrm{SO}(3) may exhibit different behavior; for instance, certain complex irreducible representations do not admit faithful real forms, leading to real representations whose complexifications decompose further but remain semisimple over \mathbb{R} due to the compactness of the group.^[18]

Maschke's criterion and finite groups

Maschke's criterion provides a fundamental condition under which every finite-dimensional representation of a finite group over a field is semisimple. Specifically, for a finite group G and a field k such that the characteristic of k does not divide the order |G|, every representation of G on a finite-dimensional k-vector space is completely reducible, meaning it decomposes as a direct sum of irreducible representations.^[19] This result ensures that the group algebra k[G] is semisimple, implying that invariant subspaces always admit invariant complements.^[20] The theorem is named after Heinrich Maschke, who proved it in his seminal 1899 paper on linear representations of finite groups.^[21] This criterion laid essential groundwork for the development of character theory, as the complete reducibility allows representations to be analyzed via sums of irreducible characters.^[19] To outline the proof, consider a representation \rho: G \to \mathrm{GL}(V) on a finite-dimensional k-vector space V, with W \subseteq V an invariant subspace. Since \mathrm{char}(k) \nmid |G|, the scalar $1/|G| exists in k. Choose any k-linear projection \pi: V \to W, and define the averaged projection

P = \frac{1}{|G|} \sum_{g \in G} \rho(g) \circ \pi \circ \rho(g^{-1}).

This operator P is idempotent (P^2 = P) and k-linear, and it commutes with the action of G because the sum averages over all group elements, making \rho(h) \circ P = P \circ \rho(h) for all h \in G. Thus, P(V) is an invariant subspace isomorphic to W, and the kernel \ker(P) provides an invariant complement, so V = P(V) \oplus \ker(P). Iterating this process yields complete reducibility.^[19]^[20] The criterion fails when \mathrm{char}(k) divides |G|, leading to non-semisimple representations known as modular representations. For instance, consider the cyclic group \mathbb{Z}/p\mathbb{Z} over the field \mathbb{F}_p of characteristic p. For example, the representation on \mathbb{F}_p^2 given by \rho(g^k) = \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} for k \in \mathbb{F}_p, where g is the generator, has a unique proper nonzero invariant subspace spanned by (0, 1)^\top, but no invariant complement exists, violating complete reducibility.^[20] More broadly, Maschke's criterion characterizes the semisimplicity of the group algebra k[G]: it holds precisely when \mathrm{char}(k) \nmid |G|, and in such cases, k[G] is a semisimple algebra isomorphic to a direct sum of matrix algebras over division rings. Conversely, when the characteristic divides the group order, k[G] is not semisimple, and representations may have non-split extensions.^[19]