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.
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