Representations of Algebras

Introduction

A representation of an algebra is an algebra homomorphism into the endomorphism algebra of a vector space, and a representation of a group is the special case of a homomorphism out of a group algebra. This article develops the theory of representations of an algebra from the module-theoretic side: irreducible representations as simple modules, Schur's lemma and the commutant of an irreducible representation, the double centralizer theorem, the structure theorem for a semisimple algebra, the explicit description of the modules over a matrix algebra, and the structure of group algebras as the principal example. It is a bridge entry of this category: it may use the group theory of the companion article Groups and the Lie theory of the companion article Lie Algebras, but it does not restate them, and a representation is always a module, never a physical object.

The conventions are those of Modules over an Algebra: $R$ is a commutative ring with identity, $A$ is a unital associative $R$-algebra, and $F$ is a field. A representation of $A$ over $F$ is a left $A$-module whose underlying additive group is an $F$-vector space; the structure theory of Simple and Semisimple Modules and the Morita theory of Morita Equivalence are used throughout and are cited rather than reproved. Where a result requires an algebraically closed field or characteristic zero, it is flagged in place.

Representations as Modules

Definition and equivalence

Let $A$ be a unital associative $F$-algebra and let $V$ be an $F$-vector space. A representation of $A$ on $V$ is a unital $F$-algebra homomorphism

$$ \rho : A \longrightarrow \operatorname{End}_F(V), $$

that is, a linear map with $\rho(ab)=\rho(a)\rho(b)$ and $\rho(1_A)=\mathrm{id}_V$. Equivalently, $V$ carries the structure of a left $A$-module by $a\cdot v=\rho(a)v$, and the equivalence is an equivalence of categories:

$$ \operatorname{Rep}_F(A) \cong \operatorname{Mod}(A). $$

Indeed a left $A$-module structure is by definition a unital ring homomorphism $A\to\operatorname{End}_{\mathbb{Z}}(V)$; the base field acts through the unit, $r\cdot v=(r1_A)v$, by Modules over an Algebra, and with this $F$-structure on $V$ the action is $F$-linear, since $r1_A$ lies in the center of $A$ and $a(r1_A)=(r1_A)a$ for every $a$, so the homomorphism lands in $\operatorname{End}_F(V)$. Conversely an $F$-algebra homomorphism $A\to\operatorname{End}_F(V)$ makes $V$ a left $A$-module by $a\cdot v=\rho(a)v$, and the two constructions are inverse to one another. The $A$-linear maps are exactly the intertwining operators of the representations, those $T$ with $T\rho(a)=\rho(a)T$ for all $a$, and the equivalence is fully faithful by construction. The dimension of the representation is $\dim_F V$; the representation is faithful if $\rho$ is injective, equivalently if $\operatorname{Ann}_A(V)=0$. The kernel of $\rho$ is a two-sided ideal, and $A/\ker\rho$ acts faithfully.

The two languages of this article — modules and representations — are therefore interchangeable, and the choice is one of emphasis. Statements about submodules are statements about invariant subspaces; statements about quotients are statements about quotient representations; statements about simple modules are statements about irreducible representations; and statements about $\operatorname{End}_A(V)$ are statements about the commutant $\rho(A)'$ inside $\operatorname{End}_F(V)$.

The regular representation

The left regular representation of $A$ is the action on $A$ by left multiplication, $\rho_{\mathrm{reg}}(a)b=ab$. It has dimension $\dim_F A$ when $A$ is finite-dimensional, and it is faithful, because $\rho_{\mathrm{reg}}(a)=\rho_{\mathrm{reg}}(a')$ implies $a=a'$ on evaluating at $1_A$. It is the representation on the regular module ${}_A A$, and its endomorphism ring is $A^{\mathrm{op}}$ by the theorem on the regular module of Modules over an Algebra. Every representation is a quotient of a direct sum of copies of the regular representation, since every module is a quotient of a free module.

