CHAPTER II: FINITE DIMENSIONAL ALGEBRAS

1. Introduction

We wish now to consider in general the structure of finite dimensional algebras A, first over a field k and then over an integral domain R. Some of what we describe will be true of wider classes of rings and algebras.

The general setting in which we will be operating for the remainder of the book will be the following. There will be a commutative ring R which will, in fact, be an integral domain, and a distinguished maximal ideal M, with a fixed isomorphism

R/M ˜→k.

We have a second field, K, which is the quotient field of R, and we will work with this triple (k, R, K), where the ring R provides a bridge between the two fields k and K. We will have an R-algebra A which is free as an R-module, and this will give us a corresponding triple of algebras (\overline{A}, A, Â), where

\overline{A} = A ⊗ k

and

Â = A ⊗ K.

The algebra A will be a deformation over R, with \overline{A} called the special fiber and Â the generic fiber.

Example 1.1. Let R be k[t], let M be the ideal (t), so that there is a natural homomorphism f : R → k with Ker f = M, given by substituting zero for t. The quotient field K of R is just k(t), a transcendental extension of degree 1. Let

A = k[t][x]/(x^{3} − t^{2}x).

Then

\overline{A} ˜→k[x]/x^{3},

while x^{3} − t^{2}x has three distinct roots over K and thus we can apply the Chinese Remainder theorem, so

Â ˜→K ⊕ K ⊕ K.

Our first concern will be to describe the structure of an algebra over a field.

Date: Nov. 9, ’03.

2. Finite dimensional algebras over a field

Let A be an finite dimensional algebra over a field k. Let us assume that k is sufficiently large that A/ Rad(A) ˜→ ⨁_{i} M_{n_{i}}(k). The diagonal idempotents of the matrix blocks provide a decomposition of the identity 1_{A} into a sum of primitive orthogonal idempotents, modulo Rad(A). We now lift these to a complete set of primitive idempotents of A by using the following theorem from ring theory:

Lemma 2.1. Let A be a k-algebra, I an ideal such that ∩_{i=1}^{∞}I^{i} = 0, and suppose that A/I ˜→S, where S is a direct sum of matrix blocks over a field k. Then f has a splitting, i.e., there is an embedding of g : S → A such the f ∘ g = id_{S}. The primitive idempotents in two different liftings are conjugate.

Corollary 2.1.1. If I = Rad(A), then two different complete sets of orthogonal primitive idempotents for a finite dimensional algebra are conjugate. Any two dempotents whose reductions modulo I lie in the same matrix block are conjugate.

Proof. If we have two different complete sets of primitive orthogonal idempotents, e_{i} and e′_{i}, then their images in S are conjugate by the Skolem-Noether theorem. Since Ker(f) = Rad(A), the invertible element \overline{u} of S by which they are conjugated lifts to a unit u of A. Conjugation of e′_{i} by u provides a set e′′_{i} of idempotents such that f(e_{i}) = f(e′′_{i}) for each i. Then, by the lemma, the two sets are conjugate.

If we have two individual idempotents {e_{1}, e_{2}} in S, lying in the same matrix blocks, then complete each e_{i} to a set of orthogonal primitive idempotents, using the fact that 1_{A} − e_{i} is an idempotent orthogonal to e_{i}, and apply the theorem. □

Example 2.1. Consider the upper triangular 2 × 2 matrices over k, which we will designate by T^{2}. The radical is the one-dimensional set of off-diagonal elements generated by E_{12}. The semisimple quotient is S ˜→k ⊕ k. The different possible pairs of orthogonal primitive idempotents are E_{11} + aE_{12}, E_{22} − aE_{12}. If u = E_{11} + aE_{12} + E_{22}, then

u^{−1}E_{11}u = E_{11} + aE_{12},

and

u^{−1}E_{22}u = E_{22} − aE_{12}.

Thus every possible pair of orthogonal primitive idempotents is conjugate to the standard pair E_{11}, E_{22}

Definition 2.1. The degree m(e_{i}) of a primitive idempotent e_{i} is the degree of the matrix block to which it belongs. It is also called the multiplicity of the idempotent, because it coincides with the number of conjugate idempotents in a decomposition of the identity I_{A} of A as a sum of orthogonal primitive idempotents.

