ON QUANTUM GROUP OF UNITARY OPERATORS QUANTUM ‘az + b’ GROUP

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras

QUANTUM ‘az + b’ GROUP ON COMPLEX PLANE

‘az + b’ is the group of affine transformations of complex plane C. The coefficients a, b ∈ C. In quantum version a, b are normal operators such that ab = q2ba, where q is the deformation parameter. We shall assume that q is a root of unity. More precisely q = e 2πi N , where N is an even natural number. To construct the group we write an explicit formula for the Kac Takesaki operator W . It is...

Bistability in the Electric Current through a Quantum-Dot Capacitively Coupled to a Charge-Qubit

We investigate the electronic transport through a single-level quantum-dot which is capacitively coupled to a charge-qubit. By employing the method of nonequilibrium Green's functions, we calculate the electric current through quantum dot at finite bias voltages. The Green's functions and self-energies of the system are calculated perturbatively and self-consistently to the second order of inte...

Haar Weight on Some Quantum Groups

We present a number of examples of locally compact quantum groups. These are quantum deformations of the group of affine transformations R (‘ax+b’ group) and C (Gz group). Starting from a modular multiplicative unitary W we find (under certain technical assumption) a simple formula expressing the (right) Haar weight on the quantum group associated with W . The formula works for quantum ‘ax + b’...

Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics

We consider pseudo-unitary quantum systems and discuss various properties of pseudounitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudo-unitary matrix is the exponential of i = √ −1 times a pseudo-Hermitian matrix, and determine the structure of the L...