An old query removed from universal enveloping algebra and archived here:

Eric: Is this a special case of universal enveloping algebra as it pertains to Lie algebras? I thought the concept of a universal enveloping algebra was more general than this. I scribbled some notes here. They are far from rigorous, but the references at the bottom of the page are certainly rigorous. I don’t remember them being confined to Lie algebras. I’m likely confused.

[Edit: Oh! I see now. From enveloping algebra you link to this page and call it

enveloping algebra of a Lie algebra. Would that be a better name for this page? Or maybeuniversal enveloping algebra of a Lie algebra? Something to make it clear this page is specific to Lie algebras?]Zoran: if you read the above article than you see that it distingusihes the enveloping algebra of a Lie algebra and universal enveloping algebra of a Lie algebra which is a universal one among all such. There is also an enveloping algebra of an associative algebra what is a different notion.

Also added to universal enveloping algebra, a link to a MathOverflow question What is the universal enveloping algebra which is looking for a rather general construction in a class of symmetric monoidal pseudoabelian categories. I also created a minimal literature section.

]]>Let $\mathfrak{g}$ be an $L_\infty$-algebra (over a characteristic zero field $k$). then, if $\mathfrak{g}$ is finite-dimensional in each degree, the $L_\infty$-algebra $inn(\mathfrak{g})$ can be defined as the $L_\infty$-algebra whose Chevalley-Eilenberg algebra is the Weil algebra $W(\mathfrak{g})$ of $\mathfrak{g}$. The Weil algebra has a remarkable freeness property:

$Hom_{dgca}(W(\mathfrak{g}),\Omega^\bullet)=Hom_{dgVect}(\mathfrak{g}^*[-1],\Omega^\bullet)=\Omega^1(\mathfrak{g}),$where $\Omega^\bullet$ is an arbitrary dgca and $\Omega^i(\mathfrak{g})$ denotes the vector space of degree $i$ elements of $\Omega^\bullet\otimes\mathfrak{g}$. The identification can then be used to define *curvature* of $\mathfrak{g}$-connections, and to write down their Bianchi identities. Indeed

and the underlying graded vector space of $inn(\mathfrak{g})$ is $\mathfrak{g}\oplus\mathfrak{g}[1]$, so that the identification together with the freeness property of $W(\mathfrak{g})$ gives the following:

for any $A\in \Omega^1(\mathfrak{g})$ there exists a unique $F_A\in \Omega^2(\mathfrak{g})$ such that the pair $(A,F_A)$ satisfies the Maurer-Cartan equation in $\Omega^\bullet(inn(\mathfrak{g})$. The element $F_A$ is the curvature of $A$ and the Maurer-Cartan equation expresses both the relation between $A$ and $F_A$ and the Bianchi identities.

To write these equations in a fully explicit form in a way that allows making contact with classical equations from differential geometry, it would be convenient to have the $L_\infty$-algebra structure on $inn(\mathfrak{g})$ spelled out in terms of lots of brackets (in terms of the lots of brackets defining the $L_\infty$-algebra sturcuture on $\mathfrak{g}$). Do we have these brackets already spelled out somewhere?

]]>New entry from Lie theory, Hausdorff series (and stub Bernoulli number, for now just one reference, the one from John Baez). It would be nice to understand also the higher analogues of Hausdorff series which play a major role in the Annals paper of Ezra Getzler on Lie infinity integration.

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