6 2. Basic notions of representation theory A nonzero representation V of A is said to be irreducible if its only subrepresentations are 0 and V itself, and it is said to be inde- composable if it cannot be written as a direct sum of two nonzero subrepresentations. Obviously, irreducible implies indecomposable, but not vice versa. Typical problems of representation theory are as follows: (1) Classify irreducible representations of a given algebra A. (2) Classify indecomposable representations of A. (3) Do (1) and (2) restricting to finite dimensional representa- tions. As mentioned above, the algebra A is often given to us by gener- ators and relations. For example, the universal enveloping algebra U of the Lie algebra sl(2) is generated by h, e, f with defining relations (2.1.1) he eh = 2e, hf fh = −2f, ef fe = h. This means that the problem of finding, say, N-dimensional represen- tations of A reduces to solving a bunch of nonlinear algebraic equa- tions with respect to a bunch of unknown N ×N matrices, for example system (2.1.1) with respect to unknown matrices h, e, f. It is really striking that such, at first glance hopelessly compli- cated, systems of equations can in fact be solved completely by meth- ods of representation theory! For example, we will prove the following theorem. Theorem 2.1.1. Let k = C be the field of complex numbers. Then: (i) The algebra U has exactly one irreducible representation Vd of each dimension, up to equivalence this representation is realized in the space of homogeneous polynomials of two variables x, y of degree d 1 and is defined by the formulas ρ(h) = x ∂x y ∂y , ρ(e) = x ∂y , ρ(f) = y ∂x . (ii) Any indecomposable finite dimensional representation of U is irreducible. That is, any finite dimensional representation of U is a direct sum of irreducible representations. As another example consider the representation theory of quivers.
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