An information complexity index for probability measures on ℝ with all moments

2016 ◽  
Vol 19 (03) ◽  
pp. 1650015 ◽  
Author(s):  
Luigi Accardi ◽  
Abdessatar Barhoumi ◽  
Mohamed Rhaima
Keyword(s):  
Lie Algebra ◽  
Limit Theorems ◽  
Heisenberg Algebra ◽  
Index Formula ◽  
Finite Support ◽  
Quantum Random Walks ◽  
Infinite Dimensional ◽  
Algebra Ii ◽  
Complexity Index ◽  
Poisson Measures

We prove that, each probability meassure on [Formula: see text], with all moments, is canonically associated with (i) a ∗-Lie algebra; (ii) a complexity index labeled by pairs of natural integers. The measures with complexity index [Formula: see text] consist of two disjoint classes: that of all measures with finite support and the semi-circle-arcsine class (the discussion in Sec. 4.1 motivates this name). The class [Formula: see text] coincides with the [Formula: see text]-measures in the finite support case and includes the semi-circle laws in the infinite support case. In the infinite support case, the class [Formula: see text] includes the arcsine laws, and the class [Formula: see text] appeared in central limit theorems of quantum random walks in the sense of Konno. The classes [Formula: see text], with [Formula: see text], do not seem to be present in the literature. The class [Formula: see text] includes the Gaussian and Poisson measures and the associated ∗-Lie algebra is the Heisenberg algebra. The class [Formula: see text] includes the non-standard (i.e. neither Gaussian nor Poisson) Meixner distributions and the associated ∗-Lie algebra is a central extension of [Formula: see text]. Starting from [Formula: see text], the ∗-Lie algebra associated to the class [Formula: see text] is infinite dimensional and the corresponding classes include the higher powers of the standard Gaussian.

On Level 0 Affine Lie Modules

1994 ◽  
Vol 37 (3) ◽  
pp. 310-314 ◽  
Author(s):  
D. J. Britten ◽  
F. W. Lemire
Keyword(s):  
Lie Algebra ◽  
Central Element ◽  
Heisenberg Algebra ◽  
Infinite Dimensional ◽  
Affine Lie Algebra ◽  
Central Elements ◽  
Graded Module ◽  
Nonzero Scalar ◽  
Lie Modules

AbstractIt is proven that the dimensions of the homogeneous summands of a nontrivial Z graded module for an infinité dimensional Heisenberg algebra on which a central element acts as nonzero scalar are unbounded. This result is then applied to show that the central elements of an affine Lie algebra act trivially on any indecomposable diagonalizable module whose weight spaces are of bounded dimension.

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

2009 ◽  
Vol 80 (1) ◽  
pp. 83-90 ◽  
Author(s):  
SHUDONG LIU ◽  
XIAOCHUN FANG
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Cuntz Algebra ◽  
Infinite Dimensional ◽  
Algebra Ii ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
K Theory ◽  
Extension Algebra

AbstractIn this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

LIE ALGEBRA OF THE q-POINCARÉ GROUP AND q-HEISENBERG COMMUTATION RELATIONS

1999 ◽  
Vol 13 (24n25) ◽  
pp. 2895-2902
Author(s):  
PAOLO ASCHIERI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Heisenberg Algebra ◽  
Poincaré Group ◽  
Orthogonal Groups ◽  
Poincare Group ◽  
Commutation Relations ◽  
Real Forms

We discuss quantum orthogonal groups and their real forms. We review the construction of inhomogeneous orthogonal q-groups and their q-Lie algebras. The geometry of the q-Poincaré group naturally induces a well defined q-deformed Heisenberg algebra of hermitian q-Minkowski coordinates xaand momenta pa.

scholarly journals A note on the zeroth products of Frenkel–Jing operators

2017 ◽  
Vol 16 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Slaven Kožić
Keyword(s):  
Lie Algebra ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Quantum Vertex ◽  
Algebra Theory

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

On Weight Systems Derived from Heisenberg Lie Algebras

2003 ◽  
Vol 12 (05) ◽  
pp. 589-604
Author(s):  
Hideaki Nishihara
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Weight System ◽  
Infinite Dimensional ◽  
Conway Polynomial ◽  
Knot Invariant ◽  
Polynomial Representations ◽  
Solvable Lie Algebras

Weight systems are constructed with solvable Lie algebras and their infinite dimensional representations. With a Heisenberg Lie algebra and its polynomial representations, the derived weight system vanishes on Jacobi diagrams with positive loop-degree on a circle, and it is proved that the derived knot invariant is the inverse of the Alexander-Conway polynomial.

Limit theorems and absorption problems for quantum radom walks in one dimension

2002 ◽  
Vol 2 (Special) ◽  
pp. 578-595
Author(s):  
N. Konno
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
The Other ◽  
One Dimension ◽  
Unitary Matrices ◽  
Quantum Random Walks ◽  
One Dimensional ◽  
The Difference ◽  
The One

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by $2 \times 2$ unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.

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