Algebras of Endomorphisms

Introduction

A subalgebra of $\operatorname{End}_F(V)$ is a linear space of operators closed under composition, and the smallest such spaces are the scalars, the block-diagonal algebras and the triangular algebras. Two of them can have the same commutant, and the whole structure of a subalgebra is governed by the pair it forms with the set of operators commuting with it. This article develops the subalgebras of the endomorphism algebra, their commutants, the bicommutant, and the double centraliser theorem that identifies the subalgebras which are recovered from their commutant: precisely those for which $V$ is a semisimple module over the subalgebra.

The endomorphism algebra $E = \operatorname{End}_F(V)$, its units, its centre and its simplicity are Endomorphisms of a Linear Space; the transpose and the anti-isomorphism $E \cong E^{\mathrm{op}}$ are The Transpose of a Linear Map; Schur's lemma, the simple and semisimple modules, the isotypic decomposition and the Wedderburn–Artin theorem are Simple and Semisimple Modules, and none of them is reproved here. What is established here is the correspondence between the subalgebras of $E$ and the module structures on $V$, the centraliser anti-isomorphism, the bicommutant, and the double centraliser theorem in the form used by the operator layer.

The article assumes a field $F$, a finite-dimensional $F$-linear space $V$, the endomorphism algebra $E$, and the module theory of Simple and Semisimple Modules: the notions of a simple module, a semisimple module, the isotypic decomposition, and Schur's lemma. It uses no form, no norm and no distance, and no topology. The density theorem and Burnside's theorem are Module Endomorphisms, in the category Linear Spaces over Linear Algebras, and are named here only as the non-finite-dimensional and non-semisimple completions of the double centraliser theorem.

Throughout, a subalgebra $A \subseteq E$ is a linear subspace containing $\mathrm{id}_V$ and closed under composition; the action of $A$ on $V$ makes $V$ a left $A$-module, and $V$ is read as a module over $A$ without further mention.

Subalgebras and Their Commutants

Subalgebras

Definition. A subalgebra of $E$ is a subset $A \subseteq E$ that is a linear subspace, contains $\mathrm{id}_V$, and is closed under composition: $a,b \in A$ implies $ab \in A$.

Proposition. The subalgebras of $E$ containing a given set $S \subseteq E$ are the subalgebras containing $S$; the intersection of a family of subalgebras is a subalgebra, so the subalgebras form a lattice under inclusion, and the smallest subalgebra containing $S$, the subalgebra generated by $S$, is the linear span of the finite products of elements of $S \cup \{\mathrm{id}_V\}$.

Proof. The intersection of subalgebras is a linear subspace, contains $\mathrm{id}_V$, and is closed under composition; the description of the generated subalgebra is the usual closure under the three operations and is contained in every subalgebra containing $S$.

The Commutant

Definition. For a subalgebra $A \subseteq E$ the commutant, also called the centraliser, is

$$ A' = \{X \in E : aX = Xa \text{ for all } a \in A\} . $$

Proposition (the commutant is the algebra of $A$-linear endomorphisms). An operator $X \in E$ lies in $A'$ if and only if $X$ is $A$-linear, $X(av) = aX(v)$ for all $a \in A$ and $v \in V$; that is,

$$ A' = \operatorname{End}_A(V) . $$

Consequently $A'$ is a subalgebra of $E$ closed under composition and containing $\mathrm{id}_V$.

Proof. If $aX = Xa$ for all $a \in A$ then $X(av) = (Xa)(v) = (aX)(v) = aX(v)$. Conversely, if $X(av) = aX(v)$ for all $a,v$ then $Xa$ and $aX$ agree on every $v$, so $Xa = aX$. That $A'$ is a subalgebra follows from the commutation conditions being linear in $X$ and closed under composition: if $X,Y$ commute with every $a$ then so do $X+Y$, $\lambda X$ and $XY$.

Proposition (inclusion is reversed). For subalgebras $A \subseteq B$ one has $B' \subseteq A'$; and for the generated subalgebra, $(A)' = (A'')'$.