Example 2.2. In a group algebra over a field whose characteristic does not divide the order of the group, the degree of a primitive idempotent e_{i} is the degree of the irreducible representation corresponding to the matrix block in which the e_{i} lies.

Definition 2.2. Let A be an algebra over a field k, and let e_{i} be a complete set of primitive idempotents. The projective decomposition of A with respect to this set is the module direct sum decomposition

A = ⨁ Ae_{j}.

The Peirce decomposition of A is a vector space direct sum decomposition

A = ⨁ e_{i}Ae_{j}.

Note. With regard to the dimensions of the summands in the decomposition, there is always complete symmetry with regard to idempotents in the same matrix block. If we hold e_{j} fixed and allow the e_{i} to range within a given matrix block then the dimensions will be constant, because the matrix units provide isomorphisms between the summands corresponding to different summands.

Definition 2.3. The basis graph of a finite dimensional algebra A over a field k is a graph with one vertex for each primitive idempotent, and

(1) n_{ij} arrows from e_{j} to e_{i}, for i ≠ j,

(2) n_{ii} − 1 loops at e_{i}, where

n_{ij} = dim_{k}(e_{i}Ae_{j}).

Remark 2.1. This definition is independent of the particular choice of complete primitive idempotent set, since any other set is conjugate, and conjugation, being an automorphism of the algebra, will preserve the dimensions of the summands of the Peirce decomposition.

In order that the basis graph should convey more information about the structure of the algebra, we will consider the Peirce decomposition of the powers of the radical, which gives a filtration of the Peirce components. Define

n_{ij}^{t} = dim_{k} e_{i}(J^{t}/J^{t+1})e_{j}, for t = 0, 1, ...

Definition 2.4. The weighted basis graph is obtained from the basis graph of an algebra A by assigning weight t to n_{ij}^{t} loops or arrows for t ≥ 0 and weight 0 to one arrow from e_{i} to e_{j} if i and j are different but lie in the same matrix block in A/ Rad(A). We represent nonidempotent matrix units by arrows with a solid head, and all other weighted arrows by using a number of arrowheads equal to the weight of the arrow.

Example 2.3. (Four-dimensional algebras) We draw the basis graphs of several different algebras of dimension four. Notice that the blocks are simply the connected components:

(1) k ⊕ k ⊕ k ⊕ k,

(2) M_{2}(k),

(3) k ⊕ T^{2},

(4) k < x, y > /(x^{2}, y^{2}, xy + yx),

(5) An algebra with two idempotents e_{1} and e_{2}, and two radical elements x and y satisfying

x = e_{1}xe_{2},
y = e_{2}ye_{1},

and

xy = yx = 0.

(6) k[x]/(x^{4}), In all but (4), the algebra is uniquely determined by the weighted basis graph. The weighted basis graph in (4) corresponds to an entire family of algebras of the form

k < x, y > /(x^{2}, y^{2}, xy − syx),

for s an element of k.

Example 2.4. We now give weighted basis graphs for the group algebra of S_{3} for characteristic 2,3,and 5. Let

S_{3} =< a, b|a^{2} = b^{3} = e, aba = b^{2} > .

The character table, as was given in the exercises at the end of the previous chapter, is

X = [\begin{matrix}1 & 1 & 1 \\ 1 & 1 & −1 \\ 2 & −1 & 0 \end{matrix}] , N = diag(1, 1, 2), andM = diag(1, 2, 3).

Thus

W = [\begin{matrix}1 & 2 & 3 \\ 1 & 2 & −3 \\ 1 & −1 & 0 \end{matrix}] .

Now,

W^{−1} = 1/6 [\begin{matrix}1 & 1 & 4 \\ 1 & 1 & −2 \\ 1 & −1 & 0 \end{matrix}] .

Since W represents the coefficients of the e_{i} in the basis of the Ĉ_{j}, this means that the coefficients of the Ĉ_{j} in the block idempotents e_{i} are given by the columns of W^{−1}.

