∞-Lie theory (higher geometry)
A Lie algebra is the infinitesimal approximation to a Lie group.
A Lie algebra is a vector space $\mathfrak{g}$ equipped with a bilinear skew-symmetric map $[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}$ which satisfies the Jacobi identity:
A homomorphism of Lie algebras is a linear map $\phi : \mathfrak{g} \to \mathfrak{h}$ such that for all $x,y \in \mathfrak{g}$ we have
This defines the category LieAlg of Lie algebras.
The notion of Lie algebra may be formulated internal to any linear category. This general definition subsumes variants of Lie algebras such as super Lie algebras.
Consider a commutative unital ring $k$, and a (strict for simplicity) symmetric monoidal $k$-linear category $(\mathcal{C},\otimes,1)$ with braiding $\tau$.
A Lie algebra object in $(\mathcal{C},\otimes,1,\tau)$ is
an object
morphism (the Lie bracket)
such that the following conditions hold:
skew-symmetry:
Equivalently, Lie algebra objects are the algebras over an operad over a certain quadratic operad, called the Lie operad, which is the Koszul dual of the commutative algebra operad.
Examples of types of Lie algebra objects:
If $k$ is the ring $\mathbb{Z}$ of integers and $\mathcal{C} =$ $k$Mod = Ab is the category of abelian groups equipped with the tensor product of abelian groups, then a Lie algebra object is called a Lie ring.
If $k$ is a field and $\mathcal{C} =$ Vect is the category of vector spaces over $k$ equipped with the tensor product of vector spaces then a Lie algebra object is an ordinary_Lie k-algebra.
If $k$ is a field and $\mathcal{C}$ = sVect is the category of super vector spaces over $k$, then a Lie algebra object is a super Lie algebra.
Lie algebras are equivalently groups in “infinitesimal geometry”.
For instance in synthetic differential geometry then a Lie algebra of a Lie group is just the first-order infinitesimal neighbourhood of the unit element (e.g. Kock 09, section 6).
More generally in geometric homotopy theory, Lie algebras, being 0-truncated L-∞ algebras are equivalently “infinitesimal ∞-group geometric ∞-stacks” (e.g. here-topos#FormalModuliProblems)), also called formal moduli problems (see there for more).
Notions of Lie algebras with extra stuff, structure, property includes
extra property
extra structure
extra stuff
See
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
A. L. Onishchik (ed.) Lie Groups and Lie Algebras
I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,
II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups
Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993
Eckhard Meinrenken, Lie groups and Lie algebas, Lecture notes 2010 (pdf)
Discussion with a view towards Chern-Weil theory is in chapter IV in vol III of
Discussion in synthetic differential geometry is in
Last revised on January 22, 2021 at 00:18:54. See the history of this page for a list of all contributions to it.