Non-commutative algebra of functions of 4-dimensional quantum Hall droplet

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

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FUZZY COMPLEX GRASSMANNIAN SPACES AND THEIR STAR PRODUCTS

International Journal of Modern Physics A ◽

10.1142/s0217751x03014113 ◽

2003 ◽

Vol 18 (11) ◽

pp. 1935-1958 ◽

Cited By ~ 18

Author(s):

BRIAN P. DOLAN ◽

OLIVER JAHN

Keyword(s):

Commutative Algebra ◽

Star Product ◽

Matrix Multiplication ◽

Noncommutative Algebra ◽

Infinite Dimensional ◽

Complex Projective Spaces ◽

Finite Dimensional ◽

Algebra Of Functions ◽

Finite Sum ◽

Complex Projective

We derive an explicit expression for an associative star product on noncommutative versions of complex Grassmannian spaces, in particular for the case of complex two-planes. Our expression is in terms of a finite sum of derivatives. This generalizes previous results for complex projective spaces and gives a discrete approximation for the Grassmannians in terms of a noncommutative algebra, represented by matrix multiplication in a finite-dimensional matrix algebra. The matrices are restricted to have a dimension which is precisely determined by the harmonic expansion of functions on the commutative Grassmannian, truncated at a finite level. In the limit of infinite-dimensional matrices we recover the commutative algebra of functions on the complex Grassmannians.

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DEFORMATION QUANTIZATION: IS C1 NECESSARILY SKEW?

International Journal of Modern Physics B ◽

10.1142/s0217979202011640 ◽

2002 ◽

Vol 16 (14n15) ◽

pp. 1925-1930

Author(s):

CHRISTIAN FRONSDAL

Keyword(s):

Poisson Bracket ◽

Commutative Algebra ◽

Deformation Quantization ◽

Formal Series ◽

Symplectic Space ◽

Wide Spread ◽

Algebra Of Functions ◽

Associative Product

Deformation quantization (of a commutative algebra) is based on the introduction of a new associative product, expressed as a formal series, [Formula: see text]. In the case of the algebra of functions on a symplectic space the first term in the perturbation is often identified with the antisymmetric Poisson bracket. There is a wide-spread belief that every associative *-product is equivalent to one for which C1(f,g) is antisymmetric and that, in particular, every abelian deformation is trivial. This paper shows that this is far from being the case and illustrates the existence of abelian deformations by physical examples.

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SPACETIME FROM SYMMETRY: THE MOYAL PLANE FROM THE POINCARÉ–HOPF ALGEBRA

Modern Physics Letters A ◽

10.1142/s0217732309031144 ◽

2009 ◽

Vol 24 (23) ◽

pp. 1811-1821 ◽

Cited By ~ 4

Author(s):

A. P. BALACHANDRAN ◽

M. MARTONE

Keyword(s):

Hopf Algebra ◽

Lorentz Group ◽

Commutative Algebra ◽

Poincaré Group ◽

Poincare Group ◽

Group Invariant ◽

Spacetime Algebra ◽

Algebra Of Functions ◽

Moyal Plane ◽

The Right

We show how to get a noncommutative product for functions on spacetime starting from the deformation of the coproduct of the Poincaré group using the Drinfel'd twist. Thus it is easy to see that the commutative algebra of functions on spacetime (ℝ4) can be identified as the set of functions on the Poincaré group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincaré group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features. As is known, spacetime algebra fixes the coproduct on the diffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.

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A NONCOMMUTATIVE EXTENSION OF GRAVITY

International Journal of Modern Physics D ◽

10.1142/s0218271894000332 ◽

1994 ◽

Vol 03 (01) ◽

pp. 221-224 ◽

Cited By ~ 20

Author(s):

J. MADORE ◽

J. MOURAD

Keyword(s):

Tensor Product ◽

Noncommutative Geometry ◽

Commutative Algebra ◽

Internal Space ◽

Complex Matrices ◽

Noncommutative Algebra ◽

Algebra Of Functions ◽

Klein Theory ◽

Hilbert Action ◽

Kaluza Klein

The commutative algebra of functions on a manifold is extended to a noncommutative algebra by considering its tensor product with the algebra of n×n complex matrices. Noncommutative geometry is used to formulate an extension of the Einstein-Hilbert action. The result is shown to be equivalent to the usual Kaluza-Klein theory with the manifold SUn as an internal space, in a truncated approximation.

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SIMPLICITY CRITERIA FOR RINGS OF DIFFERENTIAL OPERATORS

Glasgow Mathematical Journal ◽

10.1017/s0017089521000148 ◽

2021 ◽

pp. 1-5

Author(s):

V. V. BAVULA

Keyword(s):

Finite Type ◽

Algebraic Variety ◽

Commutative Algebra ◽

Differential Operators ◽

Arbitrary Field ◽

Perfect Field ◽

Arbitrary Characteristic ◽

Jacobian Ideal ◽

Algebra Of Functions ◽

Rings Of Differential Operators

Abstract Let K be a field of arbitrary characteristic, $${\cal A}$$ be a commutative K-algebra which is a domain of essentially finite type (e.g., the algebra of functions on an irreducible affine algebraic variety), $${a_r}$$ be its Jacobian ideal, and $${\cal D}\left( {\cal A} \right)$$ be the algebra of differential operators on the algebra $${\cal A}$$ . The aim of the paper is to give a simplicity criterion for the algebra $${\cal D}\left( {\cal A} \right)$$ : the algebra $${\cal D}\left( {\cal A} \right)$$ is simple iff $${\cal D}\left( {\cal A} \right)a_r^i{\cal D}\left( {\cal A} \right) = {\cal D}\left( {\cal A} \right)$$ for all i ≥ 1 provided the field K is a perfect field. Furthermore, a simplicity criterion is given for the algebra $${\cal D}\left( R \right)$$ of differential operators on an arbitrary commutative algebra R over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.

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