What do these tensor products actually mean? I have been self-studying tensor products from several sets of notes and numerous posts here. I apologize for what should probably be obvious. 
I would appreciate knowing what the following actually mean with a small example of each. (I do know that the $\mathbb{R}$ underset means that that is the field over which scalar multiplication occurs.)
$$\mathbb{C}\,\underset{\mathbb{R}}{\otimes}\,\mathfrak{gl}(n,\mathbb{R})$$
$$1\,\underset{\mathbb{R}}{\otimes}\,\tilde{h}_i$$
Thanks
 A: The tensor product in this case is being used as an extension of scalars.  That is: $\mathfrak{gl}$ is a real vector space, but we're defining/allowing for multiplication by complex numbers in a "natural way".
$%For ease of writing things down, let's take$
$%n = 2$
The elements of the space $\Bbb C \otimes _\Bbb R \mathfrak {gl}$ are sums of the from $\sum_{i=1}^k z_i \otimes h_i$ where for each $i$, $z_i \in \Bbb C$ and $h_i \in \mathfrak{gl}$.  The trick with the tensor product is that the product is defined to "distribute correctly".  So for instance,
$$
(2 + i) \otimes \pmatrix{1 & 2\\0 & 0} = \\
2 \otimes \pmatrix{1 & 2\\0 & 0} + 
i \otimes \pmatrix{1 & 2\\0 & 0} =\\
2\left[ 1 \otimes \pmatrix{1 & 2\\0 & 0}\right] + 
i \otimes \pmatrix{1 & 2\\0 & 0} =\\
1 \otimes \pmatrix{2 & 4\\0 & 0} + 
i \otimes \pmatrix{1 & 2\\0 & 0}
$$
In fact, what we've effectively defined are matrices with complex entries.  That is, $\Bbb C \otimes_{\Bbb R} \mathfrak{gl}(\Bbb R,n) \cong \mathfrak{gl}(\Bbb C,n)$ is an isomorphism of vector spaces (and of Lie algebras).
