On the nature of the Virasoro algebra

The multiplication in the Virasoro algebra $$[e_p, e_q] = (p - q) e_{p+q} + \theta \left(p^3 - p\right) \delta_{p + q}, \qquad p, q \in {\mathbf Z},$$ $$[\theta, e_p] = 0,$$ comes from the commutator $[e_p, e_q] = e_p * e_q - e_q * e_p$ in a quasiassociative algebra with the multiplication \renewcommand{\theequation}{$*$} \be \ba{l} \ds e_p * e_q = - {q (1 + \epsilon q) \over 1 + \epsilon (p + q)} e_{p+q} + {1 \over 2} \theta \left[p^3 - p + \left(\epsilon - \epsilon^{-1} \right) p^2 \right] \delta^0_{p+q}, \vspace{3mm}\\ \ds e_p * \theta = \theta* e_p = 0. \ea \ee The multiplication in a quasiassociative algebra ${\cal R}$ satisfies the property \renewcommand{\theequation}{$**$} \be a * (b * c) - (a * b) * c = b * (a * c) - (b * a) * c, \qquad a, b, c \in {\cal R}. \ee This property is necessary and sufficient for the Lie algebra {\it Lie}$({\cal R})$ to have a phase space. The above formulae are put into a cohomological framework, with the relevant complex being different from the Hochschild one even when the relevant quasiassociative algebra ${\cal R}$ becomes associative. Formula $(*)$ above also has a differential-variational counterpart

Investigating the Ca2+-dependent and Ca2+-independent mechanisms for mammalian cone light adaptation

Abstract Vision is mediated by two types of photoreceptors: rods, enabling vision in dim light; and cones, which function in bright light. Despite many similarities in the components of their respective phototransduction cascades, rods and cones have distinct sensitivity, response kinetics, and adaptation capacity. Cones are less sensitive and have faster responses than rods. In addition, cones can function over a wide range of light conditions whereas rods saturate in moderately bright light. Calcium plays an important role in regulating phototransduction and light adaptation of rods and cones. Notably, the two dominant Ca2+-feedbacks in rods and cones are driven by the identical calcium-binding proteins: guanylyl cyclase activating proteins 1 and 2 (GCAPs), which upregulate the production of cGMP; and recoverin, which regulates the inactivation of visual pigment. Thus, the mechanisms producing the difference in adaptation capacity between rods and cones have remained poorly understood. Using GCAPs/recoverin-deficient mice, we show that mammalian cones possess another Ca2+-dependent mechanism promoting light adaptation. Surprisingly, we also find that, unlike in mouse rods, a unique Ca2+-independent mechanism contributes to cone light adaptation. Our findings point to two novel adaptation mechanisms in mouse cones that likely contribute to the great adaptation capacity of cones over rods

Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras

This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice), reflection group $W\subset W(S)$, fundamental polyhedron \Cal M of $W$ and an acceptable (corresponding to twisting coefficients) set P({\Cal M})\subset M of vectors orthogonal to faces of \Cal M (simple roots). One can construct the corresponding Lorentzian Kac--Moody Lie algebra {\goth g}={\goth g}^{\prime\prime}(A(S,W,P({\Cal M}))) which is graded by $M$. We show that \goth g has good behavior of imaginary roots, its denominator formula is defined in a natural domain and has good automorphic properties if and only if \goth g has so called {\it restricted arithmetic type}. We show that every finitely generated (i.e. P({\Cal M}) is finite) algebra {\goth g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1))) may be embedded to {\goth g}^{\prime\prime}(A(S,W,P({\Cal M}))) of the restricted arithmetic type. Thus, Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a natural class to study. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the best automorphic properties for the denominator function if they have {\it a lattice Weyl vector $\rho$}. Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type with generalized lattice Weyl vector $\rho$ are called {\it elliptic}Comment: Some corrections in Sects. 2.1, 2.2 were done. They don't reflect on results and ideas. 31 pages, no figures. AMSTe

