Homotopy Lie algebra

There exists several different definitions of a homotopy Lie algebra, some particularly suited to certain situations more than others. The most traditional definition is via symmetric multi-linear maps, but there also exists a more succinct geometric definition using the language of formal geometry. Here the blanket assumption that the underlying field is of characteristic zero is made.

A homotopy Lie algebra on a graded vector space $V\; =\; \backslash bigoplus\; V\_i$ is a continuous derivation, $m$, of order $>1$ that squares to zero on the formal manifold $\backslash hat\{S\}\backslash Sigma\; V^*$. Here $\backslash hat\{S\}$ is the completed symmetric algebra, $\backslash Sigma$ is the suspension of a graded vector space, and $V^*$ denotes the linear dual. Typically one describes $(V,m)$ as the homotopy Lie algebra and $\backslash hat\{S\}\backslash Sigma\; V^*$ with the differential $m$ as its representing commutative differential graded algebra.

Using this definition of a homotopy Lie algebra, one defines a morphism of homotopy Lie algebras, $f\backslash colon(V,m\_V)\backslash to\; (W,m\_W)$, as a morphism $f\backslash colon\backslash hat\{S\}\backslash Sigma\; V^*\backslash to\backslash hat\{S\}\backslash Sigma\; W^*$ of their representing commutative differential graded algebras that commutes with the vector field, i.e., $f\; \backslash circ\; m\_V\; =\; m\_W\; \backslash circ\; f$. Homotopy Lie algebras and their morphisms define a category.

The more traditional definition of a homotopy Lie algebra is through an infinite collection of symmetric multi-linear maps that is sometimes referred to as the definition via higher brackets. It should be stated that the two definitions are equivalent.

A homotopy Lie algebra on a graded vector space $V\; =\; \backslash bigoplus\; V\_i$ is a collection of symmetric multi-linear maps $l\_n\; \backslash colon\; V^\{\backslash otimes\; n\}\backslash to\; V$ of degree $n-2$, sometimes called the $n$-ary bracket, for each $n\backslash in\backslash N$. Moreover, the maps $l\_n$ satisfy the generalised Jacobi identity:

:$\backslash sum\_\{i+j=n+1\}\; \backslash sum\_\{\backslash sigma\backslash in\; \backslash mathrm\{UnShuff\}(i,n-i)\}\; \backslash chi\; (\backslash sigma\; ,v\_1\; ,\backslash dots\; ,v\_n\; )\; (-1)^\{i(j-1)\}\; l\_j\; (l\_i\; (v\_\{\backslash sigma\; (1)\}\; ,\; \backslash dots\; ,v\_\{\backslash sigma\; (i)\}),v\_\{\backslash sigma\; (i+1)\},\; \backslash dots\; ,v\_\{\backslash sigma\; (n)\})=0,$

for each n. Here the inner sum runs over $(i,j)$-unshuffles and $\backslash chi$ is the signature of the permutation. The above formula have meaningful interpretations for low values of $n$; for instance, when $n=1$ it is saying that $l\_1$ squares to zero (i.e., it is a differential on $V$), when $n=2$ it is saying that $l\_1$ is a derivation of $l\_2$, and when $n=3$ it is saying that $l\_2$ satisfies the Jacobi identity up to an exact term of $l\_3$ (i.e., it holds up to homotopy). Notice that when the higher brackets $l\_n$ for $n\backslash geq\; 3$ vanish, the definition of a differential graded Lie algebra on $V$ is recovered.

Using the approach via multi-linear maps, a morphism of homotopy Lie algebras can be defined by a collection of symmetric multi-linear maps $f\_n\backslash colon\; V^\{\backslash otimes\; n\}\; \backslash to\; W$ which satisfy certain conditions.

There also exists a more abstract definition of a homotopy algebra using the theory of operads: that is, a homotopy Lie algebra is an algebra over an operad in the category of chain complexes over the $L\_\backslash infty$ operad.

