∞-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
For a vector space, the general linear Lie algebra or endomorphism Lie algebra of is the Lie algebra whose elements are linear endomorphisms and whose Lie bracket is given by the commutator of endomorphisms.
This is also the endomorphism L-∞ algebra of
If is a real vector space that carries an inner product there are the sub-Lie algebras
the
If is a complex vector space with an inner product there is
the
The Loday-Quillen-Tsygan theorem (Loday-Quillen 84, Tsygan 83) states that for any associative algebra, in characteristic zero, the Lie algebra homology of the infinite general linear Lie algebra with coefficients in is, up to a degree shift, the exterior algebra on the cyclic homology of :
(see e.g Loday 07, theorem 1.1).
The Loday-Quillen-Tsygan theorem is originally due, independently, to
and
Lecture notes include
Last revised on December 29, 2019 at 00:28:57. See the history of this page for a list of all contributions to it.