Case 1: char(k) = 2. In this case the first two blocks coalesce, giving a block defined by the fact that the third idempotent, which, mod 2,

Figure 1. Basis graphs for some four dimensional algebras

equals b + b^{2}, is zero. Since b is invertible, this means that b = 1, so we have the group algebra of C_{2} over a field of characteristic 2, which is isomorphic to k[x]/(x^{2}), with x = a − e.

Case 2: char(k) = 3. Modulo 3, all the rows of W are identical, so there is only one block. The ideal J defined by x = (b−b^{2}) is nilpotent, and its quotient is isomorphic to the semisimple algebra kC_{2}, so is the radical. The idempotents of kC_{2} lift to idempotents e_{1} = (e + a)/2 and e_{2} = (e−a)/2. Thus we have the following nonzero Peirce components: e_{1}(A/J)e_{1} = e_{1} + J, e_{2}(A/J)e_{2} = e_{2} + J, e_{1}(J/J^{2})e_{2}= e_{1}xe_{2} + J^{2}, e_{2}(J/J^{2})e_{1}= e_{2}xe_{1} + J^{2}, e_{1}(J^{2})e_{1}=e_{1}x^{2}e_{1}, and e_{2}(J^{2})e_{2}=e_{2}x^{2}e_{2}. Thus the weighted basis graph has two vertices, two arrows of weight 1, and two loops of weight 2.

Case 3: char(k) = 5. This is the same as the characteristic zero case, so there are two blocks of dimension 1 and one matrix block of degree 2 and dimension 4.

Definition 2.5. The points of a finite dimensional algebra A are the conjugacy classes of primitive idempotents, under conjugacy by the set of units A∗ of A. The set of points is denoted by P(A).

Lemma 2.2. An element u of A is a unit if and only if it is a unit modulo Rad(A).

Proof. One direction is trivial, for if v is the inverse of u in A, its image under the natural epimorphism is inverse to \overline{u} in A/ Rad(A).

In the other direction, we use induction on the powers of the radical. Let A_{i} denote A/ Rad^{i}(A). Suppose u_{i+1} is an element of A_{i+1} and u_{i} is its image under the natural epimorphism from A_{i+1} onto A_{i}. If u_{i} is invertible with inverse v_{i}, then let v be an arbitrary lifting of v_{i} to A_{i+1}. Clearly, u_{i+1}v = 1 + d, for d ∈ Rad^{i}(A). Since Rad^{i}(A)/ Rad^{i+1} is an A_{1} module, we now replace v by v′ = v(1 − d), giving u_{i+1}v′ = 1 Now suppose that

v′u_{i+1} = 1 + d′.

Multiplying on the left by v′ and using fact that it is a left inverse, we get d′v′ = 0. Multiplying the expression for v′u_{i+1} by (1 + d′) on the right, we conclude that d′ = 0, and thus v′ is the desired inverse. □

Example 2.5. Let A be the fifth algebra in Example 2.3 with k = ℚ. Thus 1_{A} = e_{1} + e_{2}, x = e_{1}xe_{2} and y = e_{2}ye_{1}. Let u_{1} = (1/2)e_{1} + (1/3)e_{2} + 5x + 7y. Set v = 2e_{1} + 3e_{2}. Then

u_{1}v = e_{1} + e_{2} + 15x + 14y.

We thus define v′ = 2e_{1} + 3e_{2} − 30x − 42y. Then u_{1}v′ = 1_{A}, and

v′u_{1} = e_{1} + e_{2} − 10x − 21y + 10x + 21y = 1_{A}

In Section 4 of Chapter 1, we stressed the importance of writing a module as a direct sum of indecomposable submodules. One of the most important modules over the k-algebra A is A as a left module over itself. In this case, the decomposition of 1_{A} as a sum of primitive orthogonal idempotents induces a decomposition of A as a direct sum of modules Ae_{i}.

Lemma 2.3. If e_{i} is a primitive idempotent, then Ae_{i} is an indecomposable module.

