Nonassociative rings and algebras Definition

Differential graded Lie algebra

Builds on Field Vector space Chain complex Lie algebra

Definition

Let $F$ be a field of characteristic $0$ . Let $L = \bigoplus_{i \in \Z} L^i$ be a $\Z$ -graded vector space over $F$ , and write $\abs{x} = i$ when $x \in L^i$ is homogeneous. A differential graded Lie algebra structure on $L$ is a pair $([\cdot, \cdot], d)$ . Here $[\cdot, \cdot] \colon L \times L \to L$ is bilinear and $d \colon L \to L$ is linear. The conditions are, for all homogeneous $x, y, z \in L$ :

$[L^i, L^j] \subseteq L^{i + j}$ for all $i, j$ ;
$[x, y] = -(-1)^{\abs{x}\abs{y}} [y, x]$ ;
$[x, [y, z]\,] = [\,[x, y], z] + (-1)^{\abs{x}\abs{y}} [y, [x, z]\,]$ ;
$d(L^i) \subseteq L^{i + 1}$ for all $i$ , and $d \circ d = 0$ ;
$d[x, y] = [dx, y] + (-1)^{\abs{x}} [x, dy]$ .

Remarks

Example: a Lie algebra $\mathfrak{g}$ placed in degree $0$ , with $L^i = 0$ for $i \ne 0$ and $d = 0$ . All signs are $+1$ , so clauses 2 and 3 become antisymmetry and the Jacobi identity, and clauses 4 and 5 hold because both sides are $0$ .

Example: for a smooth manifold $M$ and a Lie algebra $\mathfrak{g}$ , the forms $\Omega^\bullet(M) \otimes \mathfrak{g}$ with $[\alpha \otimes a, \beta \otimes b] = (\alpha \wedge \beta) \otimes [a, b]$ and $d$ the exterior derivative form a differential graded Lie algebra.

Non-example: take $L = L^0 = \mathfrak{g}$ as above, but with $d = \id$ . This fails clause 4, because $d$ does not raise the degree and $d \circ d = \id \ne 0$ when $\mathfrak{g} \ne 0$ .

By clause 4, $(L, d)$ is a cochain complex, and by clause 5 its cohomology $H(L)$ inherits a graded Lie bracket. Some authors use a lower grading in which $d$ lowers the degree by one.

Reading path

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Chain complex concept
Field concept
Lie algebra concept
Vector space concept
Differential graded Lie algebra definition

Tag	`B2GEEB0VT2`
URI	`m2://B2GEEB0VT2`
Kind	definition
Writing	W0 stub
MSC	17B70, 17B55, 55P62
Defines	Differential graded Lie algebra
Uses	Field, Vector space, Chain complex, Lie algebra
Changed	3 November 2026

Differential graded Lie algebra

Definition

Remarks

Reading path

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