A morphism of homotopy Lie algebras is said to be a (quasi) isomorphism if its linear component $f\backslash colon\; V\backslash to\; W$ is a (quasi) isomorphism, where the differentials of $V$ and $W$ are just the linear components of $m\_V$ and $m\_W$.

An important special class of homotopy Lie algebras are the so-called minimal homotopy Lie algebras, which are characterized by the vanishing of their linear component $l\_1$. This means that any quasi isomorphism of minimal homotopy Lie algebras must be an isomorphism. Any homotopy Lie algebra is quasi-isomorphic to a minimal one, which must be unique up to isomorphism and it is therefore called its minimal model.

Because $L\_\backslash infty$-algebras have such a complex structure describing even simple cases can be a non-trivial task in most cases. Fortunately, there are the simple cases coming from differential graded Lie algebras and cases coming from finite dimensional examples.

One of the approachable classes of examples of $L\_\backslash infty$-algebras come from the embedding of differential graded Lie algebras into the category of $L\_\backslash infty$-algebras. This can be described by $l\_1$ giving the derivation, $l\_2$ the Lie algebra structure, and $l\_k\; =0$ for the rest of the maps.

One notable class of examples are $L\_\backslash infty$-algebras which only have two nonzero underlying vector spaces $V\_0,V\_1$. Then, cranking out the definition for $L\_\backslash infty$-algebras this means there is a linear map

:$d\backslash colon\; V\_1\; \backslash to\; V\_0$,

bilinear maps

:$l\_2\backslash colon\; V\_i\backslash times\; V\_j\; \backslash to\; V\_\{i+j\}$, where $0\backslash leq\; i\; +\; j\; \backslash leq\; 1$,

and a trilinear map

:$l\_3\backslash colon\; V\_0\backslash times\; V\_0\backslash times\; V\_0\; \backslash to\; V\_1$

which satisfy a host of identities.{{Cite journal|last1=Baez|first1=John C.|author1-link=John C. Baez|last2=Crans|first2=Alissa S.|date=2010-01-24|title=Higher-Dimensional Algebra VI: Lie 2-Algebras|journal=Theory and Applications of Categories|volume=12|pages=492–528|arxiv=math/0307263}} ^{pg 28} In particular, the map $l\_2$ on $V\_0\backslash times\; V\_0\; \backslash to\; V\_0$ implies it has a lie algebra structure up to a homotopy. This is given by the differential of $l\_3$ since the gives the $L\_\backslash infty$-algebra structure implies

:$dl\_3(a,b,c)\; =\; -a,b],c]\; +\; [[a,c],b]\; +\; [a,[b,c$,

showing it is a higher Lie bracket. In fact, some authors write the maps $l\_n$ as $[-,\backslash cdots,-]\_n:\; V\_\backslash bullet\; \backslash to\; V\_\backslash bullet$, so the previous equation could be read as

:$d[a,b,c]\_3\; =\; -a,b],c]\; +\; [[a,c],b]\; +\; [a,[b,c$,

showing that the differential of the 3-bracket gives the failure for the 2-bracket to be a Lie algebra structure. It is only a Lie algebra up to homotopy. If we took the complex $H\_*(V\_\backslash bullet,\; d)$ then $H\_0(V\_\backslash bullet,\; d)$ has a structure of a Lie algebra from the induced map of $[-,-]\_2$.

In this case, for $n\; \backslash geq\; 2$, there is no differential, so $V\_0$ is a Lie algebra on the nose, but, there is the extra data of a vector space $V\_n$ in degree $n$ and a higher bracket

:$l\_\{n+2\}\backslash colon\; \backslash bigoplus^\{n+2\}\; V\_0\; \backslash to\; V\_n.$

It turns out this higher bracket is in fact a higher cocyle in Lie algebra cohomology. More specifically, if we rewrite $V\_0$ as the Lie algebra $\backslash mathfrak\{g\}$ and $V\_n$ and a Lie algebra representation $V$ (given by structure map $\backslash rho$), then there is a bijection of quadruples

