Quantum Lie Algebras Research Articles

A Quantum Space and Some Associated Quantum Groups

In the present paper, we first introduce a quantum $n$-space on which the algebra of coordinates is $\eta$-commutative. Further, it is shown that there are some $\sigma$-twisted derivations acting on this algebra, and the algebra of such derivations is a quantum group. Morever, we show that a bicovariant differential calculus on this space can be constructed by using $\sigma$-twisted derivations. Finally, the quantum Lie algebra is obtained by using this bicovariant differential calculus.

Observables and dispersion relations in \u03ba-Minkowski spacetime

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator.This general noncommutative geometry construction is then exemplified in the case of κ-Minkowski spacetime. The corresponding quantum Poincaré-Weyl Lie algebra of in-finitesimal translations, rotations and dilatations is obtained. The d’Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.

Asymmetric Stochastic Transport Models with $${\mathscr {U}}_q(\mathfrak {su}(1,1))$$ U q ( su ( 1 , 1 ) ) Symmetry

By using the algebraic construction outlined in \cite{CGRS}, we introduce several Markov processes related to the ${\mathcal{U}}_q(\mathfrak{su}(1,1))$ quantum Lie algebra. These processes serve as asymmetric transport models and their algebraic structure easily allows to deduce duality properties of the systems. The results include: (a) the asymmetric version of the Inclusion Process, which is self-dual; (b) the diffusion limit of this process, which is a natural asymmetric analogue of the Brownian Energy Process and which turns out to have the symmetric Inclusion Process as a dual process; (c) the asymmetric analogue of the KMP Process, which also turns out to have a symmetric dual process. We give applications of the various duality relations by computing exponential moments of the current.

Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

First, referring to our previous work, 'Hopf cyclic cohomology in braided monoidal categories', we reduce the restriction of the ambient category C being symmetric. We let C to be non-symmetric but assume only the restriction, ψ 2 = id, on the braid map correspond to the Hopf algebra H, which is the main player in the theory. We define a family of examples of such desired braided Hopf algebras, H, living in the category of anyonic vector spaces. Next, on one hand, we will prove that these anyonic Hopf algebras are the enveloping (Hopf) algebras of particular quantum Lie algebras, which we will construct. On the other hand, we will show that, analogous to the non-super and the super case, the well known relationship between the periodic Hopf cyclic cohomology of an enveloping (super) algebra and the (super) Lie algebra homology also holds for these particular quantum Lie algebras.

Lowering–raising triples and [formula omitted

We introduce the notion of a lowering–raising (or LR) triple of linear transformations on a nonzero finite-dimensional vector space. We show how to normalize an LR triple, and classify up to isomorphism the normalized LR triples. We describe the LR triples using various maps, such as the reflectors, the inverters, the unipotent maps, and the rotators. We relate the LR triples to the equitable presentation of the quantum algebra Uq(sl2) and Lie algebra sl2.

Soliton solutions for higher grading affine Toda from quantum algebras

Based on quantum Lie algebras associated to affine Lie algebras we construct special soliton-like solutions for higher grading affine Toda models.

Quantization of universal Teichmüller space

Universal Teichmuller space Open image in new window is the quotient of the group QS(S 1) of quasisymmetric homeomorphisms of S 1 modulo Mobius transformations. The quantization problem for Open image in new window arises in the theory of non-smooth closed bosonic strings. Because of non-smoothness of strings the natural QS(S 1)-action on Open image in new window is also not smooth so there is no classical Lie algebra, associated to QS(S 1). However, using methods of non-commutative geometry, we can define a quantum Lie algebra of observables Der q (QS), yielding the quantization of Open image in new window .

Tomographic and Lie algebraic significance of generalized symmetric informationally complete measurements

Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank 1. They are interesting originally because of their connection with rank-1 SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal informationally complete (IC) measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs are optimal among minimal IC measurements with given average purity of measurement outcomes. Besides its significance to the current study, this result may help us to understand tomographic efficiencies of minimal IC measurements under the influence of noise. When minimal IC measurements are taken as bases for the Lie algebra of the unitary group, generalized SICs are uniquely characterized by the antisymmetry of the associated structure constants.