Proof. If we have a nontrivial direct sum decomposition

Ae_{i} = P′ + P′′,

then the idempotent decomposes as e′ + e′′. If one of these is zero, then one component P′ or P′′ contains a generator and thus is the entire module. If both are non-trivial, then both are idempotents by the rules of multiplication in a direct product, and they are orthogonal. This is a contradiction to the assumption that e_{i} is primitive. □

What made this proof work was the fact that if a module contained the generating element, it contained the entire module. A corresponding fact is true for module homomorphisms from Ae_{i}: the entire homomorphism is determined by the image of e_{i}. Suppose that f : M → N is an epimorphism of modules. Any map g : Ae_{i} → N then can be lifted to a map from Ae_{i} to M by taking and arbitrary element in the preimage of g(e_{i}). This is equally true of any direct sum of modules generated by primitive idempotents, and the property is so important that it is generally taken as the defining property of this class of modules.

Definition 2.6. A module P over a ring A is called projective if for every homomorphism of A-modules g : P → N and every epimorphism of A-modules f : M → N, there exists a homomorphism h : P → M such that g = f ∘ h.

P
↙ ↓ ^{g}
M →^{f}N

We have asserted that this is the defining property of modules which are direct sums of modules Ae_{i}, or, since each Ae_{i} is a direct summand of a copy of A, direct summands of free modules A^{r}. The equivalence of these properties, and of another which is also useful, is the subject of the following lemma.

Lemma 2.4. Let A be a unitary Noetherian ring. For a finitely generated A-module P, The following properties are equivalent:

(1) P is a direct sum of modules isomorphic to Ae_{i} for e_{i} a primitive idempotent,

(2) P is projective,

(3) If f : M → P is an epimorphism, then there is a monomorphism h : P → M such that id_{P} = f ∘ h.

Proof. We show the equivalence of the given properties:

• (1) ⇒ (2): Since

Hom_{A}(⨁ P_{i}, N) ˜→ ⨁ Hom_{A}(P_{i}, N),

we check property (2) when P = Ae_{i}. Letting m be any preimage of g(e_{i}), we define h by h(a · e_{i}) = a · m.

• (2) ⇒ (3): This is just a specialization to the case g = id_{P}.

• (3) ⇒ (1):The module P, like any other finitely generated module, is isomorphic to a quotient of a free module. But then, but property (3), it is a direct summand of a free module. Since A is Noetherian, 1_{A} has a decomposition as a sum of a finite number of primitive idempotents, for otherwise we could find an ascending sequence of left ideals generated by idempotents. Thus any free module A^{r} has a decomposition into factors of the form Ae_{i}. By the Krull-Schmidt Theorem, every direct summand has this form, so we get (1).

□

Lemma 2.5. For a left Artinian ring A,the following sets are all in one-to-one correspondence:

(1) P(A)

(2) Max(A) = {I ⊆ A∥Imaximal}

(3) Irr(A) = {simple modulesofA}

Proof. We have assumed that k is sufficiently large that

ϕ : A/ Rad(A) ˜→ ⨁_{i+1}^{t} M_{n_{i}}(k).

For each i, we define a maximal ideal as the preimage of the sum of the other blocks, and the maximal ideals are all of this form. The block corresponds to a well-defined simple module, an element of Irr(A), and finally, by Lemma 2.1, two primitive idempotents whose image under ϕ lie in the same block are conjugate, and the condition is also necessary, since conjugation induces an automporphism on each block. □

Dual to projective modules are injective modules. Let us first state what we mean by ”dual”. Let A be a finite dimensional algebra over a field. We surely expect A and its opposite algebra A^{op} to have essentially the same representation theory, since they differ only in a convention of left and right. If, for example, A had only a finite number of indecomposable modules, we would expect A^{op} to have the same number of indecomposable modules of each dimension.

The function which allows us to pass from modules over A to modules over A^{op} is called the dualizing functor or transpose. Let A − mod be the collection of all finitely generated A-modules, and define

tr : A − mod → A^{op} − mod

by

tr(M) = Hom_{k}(M, k).

Since M is finite dimensional as a k-vector space, then by standard properties of dual vector spaces, tr(M) is a vector space of the same dimension as M. The action of an element a of A^{op}on a linear transformation f : M → k is given by

a · f(m) = f(am),

which satisfies

(ab) · f(m) = f(ab · m) = a · f(bm) = b · (a · f)(m).