:$(\backslash mathfrak\{g\},\; V,\; \backslash rho,\; l\_\{n+2\})$ where $l\_\{n+2\}\backslash colon\; \backslash mathfrak\{g\}^\{\backslash otimes\; n+2\}\; \backslash to\; V$ is an $(n+2)$-cocycle

and the two-term $L\_\backslash infty$-algebras with non-zero vector spaces in degrees $0$ and $n$.^{pg 42} Note this situation is highly analogous to the relation between group cohomology and the structure of n-groups with two non-trivial homotopy groups. For the case of term term $L\_\backslash infty$-algebras in degrees $0$ and $1$ there is a similar relation between Lie algebra cocycles and such higher brackets. Upon first inspection, it's not an obvious results, but it becomes clear after looking at the homology complex

:$H\_*(V\_1\; \backslash xrightarrow\{d\}\; V\_0)$,

so the differential becomes trivial. This gives an equivalent $L\_\backslash infty$-algebra which can then be analyzed as before.

One simple example of a Lie-2 algebra is given by the $L\_\backslash infty$-algebra with $V\_0=\; (\backslash R^3,\backslash times)$ where $\backslash times$ is the cross-product of vectors and $V\_1=\backslash R$ is the trivial representation. Then, there is a higher bracket $l\_3$ given by the dot product of vectors

:$l\_3(a,b,c)\; =\; a\backslash cdot\; (b\backslash times\; c).$

It can be checked the differential of this $L\_\backslash infty$-algebra is always zero using basic linear algebra^{pg 45}.

Coming up with simple examples for the sake of studying the nature of $L\_\backslash infty$-algebras is a complex problem. For example,{{Cite journal|last1=Daily|first1=Marilyn|last2=Lada|first2=Tom|date=2005|title=A finite dimensional $L\_\backslash infty$ algebra example in gauge theory|url=https://projecteuclid.org/euclid.hha/1139839375|journal=Homology, Homotopy and Applications|volume=7|issue=2|pages=87–93|doi=10.4310/HHA.2005.v7.n2.a4|mr=2156308|doi-access=free}} given a graded vector space $V\; =\; V\_0\; \backslash oplus\; V\_1$ where $V\_0$ has basis given by the vector $w$ and $V\_1$ has the basis given by the vectors $v\_1,\; v\_2$, there is an $L\_\backslash infty$-algebra structure given by the following rules

:$\backslash begin\{align\}$

& l_1(v_1) = l_1(v_2) = w \\

& l_2(v_1\otimes v_2) = v_1, l_2(v_1\otimes w) = w \\

& l_n(v_2\otimes w^{\otimes n-1}) = C_nw \text{ for } n \geq 3

\end{align},

where $C\_n\; =\; (-1)^\{n-1\}(n-3)C\_\{n-1\},\; C\_3\; =\; 1$. Note that the first few constants are

:$\backslash begin\{matrix\}$

C_3 & C_4 & C_5 & C_6 \\

1 & -1 & -2 & 12

\end{matrix}

Since $l\_1(w)$ should be of degree $-1$, the axioms imply that $l\_1(w)\; =\; 0$. There are other similar examples for super{{Cite journal|last1=Fialowski|first1=Alice|last2=Penkava|first2=Michael|year=2002|title=Examples of infinity and Lie algebras and their versal deformations|url=http://www.impan.pl/get/doi/10.4064/bc55-0-2|journal=Banach Center Publications|volume=55|pages=27–42|doi=10.4064/bc55-0-2|arxiv=math/0102140|mr=1911978|s2cid=14082754}} Lie algebras.{{Cite journal|last1=Fialowski|first1=Alice|last2=Penkava|first2=Michael|year=2005|title=Strongly homotopy Lie algebras of one even and two odd dimensions|url=https://linkinghub.elsevier.com/retrieve/pii/S0021869304004818|journal=Journal of Algebra|volume=283|issue=1|pages=125–148|doi=10.1016/j.jalgebra.2004.08.023|arxiv=math/0308016|mr=2102075|s2cid=119142148}} Furthermore, $L\_\backslash infty$ structures on graded vector spaces whose underlying vector space is two dimensional have been completely classified.{{Cite thesis|last=Daily|first= Marilyn Elizabeth|date=2004-04-14|title=$L\_\backslash infty$ Structures on Spaces of Low Dimension |type=PhD |hdl=1840.16/5282 |url=https://repository.lib.ncsu.edu/handle/1840.16/5282}}

