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