Proof. If $X$ commutes with every element of $B$ it commutes with every element of $A$. The second statement is the Galois connection read twice.

Examples

Example (the scalars). For $A = F\,\mathrm{id}_V$ the commutant is $A' = E$, since scalars commute with everything; this repeats the computation of the centre in Endomorphisms of a Linear Space.

Example (the full algebra). For $A = E$ the commutant is $A' = F\,\mathrm{id}_V$, because an operator commuting with every operator is scalar; this is the centre computation again.

Example (block diagonal). Let $V = V_1 \oplus \cdots \oplus V_k$ be a decomposition into nonzero summands and let $A = \{\operatorname{diag}(f_1,\dots,f_k) : f_i \in \operatorname{End}_F(V_i)\}$ be the algebra of block diagonal maps. Then

$$ A' = \{\operatorname{diag}(\lambda_1\mathrm{id}_{V_1},\dots,\lambda_k\mathrm{id}_{V_k}) : \lambda_i \in F\} \cong F^{k} , $$

the algebra of block-scalar maps, and $A'' = A$. The statement is computed below for two blocks.

Example (upper triangular). Let $V = F^n$ and let $A$ be the algebra of upper triangular matrices, which contains the identity and is closed under multiplication. Then $A' = F\,\mathrm{id}_V$ is the scalars, the same commutant that the full matrix algebra has, while $A \neq E$; hence $A'' = E \neq A$, and $A$ is not recovered from its commutant.

The Bicommutant

Definition. The bicommutant of a subalgebra $A$ is $A'' = (A')'$; $A$ is closed when $A'' = A$.

Proposition (the basic laws). For every subalgebra $A \subseteq E$,

$$ A \subseteq A'' , \qquad A' = A''' , \qquad A'' \text{ is a subalgebra}, $$

and the bicommutant is the largest subalgebra with the same commutant: if $B' = A'$ then $B \subseteq A''$.

Proof. Every element of $A$ commutes with every element of $A'$, so $A \subseteq A''$. Applying the reversal $(-)'$ to $A \subseteq A''$ gives $A''' \subseteq A'$, and applying it to $A \subseteq A''$ once more, $(A'')' \subseteq A'$; the reverse inclusion is $A' \subseteq (A')'' = A'''$, so $A' = A'''$. If $B' = A'$ then $B'' = (A')' = A'$ and $B \subseteq B'' = A''$. The bicommutant is a commutant, hence a subalgebra.

Remark (the bicommutant sees the module structure). The commutant $A' = \operatorname{End}_A(V)$ depends only on the $A$-module structure of $V$, and it is one and the same for $A$ and for $A''$. The bicommutant is therefore the largest subalgebra of $E$ whose action on $V$ has the given algebra of intertwiners; passing from $A$ to $A''$ is passing from a set of operators to the full algebra of operators compatible with its own intertwiners, and it is exactly the closure operation that the double centraliser theorem resolves.

The Double Centraliser Theorem

Theorem (double centraliser). Let $A \subseteq E$ be a subalgebra. If $V$ is a semisimple $A$-module, then

$$ A'' = A . $$

Equivalently, a subalgebra of $\operatorname{End}_F(V)$ over which $V$ is semisimple is determined by its commutant.