• Homotopy associative algebra
• Differential graded algebra
• BV formalism
• Simplicial Lie algebra
• Hochschild homology
• Deformation quantization
• Lie n-algebra

{{reflist}}

• Deformation Theory (lecture notes) - gives an excellent overview of homotopy Lie algebras and their relation to deformation theory and deformation quantization
• {{cite journal |first1=Tom |last1=Lada |first2=Jim |last2=Stasheff |title=Introduction to sh Lie algebras for physicists |journal=International Journal of Theoretical Physics |volume=32 |issue= 7|pages=1087–1104 |year=1993 |doi=10.1007/BF00671791 |arxiv=hep-th/9209099|bibcode=1993IJTP...32.1087L |s2cid=16456088 }}

• {{cite arXiv |first1=Alex S. |last1=Arvanitakis |title=The L∞-algebra of the S-matrix |date=2019 |class=hep-th |eprint=1903.05643}}
• {{cite journal |first1=Olaf |last1=Hohm |first2=Barton |last2=Zwiebach |title=L∞ Algebras and Field Theory |journal=Fortschr. Phys. |volume=65 |issue=3–4 |pages=1700014 |year=2017 |doi=10.1002/prop.201700014 |arxiv=1701.08824|bibcode=2017ForPh..6500014H |s2cid=90628041 }} — Towards classification of perturbative gauge invariant classical fields.

• {{cite journal |first=Jonathan P. |last=Pridham |title=Derived deformations of Artin stacks |journal=Communications in Analysis and Geometry|volume=23 |issue=3 |pages=419–477 |year=2015 |doi=10.4310/CAG.2015.v23.n3.a1 |arxiv=0805.3130|s2cid=14505074|mr=3310522 }}
• {{cite journal |first=Jonathan P. |last=Pridham |title=Unifying derived deformation theories |journal=Advances in Mathematics |volume=224 |issue=3 |pages=772–826 |year=2010 |doi=10.1016/j.aim.2009.12.009 |doi-access=free |arxiv=0705.0344 |s2cid=14136532|mr=2628795 }}
• {{cite journal |last1=Hu |first1=Po |last2=Kriz |first2=Igor |last3=Voronov |first3=Alexander A.|title=On Kontsevich's Hochschild cohomology conjecture |journal=Compositio Mathematica |volume=142 |issue=1 |pages=143–168 |year=2006 |doi=10.1112/S0010437X05001521 |arxiv=math/0309369|s2cid=15153116|mr=2197407 }}

• {{Cite journal|last1=Roberts|first1=Justin|last2=Willerton|first2=Simon|year=2010|title=On the Rozansky–Witten weight systems|url=http://www.msp.org/agt/2010/10-3/p09.xhtml|journal=Algebraic & Geometric Topology|language=en|volume=10|issue=3|pages=1455–1519|doi=10.2140/agt.2010.10.1455|arxiv=math/0602653|mr=2661534|s2cid=17829444 }} (Lie algebras in the derived category of coherent sheaves.)

• {{cite web |url=https://www.mpim-bonn.mpg.de/node/8756 |title=Learning seminar on deformation theory |year=2018 |publisher=Max Planck Institute for Mathematics}} Discusses deformation theory in the context of $L\_\backslash infty$-algebras.

Category:Homotopical algebra

Category:Differential algebra

Category:Lie algebras