40 2. Basic notions of representation theory Problem 2.16.2. Classify irreducible finite dimensional representa- tions of the two-dimensional Lie algebra with basis X, Y and commu- tation relation [X, Y ] = Y . Consider the cases of zero and positive characteristic. Is the Lie theorem true in positive characteristic? Problem 2.16.3. (Hard!) For any element x of a Lie algebra g let ad(x) denote the operator g → g, y → [x, y]. Consider the Lie algebra gn generated by two elements x, y with the defining relations ad(x)2(y) = ad(y)n+1(x) = 0. (a) Show that the Lie algebras g1, g2, g3 are finite dimensional and find their dimensions. (b) (Harder!) Show that the Lie algebra g4 has infinite dimension. Construct explicitly a basis of this algebra. Problem 2.16.4. Classify irreducible representations of the Lie al- gebra sl(2) over an algebraically closed field k of characteristic p 2. Problem 2.16.5. Let k be an algebraically closed field of character- istic zero, and let q ∈ k×,q = ±1. The quantum enveloping algebra Uq(sl(2)) is the algebra generated by e, f, K, K−1 with relations KeK−1 = q2e, KfK−1 = q−2f, [e, f] = K − K−1 q − q−1 (if you formally set K = qh, you’ll see that this algebra, in an appro- priate sense, “degenerates” to U(sl(2)) as q → 1). Classify irreducible representations of Uq(sl(2)). Consider separately the cases of q being a root of unity and q not being a root of unity.

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