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In quantum geometry or noncommutative geometry a **quantum differential calculus** or **noncommutative differential structure** on an algebra over a field means the specification of a space of differential forms over the algebra. The algebra here is regarded as a coordinate ring but it is important that it may be noncommutative and hence not an actual algebra of coordinate functions on any actual space, so this represents a point of view replacing the specification of a differentiable structure for an actual space. In ordinary differential geometry one can multiply differential 1-forms by functions from the left and from the right, and there exists an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following:

1. An --bimodule over , i.e. one can multiply elements of by elements of in an associative way:

- .

2. A linear map obeying the Leibniz rule

3.

4. (optional connectedness condition)

The last condition is not always imposed but holds in ordinary geometry when the manifold is connected. It says that the only functions killed by are constant functions.

An * exterior algebra * or *differential graded algebra * structure over means a compatible extension of to include analogues of higher order differential forms

obeying a graded-Leibniz rule with respect to an associative product on and obeying . Here and it is usually required that is generated by . The product of differential forms is called the exterior or wedge product and often denoted . The noncommutative or quantum de Rham cohomology is defined as the cohomology of this complex.

A higher order differential calculus can mean an exterior algebra, or it can mean the partial specification of one, up to some highest degree, and with products that would result in a degree beyond the highest being unspecified.

The above definition lies at the crossroads of two approaches to noncommutative geometry. In the Connes approach a more fundamental object is a replacement for the Dirac operator in the form of a spectral triple, and an exterior algebra can be constructed from this data. In the quantum groups approach to noncommutative geometry one starts with the algebra and a choice of first order calculus but constrained by covariance under a quantum group symmetry.

The above definition is minimal and gives something more general than classical differential calculus even when the algebra is commutative or functions on an actual space. This is because we do *not* demand that

since this would imply that , which would violate axiom 4 when the algebra was noncommutative. As a byproduct, this enlarged definition includes finite difference calculi and quantum differential calculi on finite sets and finite groups (finite group Lie algebra theory).

1. For the algebra of polynomials in one variable the translation-covariant quantum differential calculi are parametrized by and take the form

This shows how finite differences arise naturally in quantum geometry. Only the limit has functions commuting with 1-forms, which is the special case of high school differential calculus.

2. For the algebra of functions on an algebraic circle, the translation (i.e. circle-rotation)-covariant differential calculi are parametrized by and take the form

This shows how -differentials arise naturally in quantum geometry.

3. For any algebra one has a **universal differential calculus** defined by

where is the algebra product. By axiom 3., any first order calculus is a quotient of this.

- Connes, A. (1994),
*Noncommutative geometry*, Academic Press, ISBN 0-12-185860-X - Majid, S. (2002),
*A quantum groups primer*, London Mathematical Society Lecture Note Series,**292**, Cambridge University Press, doi:10.1017/CBO9780511549892, ISBN 978-0-521-01041-2, MR 1904789

In mathematics, **Jensen's inequality**, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.

In mathematics, an **affine space** is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

In mathematics, the **Hodge star operator** or **Hodge star** is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element of the algebra is called the element's **Hodge dual**. This map was introduced by W. V. D. Hodge.

In mathematics, a **Casimir element** is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.

In mathematics, **Kähler differentials** provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

In functional analysis, a branch of mathematics, the **Borel functional calculus** is a *functional calculus*, which has particularly broad scope. Thus for instance if *T* is an operator, applying the squaring function *s* → *s*^{2} to *T* yields the operator *T*^{2}. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential

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.

In mathematical logic and type theory, the **λ-cube** is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to bind a term or type. The respective dimensions of the λ-cube correspond to:

In mathematics, the **fundamental theorem of Galois theory** is a result that describes the structure of certain types of field extensions.

In mathematics, the **Gibbs measure**, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system *X* being in state *x* as

In mathematics, a **vector-valued differential form** on a manifold *M* is a differential form on *M* with values in a vector space *V*. More generally, it is a differential form with values in some vector bundle *E* over *M*. Ordinary differential forms can be viewed as **R**-valued differential forms.

In mathematics, **delay differential equations** (**DDEs**) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called **time-delay systems**, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs. (1) Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, actuators, sensors, communication networks that are now involved in feedback control loops introduce such delays. Finally, besides actual delays, time lags are frequently used to simplify very high order models. Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control engineering. (2) Delay systems are still resistant to many *classical* controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring effects which are adequately represented by DDEs is not a general alternative: in the best situation, it leads to the same degree of complexity in the control design. In worst cases, it is potentially disastrous in terms of stability and oscillations. (3) Delay properties are also surprising since several studies have shown that voluntary introduction of delays can also benefit the control. (4) In spite of their complexity, DDEs however often appear as simple infinite-dimensional models in the very complex area of partial differential equations (PDEs).

In mathematics, a **π-system** on a set Ω is a collection *P* of certain subsets of Ω, such that

In mathematics, the **Abel–Jacobi map** is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

In probability theory, a **Markov kernel** is a map that in the general theory of Markov processes, plays the role that the transition matrix does in the theory of Markov processes with a finite state space.

In mathematical physics, the concept of **quantum spacetime** is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies.

In mathematics, a **singular trace** is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling.

In the theory of Lie groups, Lie algebras and their representation theory, a **Lie algebra extension****e** is an enlargement of a given Lie algebra **g** by another Lie algebra **h**. Extensions arise in several ways. There is the **trivial extension** obtained by taking a direct sum of two Lie algebras. Other types are the **split extension** and the **central extension**. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges.

The **Fokas method**, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called integrable systems. It is named after Greek mathematician Athanassios S. Fokas.

In mathematics, specifically in spectral theory, a **discrete spectrum** of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite.

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