The quantum Rabi model and Lie algebra representations of

The aim of the present paper is to understand the spectral problem of the quantum Rabi model in terms of Lie algebra representations of . We define a second order element of the universal enveloping algebra of , which, through the image of a principal series representation of , provides a picture equivalent to the quantum Rabi model drawn by confluent Heun differential equations. By this description, in particular, we give a representation theoretic interpretation of the degenerate part of the spectrum (i.e., Juddʼs eigenstates) of the Rabi Hamiltonian due to Kuś in 1985, which is a part of the exceptional spectrum parameterized by integers. We also discuss the non-degenerate part of the exceptional spectrum of the model, in addition to the Judd eigenstates, from a viewpoint of infinite dimensional irreducible submodules (or subquotients) of the non-unitary principal series such as holomorphic discrete series representations of .

Quantum Lie algebra solitons

We construct a special type of quantum soliton solutions for quantized affine Toda models. The elements of the principal Heisenberg subalgebra in the affinised quantum Lie algebra are found. Their eigenoperators inside the quantized universal enveloping algebra for an affine Lie algebra are constructed to generate quantum soliton solutions.

Quantum Lie Algebras and Differential Calculus on Quantum Groups

Quantum Lie Algebras and Differential Calculus on Quantum Groups

Affine and degenerate affine BMW algebras: actions on tensor space

The affine and degenerate affine Birman–Murakami–Wenzl (BMW) algebras arise naturally in the context of Schur–Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients of the affine and degenerate affine BMW algebras. In this paper, we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements—the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.

Duality on the quantum space(3) with two parameters

Abstract In 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.

DRINFELD-JIMBO QUANTUM LIE ALGEBRA

Quantum Lie algebras related to multi-parametric Drinfeld–Jimbo R-matrices of type GL(m|n) are classified.

Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies

There is a remarkable connection between quantum generating functions of field theory and formal power series associated with dimensions of chains and homologies of suitable Lie algebras. We discuss the homological aspects of this connection with its applications to partition functions of the minimal three-dimensional gravities in the space–time asymptotic to AdS 3 , which also describe the three-dimensional Euclidean black holes, the pure N = 1 supergravity, and a sigma model on N-fold generalized symmetric products. We also consider in the same context elliptic genera of some supersymmetric sigma models. These examples can be considered as a straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL ( 2 , Z ) ) to partition functions represented by means of formal power series that encode Lie algebra properties.

Balanced tripartite entanglement, the alternating group A4 and the lie algebra sl(3,ℂ)⊕ u(1)

We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group $W(E_8)$ in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, {\it Clifford group dipoles and the enactment of Weyl/Coxeter group $W(E_8)$ by entangling gates}, Preprint 0904.3691 (quant-ph)]. Then, we describe a peculiar "condensation" of $W(E_8)$ into the four-letter alternating group $A_4$, obtained from a chain of maximal subgroups. Group $A_4$ is realized from two B-type generators and found to correspond to the Lie algebra $sl(3,\mathbb{C})\oplus u(1)$. Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.

BRST charges for finite nonlinear algebras

Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a nonlinear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex.

Cremmer‐Gervais quantum Lie algebra

AbstractWe describe a quantum Lie algebra based on the Cremmer‐Gervais R‐matrix. The algebra arises upon a restriction of an infinite‐dimensional quantum Lie algebra.

Hopf algebras constructed by the FRT-construction

Given any bialgebra A and a braiding product 〈 | 〉 on A, a bialgebra U 〈 | 〉 was constructed in [R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum Lie algebra”, Comm. Algebra 19 (1991) 3295–3345], contained in the finite dual of A . This construction generalizes a (not very well known) construction of Fadeev, Reshetikhin and Takhtajan [L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97–110]. In the present paper it is proved that when U 〈 | 〉 is finite-dimensional (even if A is not), then it is a quasitriangular Hopf algebra.

Becchi-Rouet-Stora-Tyutin operators for W algebras

The study of quantum Lie algebras motivates a use of noncanonical ghosts and antighosts for nonlinear algebras, such as W-algebras. This leads, for the W3 and W3(2) algebras, to the Becchi-Rouet-Stora-Tyutin operator having the conventional cubic form.