Due May 10.

Let \(G\) be a Lie group and \(v \in T_eG\). Recall that:

  • There is a unique left-invariant vector field \(V \in \mathfrak {X}(G)\) with \(V(e) = v\).
  • The flow \(\Phi ^V : G \times \mathbb {R} \to G \) is defined everywhere for all time.

Let \(\gamma : \mathbb {R} \to G\) be the flow-line of \(V\) with \(\gamma (0) = e\).

  • Show that for each \(g \in G\), one has

    \[ \Phi _t^V (g) = g \cdot \gamma (t) . \]
  • Show that \(\gamma \) is a group homomorphism.
Let \(V\) be a vector space with basis \(\{ e_1, \ldots , e_m \}\) and \(W\) a vector space with basis \(\{ f_1, \ldots , f_n \}\). Using the following outline, show that the set
\[ B : = \{ e_i \otimes f_j \, \vert \, 1 \leq i \leq m , 1 \leq j \leq n \, \} \]
is a basis of \(V \otimes W\).
  • Show that the set \(j (V \times W) \subset V \otimes W\) in the definition of tensor product spans \(V \otimes W\).
  • Show that \(B\) spans \(V \otimes W\).
  • Show that the dimension of \(V \otimes W\) is at least the dimension of the vector space of bilinear maps \(V \times W \to \mathbb {R}\).

  • Conclude using the fact that the space of bilinear maps \(V \times W \to \mathbb {R}\) has dimension

    \[ \dim (V) \dim (W) .\]
Let \(k \geq 2\) and \(V, X\) real vector spaces. A \(k\)-multilinear map is a map
\[ \phi : V \times \cdots \times V \to X \]
that when all but one coordinates are fixed, it depends linearly on the remaining coordinate. Namely, for each \(v_1, \ldots , v_{k-1} \in V\), the maps
\begin{align*} \phi ( \cdot , v_1, \ldots , v_{k-1}) & : V \to X \\ \phi ( v_1, \cdot , \ldots , v_{k-1}) & : V \to X \\ \ldots & \\ \phi ( v_1, \ldots , v_{k-1}, \cdot ) & : V \to X \end{align*}

are linear. The \(k\)-th iterated tensor product is defined inductively as \(V^{\otimes 0} = \mathbb {R}\), \(V^{\otimes 1} = V\), and

\[ V^{\otimes k} : = V^{\otimes (k-1)} \otimes V . \]
Show there is a \(k\)-multilinear function \(j_k : V\times \cdots \times V \to V^{\otimes k}\) such that the following holds:

For each \(k\)-multilinear map \(\phi : V \times \cdots \times V \to X\), there is a unique linear map \(\tilde {\phi } : V ^{\otimes k} \to X\) with \( \tilde {\phi } \circ j _k = \phi \).

Take for granted the case \(k = 2\) and use it as the base of induction.

We denote \(j_k (v_1, \ldots , v_k) \) by \(v_1 \otimes \cdots \otimes v_k\). The map \(j_{k + \ell } \) defines a product

\[ V^{\otimes k} \times V^{\otimes \ell } \to V^{\otimes (k + \ell )} , \]

making \(\mathcal {T}(V)\) a (non-commutative) algebra.

Let \(k \geq 2\) and \(V\) a real vector space with basis \(\{ e_1, \ldots , e_n \}\). Show that the set
\[ B_k : = \{ e_{i_1} \otimes \cdots \otimes e_{i_k} \, \vert \, i_1, \ldots , i_k \in \{ 1, \ldots , n \} \,\} \]
is a basis of \(V ^{ \otimes k }\).
Let \(V\) be a vector space with basis \(\{ e_1 , \ldots , e_n \}\). Recall that the exterior algebra \(\wedge ^{\ast } V\) is the quotient of the tensor algebra \(\mathcal {T}(V)\) over the ideal generated by the set
\[ I_{ext} ^2 : = \text {span}( \{ a \otimes b + b \otimes a \, \vert \, a,b \in V \} ) . \]
For \(k \in \mathbb {N}\), we denote by \(\wedge ^k V\) the image of \(V^{\otimes k}\) in \(\wedge ^{\ast } V\). For \(v_1 \otimes \cdots \otimes v_k \in V^{\otimes k}\), denote by \(\tfrac {1}{k!} v_1 \wedge \ldots \wedge v_k \in \wedge ^{k}V\) its image in the quotient. Show that the set
\[ B^{\wedge }_k : = \{ e_{i_1} \wedge \ldots \wedge e_{i_k}\, \vert \, 1 \leq i_1 < \ldots < i_k \leq n \, \} \]
is a basis of \(\wedge ^k V\).

Hint: A \(k\)-multilinear map \( \phi : V \times \cdots \times V \to X\) is said to be alternating if for any vectors \(v_1, \ldots , v_k \in V\) and any permutation \(\sigma : \{ 1, \ldots , k \} \to \{ 1, \ldots , k \}\), one has

\[ \phi (v_{\sigma (1)}, \ldots , v_{\sigma (k)}) = \text {sign} (\sigma ) \, \phi ( v_1, \ldots , v_k ) . \]
where sign\((\sigma ) \in \{ -1 , 1 \} \) denotes the sign of the permutation. Show that for any alternating \(k\)-multilinear map \(\phi : V\times \cdots \times V \to X\), there is a unique linear map \(\overline {\phi } : \wedge ^k V \to X\) with
\[ \phi (v_1, \ldots , v_k) = \overline {\phi } (v_1 \wedge \ldots \wedge v_k) \]
for all \(v_1, \ldots , v_k \in V\). First show that \(\wedge ^k V = V^{\otimes k} / I_{ext}^k\) with
\[ I_{ext}^k = \text {span} \left ( \, \, \bigcup _{i = 0}^{k-2} V^{\otimes i} \otimes I_{ext}^2 \otimes V^{k-2-i} \right ) . \]