∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
A line Lie -algebra over a ground field is the Lie n-algebra analog of the abelian (trivial) 1-dimensional Lie algebra on .
For the line Lie -algebra
is the L-∞ algebra whose Chevalley-Eilenberg algebra
is the free graded-commutative algebra on a single generator in degree equipped with the trivial differential
For then the line Lie 2-algebra is the Lie 2-algebra that comes from the differential crossed module .
For a Lie algebra a cocycle in degree -Lie algebra cohomology on is equivalently a morphism of L-∞ algebras
More generally, for an L-∞ algebra, a degree- cocycle in ∞-Lie algebra cohomology is given by such a morphism.
There is a unique (up to rescaling) indecomposable invariant polynomial on , given by the shifted copy of the generator in the Weil algebra .
Equivalently, we have
The Lie integration (see there) of is the line Lie n-group .
Last revised on July 30, 2018 at 16:05:08. See the history of this page for a list of all contributions to it.