Now we can take an injective module for A to be the dual of a projective module for A^{op}. The vector space dual interchanges submodules with quotients, epimorphisms with monomorphism, and for modules, the top with the socle. Each primitive idempotent e_{i} for A is also a primitive idempotent for A^{op}. If P_{i} is the projective module Ae_{i} and P′_{i} is the projective module A^{op}e_{i}, then I_{i} = Hom_{k}(P′_{i}, k) is the corresponding injective module for A. Since e_{i} is primitive, it lies in unique matrix block of A/ Rad(A). Let S_{i} be the simple module corresponding to that matrix block. Then Top(P_{i}) is isomorphic to the column of the matrix block generated by e_{i}, which is a simple module S_{i}. The top of P′_{i} is the simple S′_{i} corresponding to the row generated by e_{i}. Finally, I_{i} is a module whose socle is isomorphic to S_{i}.

In terms of the basis graph, P_{i} has a basis consisting of all arrows leaving e_{i}. The basis graph of A_{op} is just the basis graph of A with all arrows reversed, so in this dual graph, P′_{i} has a basis consisting of all arrows leaving e_{i}. Now we dualize, which reverse the arrows again, and thus I_{1} has a basis corresponding to all the arrows entering e_{i}. It must be stressed that while P_{i} is always a left ideal of A, I_{i} is generally not, except for cases like group algebras where A = A^{op} (and thus all injectives are projectives).

Lemma 2.6. Let A be a finite dimensional algebra over a field k. For a finitely generated A-module I, The following properties are equivalent:

(1) I is a direct sum of modules isomorphic to Hom(A^{op}e_{i}, k) for e_{i} a primitive idempotent,

(2) Ifg : N → I is a homomorphism, and f : N → M is a monomorphism, then there is a homomorphism h : M → I such that g = f ∘ h.

(3) If f : I → M is a monomorphism, then there is an epimorphism h : M → I such that id_{I} = f ∘ h.

Definition 2.7. The skeleton of a finite dimensional algebra A over a suficiently large field k is the algebra

A_{0} = e_{0}Ae_{0},

where e_{0} is an idempotent which is the sum of one idempotent from each point, taken from a complete orthogonal decomposition of the identity 1_{A}.

The skeleton is a basic algebra, that it,

A_{0}/ Rad(A_{0}) ˜→k^{t}.

The vertices in the basis graph of A_{0} are in one-to-one correspondence with the points of A and of A_{0}.

One of the most important examples of an algebra useful in representation theory is the algebra End_{A}(M), where M is a module over a left Artinian ring A. A homomorphism ϕ of modules over a ring is idempotent if and only if it is a projection map. Since the image of a homomorphism of modules is itself a module, then the idempotent is a projection onto a submodule N of M. It is primitive if and only if it cannot be written as the sum of two orthogonal projections onto submodules of N. If such orthogonal projections exist, then N is a direct sum of their images, and conversely, if N decomposes as a direct sum, then the projections onto the summands give a decomposition of N, which is therefore decomposable. Thus the primitive idempotents of End_{A}(M) correspond to the indecomposable submodules of M.

Figure 2. Weighted basis graph for an endomorphism algebra

Now let us ask what are the points of End_{A}(M)? These are conjugacy classes of primitive idempotents, under conjugacy by invertible elements of End_{A}(M). However, the invertible elements of End_{A}(M) are precisely the A-automorphisms of M. If π is a projection on an indecomposable submodule N, and α : M ˜→M is an automorphism, the α ∘ π ∘ α^{−1} is the projection on the isomorphic module α(N). Thus the points of End_{A}(M) are in one to one correspondence with isomorphism classes of indecomposable direct summands of M.

Example 2.6. Let A be the ring k[x]/(x^{3}), for k = ℚ. There are three nonzero indecomposable modules for A, these being isomorphic to the left ideals N_{3} = A, N_{2} = (x), and N_{1} = (x^{2}). Each N_{i} is of dimension i. By a general theorem in ring theory,

End_{A}(A) ˜→A^{op},

and since A in this case is commutative, A^{op} →^{∼} A. For the regular representation, there is a single point, corresponding to the conjugacy class of automorphisms of A.

