Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

We analyze the polynomial part of the Iwasawa realization of the coset representative of non compact symmetric Riemannian spaces. We start by studying the role of Kostant's principal SU(2)_P subalgebra of simple Lie algebras, and how it determines the structure of the nilpotent subalgebras. This allows us to compute the maximal degree of the polynomials for all faithful representations of Lie algebras. In particular the metric coefficients are related to the scalar kinetic terms while the representation of electric and magnetic charges is related to the coupling of scalars to vector field strengths as they appear in the Lagrangian. We consider symmetric scalar manifolds in ��‐extended supergravity in various space‐time dimensions, elucidating various relations with the underlying Jordan algebras and normed Hurwitz algebras. For magic supergravity theories, our results are consistent with the Tits‐Satake projection of symmetric spaces and the nilpotency degree turns out to depend only on the space‐time dimension of the theory. These results should be helpful within a deeper investigation of the corresponding supergravity theory, e.g. in studying ultraviolet properties of maximal supergravity in various dimensions.     

The Lie algebra $$\mathfrak{s}\mathfrak{l}_2 (\mathbb{F})$$ and quasi-deformations

The Lie algebra is “deformed” using twisted derivations satisfying a twisted Leibniz rule. Some particular algebras appearing in this deformation scheme are discussed. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation     

Modules over\mathfrak{U}_q (\mathfrak{s}\mathfrak{l}_2 )

The restricted quantum universal enveloping algebra decomposes in a canonical way into a direct sum of indecomposable left (or right) ideals. They are useful for determining the direct summands which occur in the tensor product of two simple . The indecomposable finite-dimensional are classified and located in the Auslander-Reiten quiver.     

Studying bi-partite entangled state representations via the integration over ket-bra operators in \mathfrak{Q}-ordering or \mathfrak{P}-ordering

For two particles＇ relative position and total momentum we have introduced the entangled state representation ｜μ〉, and its conjugate state｜ξ〉 In this work, for the first time; we study theln via the integration over ket bra operators in -ordering or -ordering, where Q-ordering means all Qs are to the left, of all Ps and -ordering means all Ps are to the left of all Qs. In this way we newly derive -ordered （or Q-ordered） expansion formulas of the two-mode squeezing operator which can show the squeezing effect on both the two-mode coordinate and momentum eigenstates. This tells that not only the integration over ket bra operators within normally ordered, but also within - ordered （or -ordered） are feasible and useful in developing quantum mechanical representation and transtbrlnation theory.     

Fedosov Star Products and One-Differentiable Deformations

We show that every star product on a symplectic manifold defines uniquely a 1-differentiable deformation of the Poisson bracket. Explicit formulas are given. As a corollary we can identify the characteristic class of any star product as a part of its explicit (Fedosov) expression.     

\mathfrak{Q}-, \mathfrak{P}-, and Weyl-ordering of Squeezing Operators: New Operator Identities and Applications

The basic operator ordering regarding to coordinate-momentum operator is discussed by virtue of the technique of integration within $\mathfrak{Q}$ -ordering (all Q are on the left of all P) and $\mathfrak{P}$ -ordering (all P are on the left of all Q). We derive new operator-ordering identities about $\mathfrak{Q}$ -ordering , $\mathfrak{P}$ -ordering and Weyl-ordering of both single-mode and two-mode squeezing operators. Its application in combinatorics is pointed out.     

BRST Resolution for the Principally Graded Wakimoto Module of $$\widehat{\mathfrak{s}\mathfrak{l}}_2 $$

BRST resolution is studied for the principally graded Wakimoto module of recently found in math.QA/0005203. The submodule structure is completely determined and irreducible representations can be obtained as the zero-th cohomology group.     

Principal Realization of Twisted Yangian {Y(\mathfrak{g}_{N})}

We give the principal realization of the twisted Yangians of orthogonal and symplectic types. The new bases are interpreted in terms of discrete Fourier transform over the cyclic group ${\mathbb Z_N}$ .     

