Spherical Harmonics Ylm(θ,ϕ)Y_{l}^{m}(\theta,\phi): Positive and Negative Integer Representations of su(1,1) for l-m and l+m

The azimuthal and magnetic quantum numbers of spherical harmonics $Y_{l}^{m}(\theta,\phi)$ describe quantization corresponding to the magnitude and $z$-component of angular momentum operator in the framework of realization of $su(2)$ Lie algebra symmetry. The azimuthal quantum number $l$ allocates to itself an additional ladder symmetry by the operators which are written in terms of $l$. Here, it is shown that simultaneous realization of the both symmetries inherits the positive and negative $(l-m)$- and $(l+m)$-integer discrete irreducible representations for $su(1,1)$ Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation of $su(2)$ compact Lie algebra via the $Y_{l}^{m}(\theta,\phi)$'s for a given $l$, we can also represent $su(1,1)$ noncompact Lie algebra by spherical harmonics for given values of $l-m$ and $l+m$.Comment: 11 pages, no figures, version to appear in Adv. High Energy Phy

Non Commutative Arens Algebras and their Derivations

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ we consider the non commutative Arens algebra $L^{\omega}(M, \tau)=\bigcap\limits_{p\geq1}L^{p}(M, \tau)$ and the related algebras $L^{\omega}_2(M, \tau)=\bigcap\limits_{p\geq2}L^{p}(M, \tau)$ and $M+L^{\omega}_2(M, \tau)$ which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra $M+L^{\omega}_2(M, \tau)$ is inner and all derivations of the algebras $L^{\omega}(M,\tau)$ and $L^{\omega}_2(M, \tau)$ are spatial and implemented by elements of $M+L^{\omega}_2(M, \tau).$Comment: 19 pages. Submitted to Journal of Functional analysi

Universal description of the rotational-vibrational spectrum of three particles with zero-range interactions

A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass $m$ and the third particle of the mass $m_1$ in the zero-range limit of the interaction between different particles is given for arbitrary values of the mass ratio $m/m_1$ and the total angular momentum $L$. If the two-body scattering length is positive, a number of vibrational states is finite for $L_c(m/m_1) \le L \le L_b(m/m_1)$, zero for $L>L_b(m/m_1)$, and infinite for $L<L_c(m/m_1)$. If the two-body scattering length is negative, a number of states is either zero for $L \ge L_c(m/m_1)$ or infinite for $L<L_c(m/m_1)$. For a finite number of vibrational states, all the binding energies are described by the universal function $\epsilon_{LN}(m/m_1) = {\cal E}(\xi, \eta)$, where $\xi=\displaystyle\frac{N-1/2}{\sqrt{L(L + 1)}}$, $\eta=\displaystyle\sqrt{\frac{m}{m_1 L (L + 1)}}$,and $N$ is the vibrational quantum number. This scaling dependence is in agreement with the numerical calculations for $L > 2$ and only slightly deviates from those for $L = 1, 2$. The universal description implies that the critical values $L_c(m/m_1)$ and $L_b(m/m_1)$ increase as $0.401 \sqrt{m/m_1}$ and $0.563 \sqrt{m/m_1}$, respectively, while a number of vibrational states for $L \ge L_c(m/m_1)$ is within the range $N \le N_{max} \approx 1.1 \sqrt{L(L+1)}+1/2$

A characterization of finite dimensional nilpotent Lie superalgebras

Let $L$ be a nilpotent Lie superalgebras of dimension $(m\mid n)$ for some non-negative integers $m$ and $n$ and put $s(L) = \frac{1}{2}[(m + n - 1)(m + n -2)]+ n+ 1 - \dim \mathcal{M}(L)$, where $\mathcal{M}(L)$ denotes the Schur multiplier of $L$. Recently, the author has shown that $s(L) \geq 0$ and the structure of all nilpotent Lie superalgebras has been determined when $s(L) = 0$ \cite{Nayak2018}. The aim of this paper is to classify all nilpotent Lie superalgebras $L$ for which $s(L) = 1$ and $2$.Comment: 19 page