Let us now take a second module N′_{1} isomorphic to N_{1} and construct a module M = N_{1} ⊕N′_{1}. Since x acts as 0 on N_{1} and N′_{1}, the A-module endomorphisms are just the k-linear transfomations of a vector space of dimension 2, to

End_{A}(M) ˜→M_{2}(k).

A primitive idempotent is just a projection of a two dimensional vector space on any one dimensional vector space. By the Skolem-Noether theorem these are all conjugate, and thus there is a single point.

If we let M = N_{1}⊕N′_{1}⊕N_{2}⊕N_{3}, then End_{A}(M) has three points, corresponding to the three isomorphism classes of indecomposables. The basis graph is given in Figure 2. Note that the natural endomorphism from N_{2} onto its radical factors either through an epimorphism onto N_{1} or a monomorphism into N_{3}. As we shall see later, the only endomporphisms between indecomposables which do not factor through an indecomposables are the epimorphisms and the monomorphisms.

Exercises (1) Find the skeleton of each of the algebras in Example 2.3.

(2) If A = A_{1} ⊗_{k} A_{2} is the tensor product of two k-algebras, prove that Rad(A) = Rad(A_{1}) ⊗ A_{2} + A_{1} ⊗ Rad(A_{2}). Conclude that

P(A) = P(A_{1}) × P(A_{2}).

(3) Find the indecomposable injective and projective modules for T^{3}.

(4) Calculate the basis graph of the algebra End_{A}(M), when A = k[x]/(x^{2}) and M is the direct sum of the two indecomposable modules of dimensions 1 and 2.

3. Morita equivalence and the quiver

Two algebras A and B with the same skeleton have equivalent representation theories, in the sense that there are functors

F : A − mod → B − mod,

and

G : B − mod → A − mod,

such that the composition in either order maps a module to an isomporphic module, i.e., the two module categories are equivalent. Furthermore, the two functors are given by tensoring with bimodules, that is, there are bimodules _{A}X_{B} and _{B}Y_{A},such that, for an A-module M and a B-module N, we have

F(M) = Y ⊗_{A} M,

and

G(N) = X ⊗_{B} N.

We will give the formal definition in terms of these bimodules:

Definition 3.1. Let A and B be finite dimensional algebras over a sufficiently large field k. The A and B are Morita equivalent if there exists bimodules _{A}X_{B} and _{B}Y_{A}, each projective as left and as right modules, such that

X ⊗_{B} Y →^{∼} A,

and

Y ⊗_{Z} X →^{∼} B.

The isomorphism is is an isomorphism of bimodules, but since the algebra structure of A is completely determined by its A − A bimodule structure, and similarly for B, we have full information about A and B from X and Y.

Example 3.1. Let A be a finite dimensional algebra with

A/ Rad(A) →^{∼} S = ⨁_{i=1}^{t} M_{n_{i}},

and let A_{0} = e_{0}Ae_{0} be its skeleton, where e_{0} is the idempotent formed by taking one primitive idempotent from each matrix block. Let X be Ae_{0}, and let Y be e_{0}A. Then X is a direct sum of one copy of each projective indecomposable A-module and is thus projective on the left. The module X is also a right A_{0} module, since ae_{0} · e_{0}a′e_{0} ∈ X for any a, a′ ∈ A.

Claim 1. X is projective as a right A_{0}-module.

Proof. Choose an embedding of S into A, which exists by Lemma 2.1. For each i = 1, …, t, let e^{i} be the idempotent of the matrix block of S, let {E_{jk}^{i}} be the matrix units associated with e^{i},and let E_{j_{i}j_{i}}^{i} = e^{i}e_{0} = e_{0}e^{i} be the primitive idempotent representing the block i in e_{0}. Then X = ⨁_{i} e^{i}Ae_{0}, and each primitive idempotent E_{jj}^{i} in block i can be written as E_{jj_{i}}e_{0}E_{j_{i}j}. Each summand is a sum of n_{i} copies of E_{j_{i}j_{i}}^{i}(e_{0}Ae_{0}), and is thus projective.