On the Yangian Y $$(\mathfrak{g}\mathfrak{l}_{m|n} )$$ and its quantum Berezinian

Jonathan Brundan and Alexander Kleshchev recently introduced a new family of presentations for the Yangian Y of the general linear Lie algebra . In this article, we extend some of their ideas to consider the Yangian Y of the Lie superalgebra . In particular, we give a new proof of the result by Nazarov that the quantum Berezinian is central. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Parametrizing Equivalence Classes of Invariant Star Products

Let (M,,) be a symplectic manifold endowed with a symplectic connection . Let Symp(M,) be the group of symplectic transformations of (M,) and Aff(M,) be the group of affine transformations of the affine manifold (M, ). In this Letter, we show that, for any subgroup G of Symp(M,) Aff(M,), the set of G-equivalence classes of G-invariant star products on (M,) is canonically parametrized by the set of sequences of elements belonging to the second de Rham cohomology space of the G-invariant de Rham complex on M.     

Star product on complex sphere $$\mathbb {S}^{2n}$$

We construct a \(U_q\bigl (\mathfrak {s}\mathfrak {o}(2n+1)\bigr )\)-equivariant local star product on the complex sphere \(\mathbb {S}^{2n}\) as a non-Levi conjugacy class \(SO(2n+1)/SO(2n)\).     

Generalizing the $$\mathfrak {bms}_{3}$$ and 2D-conformal algebras by expanding the Virasoro algebra

By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the \(\mathfrak {bms}_{3}\) algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite-dimensional lifts of the so-called \(\mathfrak {B}_{k}\), \(\mathfrak {C}_{k}\) and \(\mathfrak {D}_{k}\) algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Ka?–Moody algebras using modified Sugawara constructions. Applications in the context of three-dimensional gravity are briefly discussed.     

Degeneration of Bethe subalgebras in the Yangian of $$\mathfrak {gl}_n$$

We study degenerations of Bethe subalgebras B(C) in the Yangian \(Y(\mathfrak {gl}_n)\), where C is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parameterizes all possible degenerations, is the Deligne–Mumford moduli space of stable rational curves \(\overline{M_{0,n+2}}\). All subalgebras corresponding to the points of \(\overline{M_{0,n+2}}\) are free and maximal commutative. We describe explicitly the “simplest” degenerations and show that every degeneration is the composition of the simplest ones. The Deligne–Mumford space \(\overline{M_{0,n+2}}\) generalizes to other root systems as some De Concini–Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini–Procesi resolution.     

A Remark on Nonequivalent Star Products via Reduction for CPn

In this Letter, we construct nonequivalent star products on CPⁿ by phase space reduction. It turns out that the nonequivalent star products occur very natural in the context of phase space reduction by deforming the momentum map of the U(1)-action on Cⁿ⁺¹\{0}; into a quantum momentum map and the corresponding momentum value into a quantum momentum value such that the level set, i.e. the constraint surface, of the quantum momentum map coincides with the classical one. All equivalence classes of star products on CPⁿ are obtained by this construction.     

High-speed planar laser-induced fluorescence of the CH radical using the C^{ 2} \varSigma ^{ + } {-}X^{ 2} \varPi \left( {0,0} \right) band

The potential for kHz-rate or high-speed planar laser-induced fluorescence (PLIF) of the Methylidyne (CH) radical using its \(C^{ 2} \varSigma ^{ + } {-}X^{ 2} \varPi\) (v′ = 0, v′′ = 0) band was investigated. We show that due to its strength and the excitation wavelength (~314 nm), which is conveniently generated by a dye laser operating with a red dye, the C–X(0,0) band is a good choice for CH LIF studies wherein suppression of background scattering is not required. While interference from polycyclic aromatic hydrocarbons is small, that caused by hydroxyl (OH) can be significant. Nonetheless, the OH lines can be avoided, and we observe good CH image fidelity. Most importantly, we show that due to the favorable properties of the CH C–X band (i.e., good signal strength and convenient wavelength), it can be used for PLIF at kHz acquisition rates using a continuously pulsing laser system. This is demonstrated in laminar and turbulent CH₄-air flames with a laser system operating at 10 kHz and delivering ~0.2 mJ/pulse at 314 nm.     

Oscillatory Integral Formulae for Left-invariant Star Products on a Class of Lie Groups

We use star representation geometric methods to obtain explicit oscillatory integral formulae for strongly invariant star products on Iwasawa subgroups AN of SU(1,n)     

Tangential Star Products

We etablish a necessary and sufficient condition under which there exists a tangential and well graded star product, differential or not, on the dual of a nilpotent Lie algebra . We also give enlightening examples with explicit computations.