A CLASSICAL REALIZATION OF QUANTUM ALGEBRAS

1990 ◽  
Vol 05 (28) ◽  
pp. 2325-2333 ◽  
Author(s):  
ALEXIOS P. POLYCHRONAKOS
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Group Homomorphism ◽  
Quantum Algebras ◽  
Quantum Deformation ◽  
Standard Quantum ◽  
Algebra For All ◽  
Special Case ◽  
Classical Lie Algebra ◽  
Construction Works

We construct a realization of a deformation of the Lie algebra of a group in terms of the generators of the classical Lie algebra of the group. The construction works for arbitrary (odd) deforming functions and, as a special case, it reproduces the standard quantum deformation of the algebra. For all these functions it gives a co-multiplication, that is, a group homomorphism, and provides an antipode and a co-unit. It therefore promotes any arbitrary deformation into a Hopf algebra.

scholarly journals Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

2020 ◽  
Author(s):  
Piotr M. Hajac ◽  
Tomasz Maszczyk
Keyword(s):  
Cyclic Homology ◽  
Quantum Deformation ◽  
Homotopy Formula ◽  
Standard Quantum ◽  
Unital Algebra ◽  
Hopf Fibrations ◽  
Chain Homotopy ◽  
Cartan Model ◽  
Extension Algebra ◽  
Hochschild Complex

AbstractViewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

Duality on the quantum space(3) with two parameters

Open Physics ◽  
2012 ◽  
Vol 10 (5) ◽  
Author(s):  
Muttalip Özavşar ◽  
Gürsel Yeşilot
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Differential Forms ◽  
Quantum Space ◽  
Dual Algebra ◽  
Commutation Relations ◽  
Dual Hopf Algebra ◽  
Invariant Differential ◽  
Leibniz Rules ◽  
Two Parameters

AbstractIn this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.

Quantum algebras and parity-dependent spectra

2007 ◽  
Vol 463 (2086) ◽  
pp. 2415-2427 ◽  
Author(s):  
C.V Sukumar ◽  
Andrew Hodges
Keyword(s):  
Quantum Optics ◽  
Harmonic Oscillator ◽  
Combinatorial Theory ◽  
Squeezed States ◽  
Quantum Algebras ◽  
Matrix Elements ◽  
Euler Numbers ◽  
Simple Harmonic Oscillator ◽  
Non Gaussian ◽  
Special Case

We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.

scholarly journals A categorical reconstruction of crystals and quantum groups at q = 0

2019 ◽  
Vol 70 (3) ◽  
pp. 895-925
Author(s):  
Craig Smith
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Analogous Result ◽  
Crystal Bases ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Monoidal Structure

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

THE BANACH-LIE *-ALGEBRA OF MULTIPLICATION OPERATORS ON A W*-ALGEBRA

2011 ◽  
Vol 04 (02) ◽  
pp. 235-261
Author(s):  
Maysaa Alqurashi ◽  
Najla A. Altwaijry ◽  
C. Martin Edwards ◽  
Christopher S. Hoskin
Keyword(s):  
State Space ◽  
Lie Algebra ◽  
The State ◽  
Multiplication Operators ◽  
Dual Cone ◽  
Positive Real ◽  
Linear Isometry ◽  
Real Linear ◽  
Special Case

The hermitian part [Formula: see text] of the Banach-Lie *-algebra [Formula: see text] of multiplication operators on the W *-algebra A is a unital GM-space, the base of the dual cone in the dual GL-space [Formula: see text] of which is affine isomorphic and weak*-homeomorphic to the state space of [Formula: see text]. It is shown that there exists a Lie *-isomorphism ϕ from the quotient (A ⊕∞ Aop)/K of an enveloping W *-algebra A ⊕∞ Aop of A by a weak*-closed Lie *-ideal K onto [Formula: see text], the restriction to the hermitian part ((A ⊕∞ Aop)/K)h of which is a bi-positive real linear isometry, thereby giving a characterization of the state space of [Formula: see text]. In the special case in which A is a W *-factor this leads to a further identification of the state space of [Formula: see text] in terms of the state space of A. For any W *-algebra A, the Banach-Lie *-algebra [Formula: see text] coincides with the set of generalized derivations of A, and, as an application, a formula is obtained for the norm of an element of [Formula: see text] in terms of a centre-valued 'norm' on A, which is similar to that previously obtained by non-order-theoretic methods.

A CLOSED-FORM SOLUTION OF THE EVOLUTION OPERATOR IN TWO-LEVEL SYSTEMS

1994 ◽  
Vol 08 (08n09) ◽  
pp. 505-508 ◽  
Author(s):  
XIAN-GENG ZHAO
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Evolution Operator ◽  
Closed Form Solution ◽  
Form Solution ◽  
Time Dependent ◽  
Arbitrary Time ◽  
Two Level Systems ◽  
Special Case

It is demonstrated by using the technique of Lie algebra SU(2) that the problem of two-level systems described by arbitrary time-dependent Hamiltonians can be solved exactly. A closed-form solution of the evolution operator is presented, from which the results for any special case can be deduced.

Sign in / Sign up

Export Citation Format

Share Document