Connection (algebraic framework) - AbsoluteAstronomy.com

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

(e.g.,
noncommutative geometry

Noncommutative geometry

Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...

and supergeometry

Supergeometry

Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, BRST theory, or supergravity.Supergeometry is...

) is mainly
phrased in algebraic terms of modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

and
algebras. Connections on modules are
generalization of a linear connection

Connection (vector bundle)

In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...

on a smooth vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

written as a Koszul connection on the

-module of sections of

Commutative algebra

Let

be a commutative ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

and

-module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

. There are different equivalent definitions
of a connection on

. Let

be the module of derivations

Derivation (abstract algebra)

In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D = b + a.More...

of a ring

. A
connection on an

-module

is defined
as an

-module morphism

such that the first order differential operator

Differential calculus over commutative algebras

In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms...

obey the Leibniz rule

Connections on a module over a commutative ring always exist.

The curvature of the connection

is defined as
the zero-order differential operator

on the module

for all

.

If

is a vector bundle, there is one-to-one
correspondence between linear
connections

Connection (vector bundle)

and the
connections

on the

-module of sections of

. Strictly speaking,

corresponds to
the covariant differential

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

of a
connection on

Graded commutative algebra

The notion of a connection on modules over commutative rings is
straightforwardly extended to modules over a graded
commutative algebra

Superalgebra

In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....

. This is the case of
superconnections

Supergeometry

in supergeometry

Supergeometry

of
graded manifold

Graded manifold

Graded manifolds are extensions of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra. Graded manifolds are not supermanifolds though there is a certain correspondence between the graded manifolds and the DeWitt supermanifolds. Both graded manifolds and...

s and supervector bundles

Supergeometry

.
Superconnections always exist.

Noncommutative algebra

is a noncommutative ring, connections on left
and right

-modules are defined similarly to those on
modules over commutative rings. However
these connections need not exist.

In contrast with connections on left and right modules, there is a
problem how to define a connection on an

-bimodule

Bimodule

In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible...

over noncommutative rings

and

. There are different definitions
of such a connection. Let us mention one of them. A connection on an

-bimodule

is defined as a bimodule
morphism

which obeys the Leibniz rule

External links

Sardanashvily, G.
Gennadi Sardanashvily
Gennadi Sardanashvily is a theoretical physicist, a principal research scientist of Moscow State University.- Biography :...

, Lectures on differential geometry of modules and rings, arXiv: 0910.1515

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Commutative algebra

Graded commutative algebra

Noncommutative algebra

See also

External links