Basic operations

The direct sum of representations $\rho$ on $V$ and $\sigma$ on $W$ is the block-diagonal action on $V\oplus W$, the dual (or contragredient) is the action on $V^*=\operatorname{Hom}_F(V,F)$ given by $(\rho^*(a)f)(v)=f(\rho(a)v)$, which makes $V^*$ a right $A$-module, equivalently a left $A^{\mathrm{op}}$-module, and a left $A$-module only when $A$ is commutative or when an anti-automorphism is available to convert the two sides, as the antipode $g\mapsto g^{-1}$ does for a group algebra; and the tensor product $\rho\otimes\sigma$ on $V\otimes_F W$ is defined only when $A$ carries a comultiplication; it is defined for group algebras by $\Delta(g)=g\otimes g$ of Representations of Groups, but not for a general algebra. This asymmetry between the group-algebra case and the general case is a genuine feature of the subject and not a convention.

Irreducible Representations and Schur's Lemma

Irreducibility

A representation is irreducible if $V$ has no invariant subspaces other than $0$ and $V$, that is, if $V$ is a simple module. By the classification of simple modules in Simple and Semisimple Modules, the irreducible representations of $A$ are, up to isomorphism, the quotients

$$ V=A/\mathrm{M} $$

by the maximal left ideals $\mathrm{M}\subseteq A$. When $A$ is commutative the maximal left ideals are the maximal ideals, and the irreducible representations are the quotients by maximal ideals. When $A$ is noncommutative, distinct maximal left ideals may give isomorphic irreducible representations, and the classification up to isomorphism requires the radical: the irreducible representations of $A$ are exactly the irreducible representations of $A/J(A)$, inflated along the quotient $A\to A/J(A)$, by Change of Rings.

Schur's lemma

Theorem (Schur). Let $V$ and $W$ be irreducible representations of $A$. Then every intertwining operator $T: V\to W$ is zero or an isomorphism, and

$$ \operatorname{End}_A(V)=\rho(A)' \text{ is a division algebra}, \qquad \operatorname{Aut}_A(V)=\operatorname{End}_A(V)^{\times}. $$

The lemma is proved in Automorphisms of Modules over an Algebra and is the reason irreducible representations behave as atoms: they have no endomorphisms except zero and invertible ones. The division algebra $D=\operatorname{End}_A(V)$ is a genuine invariant of the irreducible representation. Two consequences are used constantly.

Two irreducible representations are isomorphic if and only if there is a nonzero intertwining operator between them.
If $F$ is algebraically closed and $V$ is finite-dimensional, then $D=F$. Indeed $D$ is a finite-dimensional division algebra over the algebraically closed field $F$, and the only such is $F$ itself.

The second statement is the reason that over $\mathbb{C}$ every irreducible representation of a finite-dimensional algebra or a finite group has scalar commutant, and its failure over $\mathbb{R}$ and over other fields is exactly where real and arithmetic phenomena enter; the specialisations are not treated here.

Density and the Double Centralizer

The size of the image of $A$ inside $\operatorname{End}_F(V)$ is determined by the commutant. The precise statement is the density theorem of Automorphisms of Modules over an Algebra, and its consequence for finite-dimensional representations is the double centralizer theorem.

Theorem (Double centralizer). Let $V$ be a finite-dimensional semisimple left $A$-module and put $D=\operatorname{End}_A(V)$. Then the natural map

$$ A/\operatorname{Ann}_A(V) \longrightarrow \operatorname{End}_D(V) $$

is an isomorphism of $F$-algebras. In particular, if $V$ is a finite-dimensional irreducible module with $D=\operatorname{End}_A(V)$ and $n=\dim_D V$, then

$$ A/\operatorname{Ann}_A(V) \cong \operatorname{End}_D(V) \cong M_n(D^{\mathrm{op}}). $$

Proof. The density theorem gives that the image of $A$ in $\operatorname{End}_D(V)$ is dense for the finite topology; when $V$ is finite-dimensional over $D$ that topology is discrete, so the image is everything. The kernel of the map is precisely $\operatorname{Ann}_A(V)$, giving the first isomorphism; the second is the choice of a $D$-basis, and $\operatorname{End}_D(V)\cong M_n(D^{\mathrm{op}})$ for $n=\dim_D V$, the $D$-linear endomorphisms of $D^n$ being the right multiplications by matrices over $D$, as in Automorphisms of Modules over an Algebra.