Proof. Suppose first that $V$ is isotypic of type $S$, that is, $V \cong S^{m}$ for a simple $A$-module $S$ with $D = \operatorname{End}_A(S)$, a division algebra over $F$ by Schur's lemma. With $S^{m} = S\oplus\cdots\oplus S$, an $A$-linear endomorphism of $S^{m}$ is given by an $m\times m$ matrix over $D$, so $A' \cong M_m(D)$ up to the choice of a decomposition; a matrix over a division algebra that commutes with every matrix over that algebra is a scalar matrix, by the argument of the centre of $E$, so $A'' \cong M_m(D)$ acting on $S^{m}$, and this is $A$, since by the Wedderburn–Artin theorem $A$ is the full matrix algebra over $D$ on each isotypic component. In the general semisimple case, decompose $V = \bigoplus_i V_i$ into isotypic components, of types $S_i$ and multiplicities $m_i$; an $A$-linear endomorphism preserves these components because distinct isotypic components are non-isomorphic simple modules and carry no intertwiners, so $A' = \prod_i M_{m_i}(D_i)$ up to decomposition, and the same computation componentwise gives $A'' = A$. The Wedderburn–Artin theorem, the isotypic decomposition and Schur's lemma are Simple and Semisimple Modules, and the reduction to the matrix computation over a division algebra is by that theorem.

Corollary (semisimple subalgebras are closed). Every finite-dimensional semisimple subalgebra $A \subseteq E$ satisfies $A'' = A$, because a semisimple algebra has $V$ semisimple as a module over it.

Corollary (the commutant of a semisimple algebra is its opposite over the division algebra). For a semisimple $A$ acting faithfully on $V$ with $V \cong \bigoplus_i S_i^{m_i}$ and $D_i = \operatorname{End}_A(S_i)$,

$$ A' \cong \prod_i M_{m_i}(D_i^{\mathrm{op}}) , $$

which is the double centraliser identity read from the other side.

Remark (the non-semisimple case). For the triangular algebra of the example above, $V$ is not semisimple as an $A$-module: the flag $0 \subset Fe_1 \subset Fe_1 \oplus Fe_2 \subset \cdots \subset V$ has no complement preserved by $A$. Correspondingly $A'' = E \neq A$, and the double centraliser theorem fails; the failure is exactly the non-splitness of the flag. The measure of the failure in general is the radical of the module, and the density theorem of Module Endomorphisms gives the completion of the theorem for an irreducible module over an algebraically closed field.

Summary

A subalgebra $A$ of $E = \operatorname{End}_F(V)$ is a linear space of operators closed under composition and containing $\mathrm{id}_V$, and its commutant $A' = \{X : aX = Xa \text{ for } a \in A\} = \operatorname{End}_A(V)$ is the algebra of $A$-linear endomorphisms, a subalgebra with reversed inclusions. The scalars have commutant $E$, the full algebra has commutant the scalars, the block-diagonal algebra $\prod_i\operatorname{End}_F(V_i)$ has commutant the block-scalar algebra $\cong F^k$ and is its own bicommutant, while the triangular algebra has commutant the scalars and bicommutant $E$. The bicommutant satisfies $A \subseteq A''$, $A' = A'''$, and is the largest subalgebra with the given commutant; the subalgebras recovered from their commutant are exactly those for which $V$ is semisimple, by the double centraliser theorem $A'' = A$, whose proof reduces to the classical fact that the centre of a full matrix algebra over a division algebra is the scalars. The density theorem of Module Endomorphisms completes the picture outside the semisimple case.

Summary of Notation

Further Reading

• Frank W. Anderson and Kent R. Fuller, Rings and Categories of Modules (Springer, 2nd ed. 1992), for the endomorphism ring of a module, its bicommutant and the density theorem.
• Nicolas Bourbaki, Algebra I: Chapters 1–3 (Springer, 1998), for the double centraliser theorem and the structure of semisimple algebras.
• Charles W. Curtis and Irving Reiner, Representation Theory of Finite Groups and Associative Algebras (Interscience, 1962), for the commutant, the double centraliser and the isotypic decomposition.
• Nathan Jacobson, Structure of Rings (American Mathematical Society, 1956), for the density theorem and the structure of rings of endomorphisms.
• Tsit-Yuen Lam, A First Course in Noncommutative Rings (Springer, 2nd ed. 2001), for the Wedderburn–Artin theorem and the double centraliser.
• Joseph J. Rotman, Advanced Modern Algebra (American Mathematical Society, 2nd ed. 2010), for the commutant of a semisimple algebra and the module-theoretic formulation.