October 16, 2007

Loday and Pirashvili on Lie 2-Algebras (secretly)

Posted by Urs Schreiber Zoran Škoda made me aware of

J. L. Loday, T. Pirashvili
The tensor category of linear maps and Leibniz algebras
Georgian Math. J. 5 3, 1998, 263-276 .

Even though the authors do not use that term, this is about (strict) Lie 2-algebras, namely Lie algebras internal to “Baez-Crans 2-vector spaces”, as well as more general 2-algebras: associative, Hopf, etc, all internal $2\mathrm{Vect}$.

Interestingly, they conceive $2\mathrm{Vect}$ entirely in terms of 2-term chain complexes, but consider on $2\mathrm{Term}$ the non-standard monoidal structure which makes it equivalent even as a symmetric monoidal category to $2\mathrm{Vect}$.

This non-standard monoidal structure is easy to figure out, but I think it is worthwhile making it explicit. Loday and Pirashvili make great use of it, in particular in that they prove that with that structure $2\mathrm{Term}$ becomes cartesian closed and explicitly compute the internal hom.

The issue of finding this non-standard monoidal structure on $2\mathrm{Term}$ is what Dmitry Roytenberg is referring to in the first paragraph on p. 4 of his article on weak Lie 2-algebras (pdf, html).

So it’s maybe worthwhile making explicit a couple of easy but useful facts here. That’s what I shall try to do in the following.

The 2-category $2\mathrm{Vect}$ (in the present context) is that of categories internal to vector spaces.

The 2-category $2\mathrm{Term}$ is that of chain complexes of length 2.

Notice that such chain complexes are really nothing but morphisms $V_1 \stackrel{f}{\to} V_0$ of vector spaces. So an object in $2\mathrm{Term}$ is nothing but a linear map.

Morphisms of 2-term chain complexes are nothing but commuting squares between such linear maps

$\array{ V_1 &\stackrel{f}{\to}& V_0 \\ \downarrow &&\downarrow \\ V'_1 &\stackrel{f'}{\to}& V'_0 } \,.$

Therefore Loday and Pirashvili call this category not $2\mathrm{Term}$ but

$L M$

(“Linear Maps”).

They would however probably have thought of a different name than that had they thought of the next level categorification of their setup. That they didn’t is apparently largely due to the fact, as far as I can see at least, that one of main facts they are interested in in this paper is that a couple of ordinary non-Lie algebras, like bialgebras, Leibniz algebras, become Lie objects internal to $2\mathrm{Term}$.

We will of course rephrase these statements in the form

Every Leibniz algebra $g$ gives rise to a strict Lie 2-algebra $g \to g_{\mathrm{Lie}} \,.$

Here $g_{\mathrm{Lie}}$ is the Lie algebra obtained from $g$ by antisymmetrization.

So, recall first the standard fact described in HDA VI:

There is a (slightly non-canonical) equivalence $2\mathrm{Vect}\simeq 2\mathrm{Term} \,.$

Given a 2-vector space $V$, we define the corresponding 2-term chain complex to be $\mathrm{ker}(s) \stackrel{t}{\to} \mathrm{Obj}(V) \,.$

The 2-category $2\mathrm{Vect}$ has an obvious monoidal structure obtained by tensoring internal to $\mathrm{Vect}$.

Let $V_1 \otimes V_2$ be the tensor product of two 2-vector spaces. Then, since $\mathrm{ker}_{s_{V_1 \otimes V_2}} = \left\langle \array{ 0 \\ \downarrow^f \\ t(f) } \otimes \mathrm{Id}_{a_2} \;\oplus\; \mathrm{Id}_{a_1} \otimes \array{ 0 \\ \downarrow^{f'} \\ t(f') } \right\rangle$ we find that $V_1 \otimes V_2$ corresponds to the 2-term chain complex given by $\mathrm{ker}(s_1)\otimes \mathrm{Obj}(V_2) \oplus \mathrm{Obj}(V_2) \otimes \mathrm{ker}(s_2) \stackrel{t_1\otimes \mathrm{Id}\oplus \mathrm{Id}\otimes t_2}{\to} \mathrm{Obj}(V_1) \otimes \mathrm{Obj}(V_2) \,.$

Equipped with this non-obvious tensor product

$\left( \array{ V \\ \downarrow \\ W } \right) \otimes \left( \array{ V' \\ \downarrow \\ W' } \right) := \left( \array{ (V \otimes W' ) \oplus (W \otimes V') \\ \downarrow \\ W \otimes W' } \right)$

$2\mathrm{Term}$ becomes a symmetric monoidal closed category. The equivalence $2\mathrm{Vect} \simeq 2\mathrm{Term}$ extends to an equivalence of symmetric monoidal categories.

The internal Hom-object is define din the very last lines of Loday&Pirashvili’s paper.

By the equivalence, this then also determines the internal hom in Baez-Crans 2-vector spaces $2\mathrm{Vect} \,.$

Loday and Pirashvili have a couple of further interesting observations concerning various kinds of 2-algebras as algebras internal to Baez-Crans 2-vector spaces.

Posted at October 16, 2007 2:02 PM UTC

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Re: Loday and Pirashvili on Lie 2-Algebras (secretly)

I find this very intriguing. I have a couple of naive questions. First I could not see how to write down a dual. Did I miss something? Also you can iterate this and get a category whose objects are commutative cubes and with a symmetric tensor product that needs some cunning notation to write down. Do these have any significance?

Posted by: Bruce Westbury on October 17, 2007 9:46 PM | Permalink | Reply to this

Re: Loday and Pirashvili on Lie 2-Algebras (secretly)

First I could not see how to write down a dual.

Are you asking about dual 2-vector spaces? You’ll want to fix a “valuation object” first: a dual 2-vector space should be a hom-space of morphisms from a 2-vector space into some canonical 1-dimensional 2-vector space.

In the context of Baez-Crans 2-vector spaces there are maybe 1.5 different obvious choices one could make here. Fix one, and I’d think that, by the closedness result, you’d get a nice notion of dual 2-vector spaces.

Also you can iterate this and get a category whose objects are commutative cubes

Squares might have their use, too (as do crossed squares in the theory of $n$-groups) but they are expected to give rise to 3-term chain complexes (as do crossed squares in the theory of $n$-groups).

The main generalization here is:

$n\mathrm{Vect} \simeq n\mathrm{Term} \,.$

So it’s not really cubes, but $n$-term chain complexes that generalize what we are talking about here.

A Lie algebra structure on an $n$-term chain complex yields an $n$-term $L_\infty$-algebra.

See John and Alissa’s HDA VI for more.

Posted by: Urs Schreiber on October 17, 2007 11:22 PM | Permalink | Reply to this
Read the post BV for Dummies (Part V)
Weblog: The n-Category Café
Excerpt: Some elements of BV formalism, or rather of the Koszul-Tate-Chevalley-Eilenberg resolution, in a simple setup with ideosyncratic remarks on higher vector spaces.
Tracked: October 30, 2007 10:26 PM

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