The Na+/Ca2+, K+ exchanger 2 modulates mammalian cone phototransduction

Calcium ions (Ca(2+)) modulate the phototransduction cascade of vertebrate cone photoreceptors to tune gain, inactivation, and light adaptation. In darkness, the continuous current entering the cone outer segment through cGMP-gated (CNG) channels is carried in part by Ca(2+), which is then extruded back to the extracellular space. The mechanism of Ca(2+) extrusion from mammalian cones is not understood. The dominant view has been that the cone-specific isoform of the Na(+)/Ca(2+), K(+) exchanger, NCKX2, is responsible for removing Ca(2+) from their outer segments. However, indirect evaluation of cone function in NCKX2-deficient (Nckx2(−/−)) mice by electroretinogram recordings revealed normal photopic b-wave responses. This unexpected result suggested that NCKX2 may not be involved in the Ca(2+) homeostasis of mammalian cones. To address this controversy, we examined the expression of NCKX2 in mouse cones and performed transretinal recordings from Nckx2(−/−) mice to determine the effect of NCKX2 deletion on cone function directly. We found that Nckx2(−/−) cones exhibit compromised phototransduction inactivation, slower response recovery and delayed background adaptation. We conclude that NCKX2 is required for the maintenance of efficient Ca(2+) extrusion from mouse cones. However, surprisingly, Nckx2(−/−) cones adapted normally in steady background light, indicating the existence of additional Ca(2+)-extruding mechanisms in mammalian cones

Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits

We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin diagram of compact hyperbolic type is 4. Building on earlier classification results of Kac, Kobayashi-Morita, Li and Sa\c{c}lio\~{g}lu, we present the 238 hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. For each symmetrizable hyperbolic generalized Cartan matrix, we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system. For each such hyperbolic root system we determine the disjoint orbits of the action of the Weyl group on real roots. It follows that the maximal number of disjoint Weyl group orbits on real roots in a hyperbolic root system is 4.Comment: J. Phys. A: Math. Theor (to appear

A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues

It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci tensor are shown to exist in various situations: there are homogeneous examples of any complex dimension n > 1, if one eigenvalue is negative and the other positive or zero, and of any complex dimension n > 2, if the both eigenvalues are negative; there are non-homogeneous examples of complex dimension 2, if one of the eigenvalues is zero. The problem of existence of Kahler metrics whose Ricci tensor has two distinct, constant eigenvalues is related to the celebrated (still open) Goldberg conjecture. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete, Einstein, strictly almost Kahler metrics of any even real dimension greater than 4.Comment: 18 pages; final version; accepted for publication in International Journal of Mathematic

Guanylate cyclase–activating protein 2 contributes to phototransduction and light adaptation in mouse cone photoreceptors

Light adaptation of photoreceptor cells is mediated by Ca2+-dependent mechanisms. In darkness, Ca2+ influx through cGMP-gated channels into the outer segment of photoreceptors is balanced by Ca2+ extrusion via Na+/Ca2+, K+ exchangers (NCKXs). Light activates a G protein signaling cascade, which closes cGMP-gated channels and decreases Ca2+ levels in photoreceptor outer segment because of continuing Ca2+ extrusion by NCKXs. Guanylate cyclase-activating proteins (GCAPs) then up-regulate cGMP synthesis by activating retinal membrane guanylate cyclases (RetGCs) in low Ca2+ This activation of RetGC accelerates photoresponse recovery and critically contributes to light adaptation of the nighttime rod and daytime cone photoreceptors. In mouse rod photoreceptors, GCAP1 and GCAP2 both contribute to the Ca2+-feedback mechanism. In contrast, only GCAP1 appears to modulate RetGC activity in mouse cones because evidence of GCAP2 expression in cones is lacking. Surprisingly, we found that GCAP2 is expressed in cones and can regulate light sensitivity and response kinetics as well as light adaptation of GCAP1-deficient mouse cones. Furthermore, we show that GCAP2 promotes cGMP synthesis and cGMP-gated channel opening in mouse cones exposed to low Ca2+ Our biochemical model and experiments indicate that GCAP2 significantly contributes to the activation of RetGC1 at low Ca2+ when GCAP1 is not present. Of note, in WT mouse cones, GCAP1 dominates the regulation of cGMP synthesis. We conclude that, under normal physiological conditions, GCAP1 dominates the regulation of cGMP synthesis in mouse cones, but if its function becomes compromised, GCAP2 contributes to the regulation of phototransduction and light adaptation of cones

Global analysis by hidden symmetry

Hidden symmetry of a G'-space X is defined by an extension of the G'-action on X to that of a group G containing G' as a subgroup. In this setting, we study the relationship between the three objects: (A) global analysis on X by using representations of G (hidden symmetry); (B) global analysis on X by using representations of G'; (C) branching laws of representations of G when restricted to the subgroup G'. We explain a trick which transfers results for finite-dimensional representations in the compact setting to those for infinite-dimensional representations in the noncompact setting when $X_C$ is $G_C$-spherical. Applications to branching problems of unitary representations, and to spectral analysis on pseudo-Riemannian locally symmetric spaces are also discussed.Comment: Special volume in honor of Roger Howe on the occasion of his 70th birthda

Reduction and reconstruction of stochastic differential equations via symmetries

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance