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18.755 seventh problems, due Monday, November 9, 2015 This problem set is based on the notes “Associative algebras, Lie algebras, and bilinear forms” on the class web site. First, a definition. Suppose W is an n-dimensional real vector space. Write W ∗ for the dual vector space of W (consisting of all linear maps from W to R). Set V = W ⊕ W ∗ = {(w, λ) | w ∈ W, λ ∈ W ∗ }. Define a symmetric bilinear form on V by Q((w1 , λ1 ), (w2 , λ2 )) = λ2 (w1 ) + λ1 (w2 ). 1. The orthogonal group O(V ) is isomorphic to one of the groups O(p, q) defined in Problem Set 6. Which one? V 2. Let E(W ) be the algebra of endomorphisms of (W ) defined in Theorem 3.4 of the notes. (This means that E(W ) is the linear span of all possible products of various V m(w) and ι(λ). Prove that E(W ) is the algebra of all linear transformations of (W ). 3. (Same as Problem 2, but for differential operators). Write Pn for the algebra of polynomial coefficient differential operators in n variables (notes, Theorem 2.4): a typical element of Pn is something like 2x21 ∂ ∂2 − 7x2 . ∂x2 ∂x3 ∂x1 Prove that if f1 , . . . , fm are any linearly independent polynomials in n variables, and g1 , . . . , gm are any other polynomials in n variables, then there is an element D ∈ Pn such that Dfi = gi (1 = 1, . . . , m). 4. Suppose p and q are as in Problem 1, so that O(p, q) ≃ O(V ). Write down an isomorphism σ: C(V ) → E(W ), V the algebra of all linear transformations of (W ). Hint: this is the orthogonal version of Theorem 2.4 in the notes, and has the same proof. The restriction of σ to o(V ) ⊂ C(V ) (notes, Proposition 3.2) ^ σ: o(V ) → gl( W ) is the spin representation of the Lie algebra o(V ); it has dimension 2n = 2(dim V )/2 . of 5. If n > 0, find a proper (that is, not zero and not the whole space) subspace V (W ) that is preserved by σ(o(V )). 1