The opposite in the last display is the left-module convention rather than an accident: the defining module of $M_n(D_0)$ has endomorphism ring $D_0^{\mathrm{op}}$. For the division algebras occurring here, $F$ when $F$ is algebraically closed, and $\mathbb{C}$ and $\mathbb{H}$ over $\mathbb{R}$, the opposite is isomorphic to the algebra itself, so in every example the two forms agree.

So an irreducible finite-dimensional representation of $A$ is not merely a quotient of $A$; it exhibits $A/\operatorname{Ann}_A(V)$ as a full matrix algebra over a division algebra. The density theorem is what converts information about the commutant into information about the algebra, and it is the engine of the Wedderburn–Artin theorem in the form used below.

Complete Reducibility and Structure

Semisimple algebras

A representation is completely reducible if it is a direct sum of irreducible representations, that is, if $V$ is a semisimple module. The following is the central structural result, assembled from Simple and Semisimple Modules.

Theorem. For a unital algebra $A$ the following are equivalent.

$A$ is semisimple, i.e. the regular module ${}_A A$ is semisimple.
Every left $A$-module is semisimple.
Every representation of $A$ is completely reducible.
Every left ideal of $A$ is a direct summand.
$A \cong \prod_{i=1}^{r} M_{n_i}(D_i)$ for division algebras $D_i$.

Proof. The equivalence of 1, 2, 4 and 5 is the Wedderburn–Artin theorem of Simple and Semisimple Modules; the equivalence with 3 is the definition of complete reducibility in module language.

Finite-dimensional algebras

Let $A$ be finite-dimensional over $F$. Then $A/J(A)$ is semisimple and every simple module is a simple $A/J(A)$-module. Writing $S_1,\dots,S_r$ for the simple modules up to isomorphism and $D_i=\operatorname{End}_A(S_i)^{\mathrm{op}}$, one has

$$ A/J(A) \cong \prod_{i=1}^{r} M_{n_i}(D_i), \qquad n_i=\dim_{D_i} S_i, $$

and the irreducible representations of $A$ are $S_1,\dots,S_r$. If $F$ is algebraically closed, then $D_i=F$ for all $i$ and

$$ A/J(A) \cong \prod_{i=1}^{r} M_{n_i}(F), \qquad \sum_{i=1}^{r} n_i^2 \leq \dim_F A, $$

with equality if and only if $A$ is semisimple. The inequality holds because $A/J(A)$ is a quotient of $A$ and dimension does not increase under a surjection; equality holds when $J(A)=0$. This is the quantitative form of the structure theorem: the sum of the squares of the degrees of the irreducible representations is at most the dimension of the algebra, with equality exactly in the semisimple case.

Modules over Matrix Algebras

Let $A=M_n(F)$ with $n \geq 1$ and let $S=F^n$ be the defining representation, matrices acting on column vectors.

Theorem. Every left $M_n(F)$-module is a direct sum of copies of $S$. Equivalently, $S$ is the unique irreducible representation of $M_n(F)$, up to isomorphism, and

$$ M_n(F) \cong S^{\oplus n} $$

as a left module over itself.

Proof. The regular module decomposes into the $n$ column spaces, each isomorphic to $S$, so ${}_{M_n(F)}M_n(F)\cong S^{\oplus n}$ and $S$ is a direct summand of the regular module, hence projective. The algebra $M_n(F)$ is semisimple, so every module is a direct sum of simple modules and every simple module is a quotient of the regular module. Since the regular module is a sum of copies of $S$, its only composition factor is $S$, so $S$ is the only simple module up to isomorphism. Hence every module is a direct sum of copies of $S$, and the regular module is the case $M_n(F)\cong S^{\oplus n}$.

