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Register a new userDetermination of quantum symmetries for higher ADE systems from the modular T matrix
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We show that the Ocneanu algebra of quantum symmetries, for an ADE diagram (or for higher Coxeter-Dynkin systems, like the Di Francesco - Zuber system) is, in most cases, deduced from the structure of the modular T matrix in the A series. We recover in this way the (known) quantum symmetries of su(2) diagrams and illustrate our method by studying those associated with the three genuine exceptional diagrams of type su(3), namely E5, E9 and E21. This also provides the shortest way to the determination of twisted partition functions in boundary conformal field theory with defect lines.
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For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions
The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanus cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A3 is described in detail.
A new general formalism for determining the electric multipole polarizabilities of quantum (atomic and nuclear) bound systems based on the use of the transition matrix in momentum space has been developed. As distinct from the conventional approach with the application of the spectral expansion of the total Greens function, our approach does not require preliminary determination of the entire unperturbated spectrum; instead, it makes possible to calculate the polarizability of a few-body bound complex directly based on solving integral equations for the wave function of the ground bound state and the transition matrix at negative energy, both of them being real functions of momenta. A formula for the multipole polarizabilities of a two-body bound complex formed by a central interaction potential has been derived and studied. To test, the developed $T$-matrix formalism has been applied to the calculation of the dipole, quadrupole and octupole polarizabilities of the hydrogen atom.
We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed $D=4$ quantum inhomegeneous conformal Hopf algebras $mathcal{U}_{theta }(su(2,2)ltimes T^{4}$) and $mathcal{U}_{bar{theta}}(su(2,2)ltimesbar{T}^{4}$), where $T^{4}$ describe complex twistor coordinatesand $bar{T}^{4}$ the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently we introduce the quantum deformations of $D=4$ Heisenberg-conformal algebra (HCA) $su(2,2)ltimes H^{4,4}_hslash$ ($H^{4,4}_hslash=bar{T}^4 ltimes_hslash T_4$ is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length $lambda_p$ will be called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We shall describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and by the quantization map in $H_hslash^{4,4}$. We introduce as well generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra $mathcal{U}_theta(su(2,2)ltimes T^4).$
Using anisotropic R-matrices associated with affine Lie algebras $hat g$ (specifically, $A_{2n}^{(2)}, A_{2n-1}^{(2)}, B_n^{(1)}, C_n^{(1)}, D_n^{(1)}$) and suitable corresponding K-matrices, we construct families of integrable open quantum spin chains of finite length, whose transfer matrices are invariant under the quantum group corresponding to removing one node from the Dynkin diagram of $hat g$. We show that these transfer matrices also have a duality symmetry (for the cases $C_n^{(1)}$ and $D_n^{(1)}$) and additional $Z_2$ symmetries that map complex representations to their conjugates (for the cases $A_{2n-1}^{(2)}, B_n^{(1)}, D_n^{(1)}$). A key simplification is achieved by working in a certain unitary gauge, in which only the unbroken symmetry generators appear. The proofs of these symmetries rely on some new properties of the R-matrices. We use these symmetries to explain the degeneracies of the transfer matrices.
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