□

The proof that Y is a left projective A_{0} module is dual. That Y ⊗_{A}X is isomorphic to A_{0} is clear, since A ⊗_{A} A →^{∼} A. To show that X ⊗_{A_{0}} Y is isomporphic to A, we use the method in the claim on both sides.

Example 3.2. The matrix algebras A = M_{s}(k) and B = M_{t}(k) have the same skeleton, which is just k. To see directly that they are Morita equivalent, we set

X = M_{st},

and

Y = M_{ts}.

Then

X ⊗_{B} Y →^{∼} A,

with the isomorphism given by matrix multiplication.

Definition 3.2. The quiver Q of a finite dimensional algebra A is the directed graph consisting of the vertices and weight one arrows of the skeleton A_{0} of A in the weighted basis graph of A_{0}.

Definition 3.3. The path algebra over a field k of a directed graph Q is the algebra k[Q] which has as k-basis all the paths in Q, i.e., finite strings of arrows which are compatible in that one starts where the previous one ends. The product of two paths is zero if they are not compatible and is the composition if they are compatible.

Example 3.3. (1) The quiver of k[x]/(x^{7}) consists of a point and a loop α. The path algebra is infinite dimensional, with a basis consisting of all the powers of α.

(2) Over a field of characteristic 5, the quiver of D_{5} consists of two points and one arrow in each direction. The path algebra is again infinite dimensional.

Figure 3. Quivers of some algebras

(3) If the quiver has no oriented cycles, i.e., no paths which return to the starting vertex, then the path algebra will be finite dimensional. The quiver of the algebra of upper triangular matrices T^{n} consists of a linear string of arrows on n vertices. In this case, the path algebra of the quiver has the same dimension as the algebra. See Figure 3.

One of the most important quivers in representation theory is the Auslander-Reiten quiver. A general definition is outside the material of this book, but in the particular case when the algebra A is of finite representation type, i.e., has a finite number of isomorphism classes of indecomposable modules, then, if we let M be the direct sum of one copy of each indecomposable, the Auslander-Reiten quiver is just the quiver of the algebra End_{A}(M).

The vertices of the quiver correspond to the indecomposable modules. We generally assume that A is basic, with primitive idempotents e_{i}, and, when this suffices to distinguish the modules, we label the vertex corresponding to an indecomposable N by the dimension vector d, an integral vector whose components are

d_{i} = dim_{k}(e_{i}N).

The arrows correspond to the irreducible maps, i.e., maps which do not factor through any other indecomposable. In the finite representation case, every other homomorphism between indecomposables is a linear combination of the compositions of irreducible maps.

The following very basic results are true for algebras other than those of finite representation type:

Lemma 3.1. Every irreducible map is either a monomorphism or an epimorphism.

Lemma 3.2. The irreducible maps to a projective indecomposable P are the embeddings of the indecomposable summands of Rad(P) into P.

Lemma 3.3. The irreducible maps from an injective indecomposable I are the natural epimorphisms of I onto the indecomposable summands of the socle quotient I/ Soc(I).

Figure 4. Auslander-Reiten quiver for T^{3}

Example 3.4. (Upper triangular matrices) Let our algebra A be T^{n}, the upper triangular 3 × 3 matrices. The dimension vectors of the indecomposables are all 3-vectors with entries 0 or 1 such that the nonzero entries form a connected interval. The irreducible maps are precisely the maps into the radical or onto the socle quotient. In this case only indecomposable which is not either projective or injective is the module with dimension vector (0,1,0) but that is rare. Usually most modules are neither. We indicate the top of a projective module by a bar, and the socle of an injective module by an underline. The Auslander-Reiten quiver is given in Figure 4.

Since A is Morita equivalent to its skeleton, and Morita equivalence preserves the representation theory, the quiver is the central object in algebra representation theory. It is not generally used in group representation theory because it does not preserve the action of the group. The algebra corresponding to the skeleton in group respresentation theory is the source algebra, which will decribed later.

