Math 7290 Fall 2010 Introduction to Lie Theory P. Achar Summary: Math 7290 Fall 2010 Introduction to Lie Theory P. Achar Problem Set 3 Due: October 19, 2010 1. Prove that the set of diagonal matrices in sln is a Cartan subalgebra. (We used this example in class many times, but we never actually proved this statement.) 2. Let X be an affine variety in CN . Prove that X has only finitely many connected components. (Hint: Show that for any component X1 X, there is a polynomial f C[x1, . . . , xN ] such that f|X X1 = 0 but f|X1 = 0. Then use the fact that C[x1, . . . , xN ] is a noetherian ring.) 3. Let X CN be a variety defined by equations f1(x1, . . . , xN ) = f2(x1, . . . , xN ) = · · · = fk(x1, . . . , xN ) = 0. Assume that this set of equations is reduced, meaning that every polynomial p C[x1, . . . , xN ] that vanishes on all points of X is in the ideal generated by f1, . . . , fk. Consider a point a = (a1, . . . , aN ) X. Show that the tangent space TzX can be identified with the following subspace of CN : (b1, . . . , bN ) for all j, N Collections: Mathematics