Corollary. For the endomorphism and automorphism groups,

$$ \operatorname{End}_{M_n(F)}(S)\cong F, \qquad \operatorname{Aut}_{M_n(F)}(S)\cong F^{\times}, \qquad \operatorname{Aut}_{M_n(F)}(M_n(F))\cong GL_n(F). $$

The contrast between the last two is the contrast of Automorphisms of Modules over an Algebra: the defining representation is rigid, with only scalar intertwining operators, while the regular representation carries the full general linear group. For a general algebra $A$ the module theory of $M_n(A)$ is the module theory of $A$, since $A$ and $M_n(A)$ are Morita equivalent by Morita Equivalence, with the defining module of $M_n(A)$ playing the role of the regular module of $A$.

For the biquaternion algebra $\mathbb{B}\cong M_2(\mathbb{C})$, the theorem applies with $n=2$, $F=\mathbb{C}$: the unique irreducible representation is $S=\mathbb{C}^2$, every representation is a direct sum of copies of $S$, and $\mathbb{B}\cong S\oplus S$. This is the structural statement underlying.

The Structure of Group Algebras

Let $G$ be a finite group and $F$ a field, so that $A=F[G]$ is finite-dimensional over $F$. The general theory specialises as follows; the representation-theoretic consequences are developed in Representations of Groups, and only the algebra structure is recorded here.

Semisimple case. If $\operatorname{char} F \nmid |G|$, Maschke's theorem gives $J(F[G])=0$, so

$$ F[G] \cong \prod_{i=1}^{r} M_{n_i}(D_i) $$

with $D_i$ finite-dimensional division algebras over $F$ and $|G|=\sum_i n_i^2 \dim_F D_i$. Over an algebraically closed field the $D_i$ are $F$, the number $r$ of irreducible representations equals the number of conjugacy classes of $G$, and $|G|=\sum_i n_i^2$.

Modular case. If $\operatorname{char} F=p$ divides $|G|$, the radical is nonzero and

$$ F[G]=\bigoplus_b B_b $$

decomposes into indecomposable two-sided summands, the **blocks**. The irreducible representations of $F[G]$ are the irreducible representations of $F[G]/J(F[G])$, and the blocks are compared up to Morita equivalence as in *Morita Equivalence*.

The regular representation. The left regular representation of $G$ is the module ${}_{F[G]}F[G]$; in the semisimple case it decomposes as $\bigoplus_i S_i^{\oplus n_i}$, and its endomorphism algebra is $F[G]^{\mathrm{op}}\cong F[G]$.
Abelian groups. If $G$ is abelian and $F$ is algebraically closed of characteristic not dividing $|G|$, then $F[G]\cong F^{|G|}$ is commutative semisimple, and every irreducible representation is one-dimensional.

The structure theorem therefore reduces the representation theory of a finite group to the module theory of a product of matrix algebras over division algebras, and the division algebras are the only place where the base field enters beyond its characteristic.

Idempotents and Projective Representations

An idempotent $e \in A$ gives a decomposition ${}_A A=Ae\oplus A(1_A-e)$, so the representation $Ae$ is a direct summand of the regular representation and hence projective. Every direct summand of the regular representation arises in this way, by the proposition of Modules over an Algebra. In a semisimple algebra every module is projective and every simple module is a summand of the regular representation, so every irreducible representation is of the form $Ae$ for a primitive idempotent $e$. For a matrix algebra the idempotents $E_{ii}$ of the standard basis are primitive, and $M_n(F)E_{11}$ is the defining module $S$.

Conversely, a projective indecomposable module $P$ over a finite-dimensional algebra has a unique simple quotient $P/\operatorname{rad}(P)$, and the assignment of a projective indecomposable to its simple quotient is a bijection between the isomorphism classes of projective indecomposables and the isomorphism classes of simple modules. The multiplicities in the regular module are the dimensions $\dim_{D_i}S_i=n_i$, and in the modular case they are recorded by the Cartan matrix of the algebra, whose entry $c_{ij}$ is the multiplicity of $S_i$ among the composition factors of the projective cover of $S_j$. This is the standard organisation of the modular representation theory referred to in §The Structure of Group Algebras.

