16 2. Basic notions of representation theory

because a(b −b) ∈ aI ⊆ I = ker π, as I is also a left ideal. Thus, A/I

is an algebra.

Similarly, if V is a representation of A and W ⊂ V is a subrep-

resentation, then V/W is also a representation. Indeed, let π : V →

V/W be the quotient map, and set ρV/W (a)π(x) := π(ρV (a)x).

Above we noted that left ideals of A are subrepresentations of the

regular representation of A, and vice versa. Thus, if I is a left ideal

in A, then A/I is a representation of A.

Problem 2.5.1. Let A = k[x1,...,xn] and let I = A be any ideal

in A containing all homogeneous polynomials of degree ≥ N. Show

that A/I is an indecomposable representation of A.

Problem 2.5.2. Let V = 0 be a representation of A. We say that a

vector v ∈ V is cyclic if it generates V , i.e., Av = V . A representation

admitting a cyclic vector is said to be cyclic. Show the following:

(a) V is irreducible if and only if all nonzero vectors of V are

cyclic.

(b) V is cyclic if and only if it is isomorphic to A/I, where I is a

left ideal in A.

(c) Give an example of an indecomposable representation which

is not cyclic.

Hint: Let A = C[x, y]/I2, where I2 is the ideal spanned by ho-

mogeneous polynomials of degree ≥ 2 (so A has a basis 1,x,y). Let

V =

A∗

be the space of linear functionals on A, with the action of A

given by (ρ(a)f)(b) = f(ba). Show that V provides such an example.

2.6. Algebras defined by generators and

relations

If f1,...,fm are elements of the free algebra k x1,...,xn , we say

that the algebra A := k x1,...,xn /{f1,...,fm} is generated by

x1,...,xn with defining relations f1 = 0, . . . , fm = 0.