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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