Quantum symmetric spaces

Let $G$ be a semisimple Lie group, ${\frak g}$ its Lie algebra. For any symmetric space $M$ over $G$ we construct a new (deformed) multiplication in the space $A$ of smooth functions on $M$. This multiplication is invariant under the action of the Drinfeld--Jimbo quantum group $U_h{\frak g}$ and is commutative with respect to an involutive operator $\tilde{S}: A\otimes A \to A\otimes A$. Such a multiplication is unique. Let $M$ be a k\"{a}hlerian symmetric space with the canonical Poisson structure. Then we construct a $U_h{\frak g}$-invariant multiplication in $A$ which depends on two parameters and is a quantization of that structure.Comment: 16 pp, LaTe

Polyvector Super-Poincare Algebras

A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the maximal solvable ideal of g, W_0 is generated by W_1 and commutes with W. Choosing W_1 to be a spinorial so(V)-module (a sum of an arbitrary number of spinors and semispinors), we prove that W_0 consists of polyvectors, i.e. all the irreducible so(V)-submodules of W_0 are submodules of \Lambda V. We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of so(V)-invariant \Lambda^k V-valued bilinear forms on the spinor module S.Comment: 41 pages, minor correction

The EPR experiment in the energy-based stochastic reduction framework

We consider the EPR experiment in the energy-based stochastic reduction framework. A gedanken set up is constructed to model the interaction of the particles with the measurement devices. The evolution of particles' density matrix is analytically derived. We compute the dependence of the disentanglement rate on the parameters of the model, and study the dependence of the outcome probabilities on the noise trajectories. Finally, we argue that these trajectories can be regarded as non-local hidden variables.Comment: 11 pages, 5 figure

Lmo4 Establishes Rostral Motor Cortex Projection Neuron Subtype Diversity

The mammalian neocortex is parcellated into anatomically and functionally distinct areas. The establishment of area-specific neuronal diversity and circuit connectivity enables distinct neocortical regions to control diverse and specialized functional outputs, yet underlying molecular controls remain largely unknown. Here, we identify a central role for the transcriptional regulator Lim-only 4 (Lmo4) in establishing the diversity of neuronal subtypes within rostral mouse motor cortex, where projection neurons have particularly diverse and multi-projection connectivity compared with caudal motor cortex. In rostral motor cortex, we report that both subcerebral projection neurons (SCPN), which send projections away from the cerebrum, and callosal projection neurons (CPN), which send projections to contralateral cortex, express Lmo4, whereas more caudal SCPN and CPN do not. Lmo4-expressing SCPN and CPN populations are comprised of multiple hodologically distinct subtypes. SCPN in rostral layer Va project largely to brainstem, whereas SCPN in layer Vb project largely to spinal cord, and a subset of both rostral SCPN and CPN sends second ipsilateral caudal (backward) projections in addition to primary projections. Without Lmo4 function, the molecular identity of neurons in rostral motor cortex is disrupted and more homogenous, rostral layer Va SCPN aberrantly project to the spinal cord, and many dual-projection SCPN and CPN fail to send a second backward projection. These molecular and hodological disruptions result in greater overall homogeneity of motor cortex output. Together, these results identify Lmo4 as a central developmental control over the diversity of motor cortex projection neuron subpopulations, establishing their area-specific identity and specialized connectivity.Stem Cell and Regenerative Biolog

On quantum group SL_q(2)

We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation we develop a general method of constructing quantum groups with similar property. We also describe this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups. We carry our method in detail for root systems of type SL(2); as a byproduct we find a new series of quantum groups - metaplectic groups of SL(2)-type. Representations of these groups can provide interesting examples of bimodule categories over monoidal category of representations of SL_q(2).Comment: plain TeX, 19 pages, no figure

Dipper-Donkin algebra as global symmetry of quantum chains

We analize the role of GL_2, a quantum group constructed by Dipper-Donkin, as a global symmetry for quantum chains, and show the way to construct all possible Hamiltonians for four states quantum chains with GL_2 global symmetry. In doing this, we search all inner actions of GL_2 on the Clifford algebra C(1,3) and show them. We also introduce the corresponding operator algebras, invariants and Hamiltonians, explicitly.Comment: 30 pages, 3 Figures, LaTex2

Solvable Lie algebras with triangular nilradicals

All finite-dimensional indecomposable solvable Lie algebras $L(n,f)$, having the triangular algebra T(n) as their nilradical, are constructed. The number of nonnilpotent elements $f$ in $L(n,f)$ satisfies $1\leq f\leq n-1$ and the dimension of the Lie algebra is $\dim L(n,f)=f+{1/2}n(n-1)$

Projective connections in CR geometry

Holomorphic invariants of an analytic real hypersurface in ℂ n+1 can be computed by several methods, coefficients of the Moser normal form [4], pseudo-con-formal curvature and its covariant derivatives [4], and projective curvature and its covariant derivatives [3]. The relation between these constructions is given in terms of reduction of the complex projective structure to a real form and exponentiation of complex vectorfields to give complex coordinate systems and corresponding Moser normal forms. Although the results hold for hypersurfaces with non-degenerate Levi-form, explicit formulas will be given only for the positive definite case.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46644/1/229_2005_Article_BF01298334.pd