Invariant Differential Operators for Non-Compact Lie Groups: the Sp(n,R) Case

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras sp(n,R), in detail for n=6. Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which we call 'conformal Lie algebras', which have very similar properties to the conformal algebras of Minkowski space-time. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations for n=6, including the necessary data for all relevant invariant differential operators. In fact, this gives by reduction also the cases for n<6, since the main multiplet for fixed n coincides with one reduced case for n+1.Comment: Latex2e, 27 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:0812.2690, arXiv:0812.265

L\'evy Processes on Uq(g)U_q(g) as Infinitely Divisible Representations

L\'evy processes on bialgebras are families of infinitely divisible representations. We classify the generators of L\'evy processes on the compact forms of the quantum algebras $U_q(g)$, where $g$ is a simple Lie algebra. Then we show how the processes themselves can be reconstructed from their generators and study several classical stochastic processes that can be associated to these processes.Comment: 13 pages, LATEX file, ASI-TPA/13/99 (TU Clausthal); 6/99 (Preprint-Reihe Mathmatik, Univ. Greifswald)

Anti de Sitter Holography via Sekiguchi Decomposition

In the present paper we start consideration of anti de Sitter holography in the general case of the (q+1)-dimensional anti de Sitter bulk with boundary q-dimensional Minkowski space-time. We present the group-theoretic foundations that are necessary in our approach. Comparing what is done for q=3 the new element in the present paper is the presentation of the bulk space as the homogeneous space G/H = SO(q,2)/SO(q,1), which homogeneous space was studied by Sekiguchi.Comment: 10 pages, to appear in the Proceedings of the XI International Workshop "Lie Theory and Its Applications in Physics", (Varna, Bulgaria, June 2015

Positive Energy Unitary Irreducible Representations of the Superalgebras osp(1|2n,R)

We give the classification of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R).Comment: 20 pages, LATEX2e (revtex4,amsmath,amssymb), Plenary talk by VKD at X International Conference on Symmetry Methods in Physics, Yerevan, 13-21.8.2003; added acknowledgements; corrected misprint

Positive Energy Representations, Holomorphic Discrete Series and Finite-Dimensional Irreps

Let G be a semi-simple non-compact Lie group with unitary lowest/highest weight representations. We consider explicitly the relation between three types of representations of G: positive energy (unitary lowest weight)representations, (holomorphic) discrete series representations and non-unitary finite-dimensional irreps. We consider mainly the conformal groups SO(n,2) treating in full detail the cases n=1,3,4.Comment: 28 pages, TEX with Harvmac using amssym.def, amssym.tex, epsf.tex; v2: new texts in Sections 1 & 3, new refs; v3: added 5 figures; v4: small correction

Accurate three states model for amino acids with two chemically coupled titrating sites in explicit solvent atomistic constant pH simulations and pK_a calculations.

Correct protonation of titratable groups in biomolecules is crucial for their accurate description by molecular dynamics simulations. In the context of constant pH simulations, an additional protonation degree of freedom is introduced for each titratable site, allowing the protonation state to change dynamically with changing structure or electrostatics. Here, we extend previous approaches for an accurate description of chemically coupled titrating sites. A second reaction coordinate is used to switch between two tautomeric states of an amino acid with chemically coupled titratable sites, such as aspartate (Asp), glutamate (Glu), and histidine (His). To this aim, we test a scheme involving three protonation states. To facilitate charge neutrality as required for periodic boundary conditions and Particle Mesh Ewald (PME) electrostatics, titration of each respective amino acid is coupled to a “water” molecule that is charged in the opposite direction. Additionally, a force field modification for Amber99sb is introduced and tested for the description of carboxyl group protonation. Our three states model is tested by titration simulations of Asp, Glu, and His, yielding a good agreement, reproducing the correct geometry of the groups in their different protonation forms. We further show that the ion concentration change due to the neutralizing “water” molecules does not significantly affect the protonation free energies of the titratable groups, suggesting that the three states model provides a good description of biomolecular dynamics at constant pH

qq-Trinomial identities

We obtain connection coefficients between $q$-binomial and $q$-trinomial coefficients. Using these, one can transform $q$-binomial identities into a $q$-trinomial identities and back again. To demonstrate the usefulness of this procedure we rederive some known trinomial identities related to partition theory and prove many of the conjectures of Berkovich, McCoy and Pearce, which have recently arisen in their study of the $\phi_{2,1}$ and $\phi_{1,5}$ perturbations of minimal conformal field theory.Comment: 21 pages, AMSLate