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A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx ; or more generally an algebraic structure in which one of the principal binary operations is not commutative ; one also allows additional structures, e. g. topology or norm to be possibly carried by the noncommutative algebra of functions.
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noncommutative and algebra
His mathematical specialties were noncommutative ring theory and computational algebra and its applications, including automated theorem proving in geometry.
The study of rings which are not necessarily commutative is known as noncommutative algebra ; it includes ring theory, representation theory, and the theory of Banach algebras.
These noncommutative algebras, and the non-associative Lie algebras, were studied within universal algebra before the subject was divided into particular mathematical structure types.
This suggests that one might define a noncommutative Riemannian manifold as a spectral triple ( A, H, D ), consisting of a representation of a C *- algebra A on a Hilbert space H, together with an unbounded operator D on H, with compact resolvent, such that is bounded for all a in some dense subalgebra of A.
* The noncommutative torus, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple.
In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure.
The term " quantum group " often denotes a kind of noncommutative algebra with additional structure that first appeared in the theory of quantum integrable systems, and which was then formalized by Vladimir Drinfel'd and Michio Jimbo as a particular class of Hopf algebra.
One can think of the deformed object as an algebra of functions on a " noncommutative space ", in the spirit of the noncommutative geometry of Alain Connes.
A bounded linear functional on a C *- algebra A is said to be self-adjoint if it is real-valued on the self-adjoint elements of A. Self-adjoint functionals are noncommutative analogues of signed measures.
* The Weyl algebra is a noncommutative domain.
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring ( which may be regarded as a free commutative algebra ).
So we think of a noncommutative C *- algebra as the algebra of functions on a ' noncommutative space ' which does not exist classically.
noncommutative and is
Multiplicative notation is typically used when numerical multiplication or a noncommutative operation interprets the group operation.
: The order of columns in an SQL table is defined and significant, one consequence being that SQL's implementations of Cartesian product and union are both noncommutative.
In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals.
The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring.
For noncommutative rings, it is necessary to distinguish between three very similar concepts:
As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting.
The Poincaré symmetry of ordinary special relativity is deformed into some noncommutative symmetry and Minkowski space is deformed into some noncommutative space.
ijk = − 1 and the usual algebraic rules except the commutative law of multiplication ( a familiar example of such a noncommutative multiplication is matrix multiplication ).
Quaternion multiplication is noncommutative ( because of the cross product, which anti-commutes ), while scalar-scalar and scalar-vector multiplications commute.
Classical expressions, observables, and operations ( such as Poisson brackets ) are modified by ħ-dependent quantum corrections, as the conventional commutative multiplication applying in classical mechanics is generalized to the noncommutative star-multiplication characterizing quantum mechanics and underlying its uncertainty principle.
That is, it means " not necessarily associative " just as " noncommutative " means " not necessarily commutative ".
Following Witten, who was motivated by ideas from noncommutative geometry, it is conventional to introduce the-product defined implicitly through
They play the central role in the Langlands correspondence which studies finite dimensional representations of the Galois group of the field and which is one of noncommutative extensions of class field theory.
An operation that does not satisfy the above property is called noncommutative.
* Concatenation, the act of joining character strings together, is a noncommutative operation.
* Subtraction is noncommutative, since
* Division is noncommutative, since
* Matrix multiplication is noncommutative since
Noncommutative geometry ( NCG ) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces which are locally presented by noncommutative algebras of functions ( possibly in some generalized sense ).
noncommutative and associative
In analogy to the duality between affine schemes and commutative rings, we define a category of noncommutative affine schemes as the dual of the category of associative unital rings.
One of the main starting points of the Alain Connes ' direction in noncommutative geometry is his spectacular discovery ( and independently by Boris Tsygan ) of a very important new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely the cyclic homology and its relations to the algebraic K-theory ( primarily via Connes-Chern character map ).
In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain ( co ) homology theories for associative algebras which generalize the de Rham ( co ) homology of manifolds.
It is readily verified, using the chain rule, that this constitutes an associative noncommutative operation on the space of jets at the origin.
noncommutative and which
Thus, general operator algebras are often regarded as a noncommutative generalizations of these algebras, or the structure of the base space on which the functions are defined.
This point of view is elaborated as the philosophy of noncommutative geometry, which tries to study various non-classical and / or pathological objects by noncommutative operator algebras.
* The set of all integral quaternions is a noncommutative ring which is a subring of quaternions, hence a noncommutative domain.
Noncommutative logic is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus ( see External links below ).
By extension, the term noncommutative logic is also used by a number of authors to refer to a family of substructural logics in which the exchange rule is inadmissible.
The oldest noncommutative logic is the Lambek calculus, which gave rise to the class of logics known as categorial grammars.
Alessio Guglielmi proposed a variation of Retoré's calculus, BV, in which the two noncommutative operations are collapsed onto a single, self-dual, operator, and proposed a novel proof calculus, the calculus of structures to accommodate the calculus.
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