Because the skeleton is a basic algebra, one can make use in algebra representation theory of the following useful result:

Theorem 3.1. If A is a basic algebra over a field, then A is isomorphic to a quotient of the path algebra k[Q∗] of the dual of the quiver of A, by an ideal contained in Rad(A)^{2}.

Exercises (1) Calculate the quiver of T^{n}.

(2) Prove that M_{st}(k) ⊗ M_{t}(k)M_{ts}(k) →^{∼} M_{s}(k), where the map is given by matrix multiplication.

(3) Given natural numbers r and n and a partition

ν = (n_{1}, …, n_{r})

of n into r parts, we defined, in Chapter 1, the algebra R_{ν} of upper triangular block matrices. If m is another natural number, and

μ = (m_{1}, …, m_{r})

is a partition of m into r parts, then we can define a R_{ν} − R_{μ} bimodule M_{νμ} of upper triangular block matrices where the ij block has dimension n_{i} × m_{j}. Prove that M_{νμ} and M_{μν} determine a Morita equivalence between the algebras R_{ν} and R_{μ}.

(4) If A = A_{1} ⊗ A_{2} is the tensor product of two algebras with quivers Q_{1} and Q_{2}, and points sets P(A_{1}) and P(A_{2}), show that the quiver Q of A, which was shown before to have vertex set P(A_{1}) × P(A_{2}), is the join of the two graphs Q_{1} × P(A_{2}), with the graph P(A_{1}) × Q_{2}.

(5) Over a sufficiently large field of characteristic 2, show that the group algebra of C_{6} ×C_{2} consiste of three points, each with two loops.

(6) Find the kernal of the map from the path algebra of the dual quiver onto the algebras in Example 3.3.

4. Finite rank lattices over an integral domain

As we indicated in the introduction to this chapter, we will generally be considering not just finite dimensional algebras over a field, but also algebras over an integral domain R. We will always require that our algebra A be free of finite rank over the commutative ring R. The fact that A is free as an R-algebra, i.e., that A →^{∼} R^{n}, will be denoted by saying that A is an R-lattice. We will designate a particular maximal ideal M, and let k = R/M, K = QuotF ield(R). This then induces two algebras, the special fiber

\overline{A} = A ⊗_{R} k,

and the generic fiber

Â = A ⊗_{R} K.

Example 4.1. Let R be k[t], let M = (t), and let K = k(t). Let Q be the quiver of (5) in Example 2.3. Let A be the quotient of the path algebra R[Q] by the ideal I generated by (xyx − t^{2}x, yxy − t^{2}y), where we recall that e_{1}xe_{2} = x and e_{2}ye_{2} = y. When t = 0 this is an indecomposable algebra, and if char(k) = 3, it is isomorphic to the group algebra of S_{3}. When t ≠ 0, multiplying the relation xyx − t^{2}x on the left by y gives a new primitive idempotents

e′_{1} = (1/t^{2})xy,

and similarly the other relation gives an idempotent

e′_{2} = (1/t^{2})yx.

We get a full set of matrix units by setting

E_{12} = (1/t)x, andE_{21} = (1/t)y.

Since e′_{1}e_{1} = e′_{1} = e_{1}e′_{1} we have another idempotent e′′_{1} = e_{1} − e′_{1} and similarly e′′_{2} = e_{2} − e′_{2}. Thus the algebra when we substitute any value of t except 0 is isomorphic to

k ⊕ k ⊕ M_{2}(k),

which we will call the general fiber, and the generic fiber is

K ⊕ K ⊕ M_{2}(K),

which corresponds to the group algebra of S_{3} over a field of characteristic other than 2 or 3.

One of the most important things to note is that the primitive idempotents of the special fiber \overline{A} are the residues mod M of idempotents e_{1} and e_{2} of A, but in A these ”lifted” idempotents are no longer primitive. Rather,

e_{1} = e′_{1} + e′′_{1},

and

e_{2} = e′_{2} + e′′_{2}.

The basis graphs for \overline{A} and Â are given in Fig. 4. Both have dimension 6. When the four idempotents coallesce into two, we get two new radical elements, xy and yx, both of which appear as loops.