I have been looking at the Chevalley Eilenberg complex $CE_*(\mathfrak g)$ of a Lie algebra $\mathfrak g$ over a field $k$.
$$ \wedge^3\mathfrak g\longrightarrow \wedge^2\mathfrak g\longrightarrow \wedge^1\mathfrak g \longrightarrow k$$
Let $A$ be a $k$-algebra and $M(A)$ the limit of matrix rings $M_r(A)$ with coefficients in $A$. Let $\mathfrak{gl}(A)$ be the same as $M(A)$, but treated as a Lie algebra with standard commutator of matrices.
I am trying to understand the product structure on $H_\bullet(\mathfrak{gl}(A))$.
Loday says it is induced by the direct sum of matrices
$$ \oplus : \mathfrak{gl}(A)\times \mathfrak{gl}(A) \longrightarrow \mathfrak{gl}(A) $$
However, the exact details are not clear to me. After much searching, I found Dale Husemoller's lecture notes which sketches here (see Page 54)
http://www.math.tifr.res.in/~publ/ln/tifr83.pdf
that the process is as follows :
There is an isomorphism (only quasi-isomorphism perhaps?)
$$CE_*(\mathfrak g_1)\otimes CE_*(\mathfrak g_2)\longrightarrow CE_*(g_1\oplus g_2)$$
presumably for two Lie algebras $\mathfrak g_1$, $\mathfrak g_2$. This then leads to
$$ CE_*(\mathfrak{gl}_n(A))\otimes CE_*(\mathfrak{gl}_n(A))\longrightarrow CE_*(\mathfrak{gl}_n(A)\oplus \mathfrak{gl}_n(A))\overset{\oplus}{\longrightarrow} CE_*(\mathfrak{gl}_{2n}(A))$$ and finally composed with the inclusion $$CE_*(\mathfrak{gl}_{2n}(A))\longrightarrow CE_*(\mathfrak{gl}(A))$$ Then limit is taken over all $n$.
Here, again the explicit isomorphism $$CE_*(\mathfrak g_1)\otimes CE_*(\mathfrak g_2)\longrightarrow CE_*(g_1\oplus g_2)$$ is not clear to me. Is there a reference where this map or the product structure on $H_\bullet(\mathfrak{gl}(A))$ has been described clearly?
Thanks
