scholarly journals FUZZY COMPLEX GRASSMANNIAN SPACES AND THEIR STAR PRODUCTS

2003 ◽  
Vol 18 (11) ◽  
pp. 1935-1958 ◽  
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.

scholarly journals COHOMOLOGY RING OF THE BRST OPERATOR ASSOCIATED TO THE SUM OF TWO PURE SPINORS

2013 ◽  
Vol 28 (23) ◽  
pp. 1350107 ◽  
Author(s):  
ANDREI MIKHAILOV ◽  
ALBERT SCHWARZ ◽  
RENJUN XU
Keyword(s):  
Cohomology Ring ◽  
Type Ii ◽  
Pure Spinor ◽  
Multiplicative Structure ◽  
Dimensional Algebra ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Brst Operator ◽  
Finite Dimensional ◽  
Algebra Of Functions

In the study of the Type II superstring, it is useful to consider the BRST complex associated to the sum of two pure spinors. The cohomology of this complex is an infinite-dimensional vector space. It is also a finite-dimensional algebra over the algebra of functions of a single pure spinor. In this paper we study the multiplicative structure.

Geometric Hamiltonian formulation of quantum mechanics in complex projective spaces

2015 ◽  
Vol 12 (08) ◽  
pp. 1560015 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Dynamics ◽  
Hamiltonian Formulation ◽  
Hamiltonian Vector Field ◽  
Complex Projective Spaces ◽  
Finite Dimensional ◽  
Probability Densities ◽  
Quantum Observables ◽  
Complex Projective

In finite dimension (at least), Quantum Mechanics can be formulated as a proper Hamiltonian theory where a notion of phase space is given by the projective space P(H) constructed on the Hilbert space H of the considered quantum theory. It is well-known P(H) can be equipped with a structure of Kähler manifold, in particular we have a symplectic form and a Poisson structure; Quantum dynamics can be described in terms of a Hamiltonian vector field on P(H). In this paper, exploiting the notion and properties of so-called frame functions, I describe a general prescription for associating quantum observables to real functions on P(H), classical-like observables, and quantum states to probability densities on P(H), Liouville densities, in order to obtain a complete and meaningful Hamiltonian formulation of a finite-dimensional quantum theory.

scholarly journals On Varieties of Lie Algebras of Maximal Class

2015 ◽  
Vol 67 (1) ◽  
pp. 55-89 ◽  
Author(s):  
Tatyana Barron ◽  
Dmitry Kerner ◽  
Marina Tvalavadze
Keyword(s):  
Lie Algebras ◽  
Technical Condition ◽  
Partial Answer ◽  
Maximal Class ◽  
Graded Lie Algebras ◽  
Projective Varieties ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Filiform Lie Algebras ◽  
Complex Projective

AbstractWe study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over ℂ, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on ℕ-graded Lie algebras of maximal class. As shown by A. Fialowski there are only three isomorphism types of ℕ-graded Lie algebras of maximal class generated by L1 and L2, L = 〈L1; L2〉. Vergne described the structure of these algebras with the property L = 〈L1〉. In this paper we study those generated by the first and q-th components where q > 2, L = 〈L1; Lq〉. Under some technical condition, there can only be one isomorphism type of such algebras. For q = 3 we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

A NONCOMMUTATIVE EXTENSION OF GRAVITY

1994 ◽  
Vol 03 (01) ◽  
pp. 221-224 ◽  
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.

scholarly journals Frame functions in finite-dimensional quantum mechanics and its Hamiltonian formulation on complex projective spaces

2016 ◽  
Vol 13 (02) ◽  
pp. 1650013 ◽  
Author(s):  
Valter Moretti ◽  
Davide Pastorello
Keyword(s):  
Quantum Mechanics ◽  
Projective Space ◽  
Hamiltonian Formulation ◽  
Point Of View ◽  
Algebra Structure ◽  
Complex Projective Spaces ◽  
Finite Dimensional ◽  
Quantum Observables ◽  
Complex Projective

This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by Gleason to prove his celebrated theorem. In particular, the problem of associating quantum states with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical-like observables (i.e. real scalar functions on the projective space) representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical-like objects representing quantum ones result to be frame functions. The requirements of [Formula: see text] covariance and (convex) linearity play a central role in the proof of those theorems. A new characterization of classical-like observables describing quantum observables is presented, together with a geometric description of the [Formula: see text]-algebra structure of the set of quantum observables in terms of classical-like ones.

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