In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
a Poisson supermanifold is a differential
supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
Informal definition
An informal definition is com ...
M such that the
supercommutative algebra
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have
:yx = (-1)^xy ,
where , ''x'', denotes the grade of the element and is 0 or 1 ( ...
of
smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s over it (to clarify this: M is not a point set space and so, doesn't "really" exist, and really, this algebra is all we have),
is equipped with a
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
called the
Poisson superbracket turning it into a
Poisson superalgebra In mathematics, a Poisson superalgebra is a Z2- graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra ''A'' with a Lie superbracket
: cdot,\cdot: A\otimes A\to A
such that (''A'', ·,· i ...
.
Every
symplectic supermanifold is a Poisson supermanifold but not vice versa.
See also
*
Poisson manifold
In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule
: \ = \h + g \ .
Equivalentl ...
*
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central i ...
*
Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some ge ...
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Symplectic geometry