Rolling of Coxeter polyhedra along mirrors

The topic of the paper are developments of $n$-dimensional Coxeter polyhedra. We show that the surface of such polyhedron admits a canonical cutting such that each piece can be covered by a Coxeter $(n-1)$-dimensional domain.Comment: 20pages, 15 figure

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

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

Sexual distress and quality of life among women with bipolar disorder

Abstract Background Information on the association between bipolar disorder (BD), sexual satisfaction, sexual function, sexual distress and quality of life (QoL) is sparse. This study aims, in women with BD, to (i) investigate sexual dysfunction, sexual distress, general sexual satisfaction and QoL; (ii) explore whether sexual distress was related to affective symptoms and (iii) investigate whether QoL was associated with sexual distress. The study is a questionnaire survey in an outpatient cohort of women with BD using: Changes in Sexual Functioning Questionnaire, Female Sexual Distress Scale, Altman Self-Rating Mania Scale (ASRM), Major Depression Inventory (MDI) and The World Health Organisation Quality of Life-Brief. Results In total, 61 women (age range 19–63, mean 33.7 years) were recruited. Overall, 54% reported sexual distress (n = 33) and 39% were not satisfied with their sexual life (n = 24). Women with BD were significantly more sexually distressed in comparison with Danish women from the background population but they did not have a higher prevalence of impaired sexual function. Better sexual function was positively associated with ASRM scores while MDI scores were associated with more distress. Finally, the group of non-sexually distressed women with BD reported higher QoL scores compared with the sexually distressed group. Conclusions Women with BD exhibited a high prevalence of sexual distress and their sexual function seemed associated with their actual mood symptoms and perception of QoL

Affine spherical homogeneous spaces with good quotient by a maximal unipotent subgroup

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.Comment: v2: title and abstract changed; v3: 16 pages, minor correction

Hypermatrix factors for string and membrane junctions

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric products of two vector spaces, and hence are matrix representations. We consider the analogous products of three vector spaces and study when they appear as summands in Lie algebra decompositions. The Z3-grading of the exceptional Lie algebras provide such summands and provides representations of classical groups on hypermatrices. The main natural application is a formal study of three-junctions of strings and membranes. Generalizations are also considered.Comment: 25 pages, 4 figures, presentation improved, minor correction

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

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

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