Path Algebras and Representation Type

The representations of an arbitrary finite-dimensional algebra can be reduced to those of a combinatorially given quotient of a free algebra, and this is the second device, after the structure theorem, by which representations of algebras are classified.

Quivers and their representations

A quiver $Q$ consists of a set of vertices $Q_0$, a set of arrows $Q_1$, and maps $s,t:Q_1\to Q_0$ assigning to an arrow its source and target; it is finite when both sets are finite. A representation of $Q$ over $F$ assigns to each vertex $x$ a vector space $V_x$ and to each arrow $a:x\to y$ a linear map $V_a:V_x\to V_y$; a morphism of representations is a family of linear maps $V_x\to W_x$ making all the squares commute. The dimension vector of a representation is

$$ \underline{\dim}\,V=(\dim_F V_x)_{x\in Q_0}\in\mathbb{N}^{Q_0}. $$

The path algebra

The path algebra $FQ$ has as an $F$-basis the paths of $Q$, including the trivial path $e_x$ of length zero at each vertex, and its product is concatenation of composable paths and zero otherwise. It is an associative algebra with unit $\sum_{x\in Q_0}e_x$, the $e_x$ are orthogonal idempotents, and the decomposition

$$ FQ=\bigoplus_{x,y\in Q_0}e_xFQe_y $$

writes $FQ$ as a direct sum of the subspaces spanned by paths from $x$ to $y$. The construction converts representations into modules:

Proposition. For every finite quiver $Q$ there is an equivalence of categories $\operatorname{Mod}(FQ)\simeq\operatorname{Rep}_F(Q)$, under which a representation $V$ corresponds to the module $\bigoplus_{x\in Q_0}V_x$ with each path acting as the composite of its arrow maps.

For a finite quiver without oriented cycles, $FQ$ is finite-dimensional, with $\dim_F FQ$ equal to the number of paths, and hereditary; the simple $FQ$-modules are the one-dimensional modules $S_x=FQe_x/\operatorname{rad}(FQ)e_x$ supported at the vertices, and the indecomposable projective modules are the $FQe_x$, of dimension vectors counting the paths starting at $x$. A representation is thus a module, and the dimension vector is its coarsest invariant.

Gabriel's theorem and the structure of basic algebras

Theorem (Gabriel). Let $Q$ be a finite connected quiver without oriented cycles and let $F$ be algebraically closed. Then $FQ$ has finitely many isomorphism classes of indecomposable representations if and only if the underlying undirected graph of $Q$ is one of the Dynkin diagrams $A_n$, $D_n$, $E_6$, $E_7$, $E_8$; in that case the indecomposable representations are in bijection with the positive roots of the corresponding root system, and their dimension vectors are exactly those roots.

The passage from a quiver to an arbitrary algebra is given by the following standard structure theorem.

Theorem. Every finite-dimensional basic algebra over an algebraically closed field $F$ is isomorphic to a quotient $FQ/I$ of a path algebra by an admissible ideal $I$, for a unique finite quiver $Q$; every finite-dimensional algebra over $F$ is Morita equivalent to such a basic algebra, and under the equivalence the simple modules correspond to the vertices of $Q$.

The theorem separates the representation theory of a finite-dimensional algebra over an algebraically closed field into a combinatorial part, the quiver and the relations generating $I$, and a Morita-theoretic part, the passage from the algebra to its basic form as in Morita Equivalence. The first part is visible already for the polynomial algebra, whose quiver has one vertex and one loop.

Representation type

A finite-dimensional algebra $A$ is of finite representation type if it has finitely many isomorphism classes of indecomposable modules, of tame representation type if its indecomposables in each dimension occur in finitely many one-parameter families, and of wild representation type if the classification of its representations contains that of the representations of a free algebra in two generators. Drozd's theorem states that over an algebraically closed field every finite-dimensional algebra is of exactly one of these three types, so the trichotomy is exhaustive.

The simplest illustrations are the quivers with two vertices and $n$ arrows from the first to the second. For $n=1$ the quiver is $A_2$, of finite representation type, with three indecomposables: the simple modules $S_1$ and $S_2$ and the indecomposable projective $P_1=FQe_1$, of dimension vector $(1,1)$. For $n=2$, the Kronecker quiver, the algebra is of tame representation type, and the indecomposables are the preprojective, preinjective and regular modules, the last parametrized by the projective line. For $n\geq3$ the algebra is of wild representation type. The three cases show that a representation is a module and that the classification of modules is in general as hard as linear algebra can be, which is why the semisimple and hereditary Dynkin cases of the structure theorem and Gabriel's theorem are the ones that admit a complete answer.

Examples

(a) Fields and division algebras. $A=F$ has a single irreducible representation, $F$ itself, of dimension one, and every representation is a direct sum of copies of $F$. A division algebra $D$ over $F$ has the single irreducible representation ${}_D D$, of dimension $\dim_F D$, and every representation is free by Modules. For $D=\mathbb{H}$ over $\mathbb{R}$, the unique irreducible representation has real dimension $4$ and commutant $\mathbb{H}$.

(b) Matrix algebras and their quotients. $A=M_n(F)$ has the unique irreducible representation $F^n$ with commutant $F$, as in §Modules over Matrix Algebras. A quotient of a matrix algebra by a two-sided ideal is either $M_n(F)$ or $0$, so no nontrivial quotient changes the representation theory.

(c) Polynomial algebras and normal forms. For $A=F[x]$, a representation is a vector space with a single linear operator, and the classification of representations is the structure theorem for finitely generated modules over the principal ideal domain $F[x]$: rational canonical form in general, and Jordan form over an algebraically closed field. The irreducible representations are the quotients $F[x]/(p)$ by irreducible polynomials $p$, one for each maximal ideal of $F[x]$; the field of fractions $F(x)$, viewed as an $F[x]$-module, is infinite-dimensional and is not irreducible, the zero ideal being prime but not maximal. This is the model case in which indecomposable representations are not all irreducible, and it is developed in the companion article Modules over a PID.

(d) Local algebras. A finite-dimensional algebra $A$ with a unique maximal left ideal — equivalently, a finite-dimensional algebra for which $A/J(A)$ is a division algebra, which over an algebraically closed $F$ amounts to $A/J(A)\cong F$ — has a single irreducible representation and is not semisimple unless $J(A)=0$. The algebra $F[x]/(x^n)$ and the group algebra of a cyclic $p$-group over a field of characteristic $p$ are the standard examples; their regular representation has length $n$ and a single composition factor.

(e) Group algebras. As in §The Structure of Group Algebras: $\mathbb{C}[S_3]\cong\mathbb{C}\oplus\mathbb{C}\oplus M_2(\mathbb{C})$ has three irreducible representations of dimensions $1,1,2$ with $1+1+4=6$; $\mathbb{C}[Q_8]\cong\mathbb{C}^4\oplus M_2(\mathbb{C})$ has five irreducible representations of dimensions $1,1,1,1,2$ with sum of squares $8$. Over $\mathbb{R}$ the two-dimensional factor for $Q_8$ is replaced by $\mathbb{H}$, so the commutant of the irreducible representation is a noncommutative division algebra and the representation is not absolutely irreducible.

(f) The biquaternion algebra. $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}\cong M_2(\mathbb{C})$ has the single irreducible representation $S=\mathbb{C}^2$ with commutant $\mathbb{C}$, and $\mathbb{B}\cong S\oplus S$. Every finite-dimensional representation is isomorphic to $S^{\oplus k}$ for a unique $k \geq 0$, so its complex dimension is $2k$; such a module is free if and only if $k$ is even, equivalently if and only if its complex dimension is divisible by four. This is developed.

Summary

A representation of a unital $F$-algebra $A$ is a unital homomorphism $A\to\operatorname{End}_F(V)$, equivalently a left $A$-module, and the two notions form equivalent categories whose morphisms are the intertwining operators. The irreducible representations are the simple modules, that is the quotients $A/\mathrm{M}$ by maximal left ideals; Schur's lemma makes the endomorphism algebra of an irreducible representation a division algebra, which is $F$ when $F$ is algebraically closed and the representation is finite-dimensional. The density theorem gives the double centralizer theorem: an irreducible finite-dimensional representation with commutant $D$ and dimension $n$ over $D$ exhibits $A/\operatorname{Ann}_A(V)$ as $M_n(D^{\mathrm{op}})$. An algebra is semisimple exactly when every representation is completely reducible, and then $A\cong\prod_i M_{n_i}(D_i)$ by Wedderburn–Artin; for a finite-dimensional algebra $A/J(A)\cong\prod_i M_{n_i}(D_i)$, with $\sum_i n_i^2\leq\dim_F A$ over an algebraically closed field and equality exactly in the semisimple case. The modules over $M_n(F)$ are the direct sums of copies of the defining representation $F^n$, the unique irreducible one, and $M_n(F)\cong S^{\oplus n}$; endomorphism and automorphism groups are those of the type $\operatorname{End}_{M_n(F)}(S)=F$, $\operatorname{Aut}_{M_n(F)}(M_n(F))=GL_n(F)$.

The structure of group algebras is the principal specialisation: over a field whose characteristic does not divide $|G|$ the group algebra is a product of matrix algebras over division algebras, with $|G|=\sum n_i^2\dim_F D_i$; over an algebraically closed field this is $\sum n_i^2=|G|$ and the number of irreducible representations is the number of conjugacy classes; in the modular case the group algebra splits into blocks. Idempotents produce the projective representations, the primitive idempotents produce the irreducible representations in the semisimple case, and the Cartan matrix records the multiplicities in the modular case. The examples — division algebras, matrix algebras, polynomial algebras with their normal forms, local algebras, group algebras, and the biquaternion algebra $\mathbb{B}\cong M_2(\mathbb{C})$ with its unique irreducible representation $S=\mathbb{C}^2$ — exhaust the range of the theory.

Summary of Notation

Symbol	Meaning
$R$	commutative ring with identity, the base ring
$F$	a field
$A$	unital associative $F$-algebra
$V$, $W$	representations, i.e. left $A$-modules
$\rho : A \to \operatorname{End}_F(V)$	a representation
$\rho_{\mathrm{reg}}$	left regular representation of $A$
$\operatorname{Rep}_F(A)\cong\operatorname{Mod}(A)$	equivalence of representations with modules
$\operatorname{End}_A(V)=\rho(A)'$	intertwining operators, the commutant
$\operatorname{Aut}_A(V)$	invertible intertwining operators
$A/\mathrm{M}$	irreducible representation from a maximal left ideal
$J(A)$	Jacobson radical
$D=\operatorname{End}_A(S)$	division algebra of an irreducible representation
$A/\operatorname{Ann}_A(V)\cong M_n(D^{\mathrm{op}})$	double centralizer form
$A\cong\prod_i M_{n_i}(D_i)$	Wedderburn–Artin decomposition
$n_i=\dim_{D_i}S_i$	degrees, with $\sum_i n_i^2\leq\dim_F A$
$M_n(F)$, $S=F^n$	matrix algebra and its defining representation
$Ae$	projective representation from an idempotent
$F[G]$, $B_b$	group algebra and its blocks
$Q=(Q_0,Q_1,s,t)$	quiver with vertices, arrows, source and target
$\operatorname{Rep}_F(Q)\simeq\operatorname{Mod}(FQ)$	quiver representations and modules over the path algebra
$FQ$, $e_x$	path algebra and the trivial path at a vertex
$\underline{\dim}\,V$	dimension vector of a quiver representation
$A_n,D_n,E_6,E_7,E_8$	Dynkin diagrams of finite representation type
$\mathbb{H}$, $\mathbb{B}$	quaternions, biquaternions
$S=\mathbb{C}^2$	irreducible representation of $\mathbb{B}\cong M_2(\mathbb{C})$