Differential algebras for quantum groups of type B, C, and D

We study higher order bicovariant differential calculi on the quantum groups O_q(N) and Sp _q (N). We show that the second antisymmetrizer exterior algebra _u is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars biinvariant 1-form. Moreover is central in _u and _u is an inner differential calculus. We show that the quadratic dual to the left-invariant algebra _{s L} is isomorphic to the reflection equation algebra. Let be an arbitrary left-covariant first order differential calculus. We show that the dimension of the space of left-invariant 2-forms in the universal exterior algebra equals the number of linearly independent quadratic-linear relations in the quantum tangent space.     

Classification of bicovariant differential calculi on quantum groups of type A,B, C and D

Under the assumptions thatq is not a root of unity and that the differentialsdu _j ⁱ of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA _n–1 ,B _n ,C _n andD _n . We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA _n–1 =SL _q (n) forn3. All these calculi have the dimensionn ². For the quantum groupsB _n ,C _n andD _n we show that except for finitely manyq there exist precisely twoN ²-dimensional bicovariant calculi forN3, whereN=2n+1 forB _n andN=2n forC _n ,D _n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker are determined. In the limitq1 two of the 2n calculi forA _n–1 and one of the two calculi forB _n ,C _n andD _n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.     

Quantum Lie algebras for B l, C l and D l

We give the quantum structure constants for the analogues of the classical Lie algebras so(n) and sp(n) for any n, as well as their quantum Killing form. We also include a summary of the method used to obtain them.     

Coxeter groups A4, B4 and D4 for two-qubit systems

The Coxeter–Weyl groups W(A₄), W(B₄), and W(D₄) have proven very useful for two-qubit systems in quantum information theory. A simple technique is employed to construct the unitary matrix representations of the groups, based on quaternionic transformation of the usual reflection matrices. The von Neumann entropy of each reduced density matrix is calculated. It is shown that these unitary matrix representations are naturally related to various universal quantum gates and they lead to entangled states. Canonical decomposition of generators in terms of fundamental gate representations is given to construct the quantum circuits.     

Quantum groups and WZNW models

An explanation of the appearance of quantum groups in chiral WZNW models is given. Invariance of the theory under quantum group action is discussed.Supported by NSF-PHY-86-57788     

Quantum groups of fermionic algebras

We consider inhomogeneous quantum groups that transform various types of fermions: standard fermions, commuting fermions and orthofermions. These quantum groups are notq-deformations.     

Quantum Heisenberg groups and sklyanin algebras

New bialgebra structures on the Heisenberg-Lie algebra and their quantizations are introduced. Some of these quantizations give rise to new multiplications in homogeneous coordinate rings of Abelian varieties, via the well-known identification of theta functions with suitable matrix coefficients of the Stone-von Neumann representations.N.A.: Forschungsstipendiat der Alexander von Humboldt-Stiftung. J. D. and A. T.: Postdoctoral fellowship, ICTP. N. A. and A. T.: This work was also partially supported by CONICET, CONICOR and FAMAF, Argentina.     

Quantum groups and diagonalization of the braid generator

It is shown that the braid generator is diagonalizable on arbitrary tensor product modules V V, V an irreducible module for a quantum group. A generalization of the Reshetikhin form for the braid generator is thereby obtained in the general case. As an application, a general closed formula is determined for link polynomials.     

A remark on matrix product operator algebras,anyons and subfactors

Letters in Mathematical Physics - We show that the mathematical structures in a recent work of...     

Toward Quantum Paraelectric,Paraelastic, and Paramagnetic 2D Materials

Quantum fluctuations are known to affect the finite-temperature properties of materials made out of light elements such as hydrogen and helium. More recently, it has also been realized that quantum effects may play a role on structural transformations of ferroic materials containing heavier atoms, provided the energy barrier separating two different phases is small when compared to thermal fluctuations. Herein, 2D ferroelectric and ferroelastic materials are showcased as potential candidates to experience quantum effects on their structural conformation at liquid helium temperatures. A brief literature overview of the path integral molecular dynamics approach, which could be useful for the discovery of quantum paraelectric, paraelastic, and paramagnetic behavior in 2D materials, is also provided.     

Angular Momentum in Quantum Physics,by L.C. Biedenharn and J.D. Louck

An account is given of the Brillouin scattering of light from liquids. Different contributions to the Brillouin spectrum are discussed and are shown to depend on liquid properties such as sound velocity, viscosity, thermal conductivity, structural relaxation and concentration effects. The importance of these measurements to liquid-state theory is emphasized. Stimulated Brillouin scattering is briefly explained. Finally, the use of the Fabry-Perot interferometer in these experiments is discussed.