Cohomological differential graded Lie algebra

Let $k$ be a characteristic-zero field (so $n{1}_k\mathrm{\ne }0$ for every integer $n>0$), and let $(\mathfrak{g},d)$ be a cochain complex of $k$-vector spaces, with $d:{\mathfrak{g}}^i\to {\mathfrak{g}}^{i+1}$. A cohomological differential graded Lie algebra (dg Lie algebra) over $k$ is such a complex together with $k$-bilinear maps $[-,-]:{\mathfrak{g}}^p{\otimes }_k{\mathfrak{g}}^q\to {\mathfrak{g}}^{p+q}$ for all integers $p,q$, such that for homogeneous $x\in {\mathfrak{g}}^p$, $y\in {\mathfrak{g}}^q$, and $z\in {\mathfrak{g}}^r$, $[x,y]=-(-1{)}^{pq}[y,x],$ $(-1{)}^{pr}[x,[y,z]]+(-1{)}^{pq}[y,[z,x]]+(-1{)}^{qr}[z,[x,y]]=0,$ and $d[x,y]=[dx,y]+(-1{)}^p[x,dy]\mathrm{.}$ A morphism of dg Lie algebras is a degree-zero cochain map preserving the bracket. It is a quasi-isomorphism when its underlying cochain map induces isomorphisms on all cohomology groups. This cohomological convention places first-order deformations, infinitesimal automorphisms, and primary obstructions in $H^1(\mathfrak{g})$, $H^0(\mathfrak{g})$, and $H^2(\mathfrak{g})$ only after the particular deformation problem has been identified with $\mathfrak{g}$.

For the coefficient construction, a nonpositive augmented commutative dg $k$-algebra is a cochain complex $R={⨁}_{i\le 0}{R}^i$ with a unital, associative, degree-zero multiplication, differential $d_R:R^i\to R^{i+1}$, and augmentation $ϵ:R\to k$ which is a unital chain-algebra map, such that for homogeneous $a,b\in R$, $ab=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ba,d_R(ab)=d_R(a)b+(-1{)}^{\mathrm{\mid }a\mathrm{\mid }}a{d}_R(b)\mathrm{.}$ Its augmentation ideal is ${\mathfrak{m}}_R:=\ker ϵ$, a differential graded ideal. This is a commutative subclass of the differential graded algebras of Cohomological differential graded algebra; that supplier alone imposes no graded-commutativity condition. The tensor product $\mathfrak{g}{\otimes }_k{\mathfrak{m}}_R$ has differential $d(x\otimes a)=dx\otimes a+(-1{)}^{\mathrm{\mid }x\mathrm{\mid }}x\otimes d_{R}a$ and bracket $[x\otimes a,y\otimes b]=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}[x,y]\otimes ab\mathrm{.}$ This is the coefficient dg Lie algebra used in the Maurer–Cartan presentation of a derived deformation functor; the definition alone does not identify an arbitrary geometric deformation functor with that presentation.

Cohomological differential graded algebra

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