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.     

Gauss Decomposition of the Yangian $$Y({\mathfrak{gl}}_{m|n})$$

We describe a Gauss decomposition for the Yangian of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y(A(m − 1, n − 1)) (for m ≠ n) defined and studied in papers by Stukopin, and suggests natural definitions for the Yangians and Y(A(n, n)). We also show that the coefficients of the quantum Berezinian generate the centre of the Yangian . This was conjectured by Nazarov in 1991.     

Parabolic Presentations of the Super Yangian Y(\mathfrakglM|N){Y(\mathfrak{gl}_{M|N})}

Associated to a composition of M and a composition of N, a new presentation of the super Yangian of the general linear Lie superalgebra Y(\mathfrakgl_M|N){Y(\mathfrak{gl}_{M|N})} is obtained.     

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.     

Ferromagnetic Ordering of Energy Levels for {U_q(\mathfrak{sl}_2)} Symmetric Spin Chains

We consider the class of quantum spin chains with arbitrary ${U_{q}(\mathfrak{sl}_{2})}$ -invariant nearest-neighbor interactions, sometimes called SU _q (2) for the quantum deformation of SU(2), for q >?0. We derive sufficient conditions for the Hamiltonian to satisfy the property we call ferromagnetic ordering of energy levels. This is the property that the ground state energy restricted to a fixed total spin subspace is a decreasing function of the total spin. Using the Perron?CFrobenius theorem, we show sufficient conditions are positivity of all interactions in the dual canonical basis of Lusztig. We characterize the cone of positive interactions, showing that it is a simplicial cone consisting of all non-positive linear combinations of ??cascade operators,?? a special new basis of ${U_{q}(\mathfrak{sl}_2)}$ intertwiners we define.     

Spinor Representations of $${U}_q (\hat {\mathfrak{o}}(N))$$ )

This Letter concerns an extension of the quantum spinor construction of . We define quantum affine Clifford algebras based on the tensor category and the solutions of q-KZ equations, and construct quantum spinor representations of .     

Universal R-matrix of the Super Yangian Double DY (gl(1|1))

Based on Drinfeld realization of super Yangian double DY (gl(1|1)), its pairing relations and universal R-matrix are given. By taking evaluation representation of universal R-matrix, another realization L^±(u) of DY (gl(1|1)) is obtained. These two realizations of DY (gl(1|1)) are related by the supersymmetric extension of Ding-fienkel map.     

Super Yangian Y(osp(1|2)) and the Universal R-matrix of Its Quantum Double

We present the Drinfel'd realisation of the super Yangian Y(osp(1|2)), including the explicit expression for the coproduct. We show in particular that it is necessary to introduce supplementary Serre relations. The construction of its quantum double is carried out. This allows us to give the universal R-matrix of DY(osp(1|2)).Member of Institut Universitaire de France     

NNLO corrections to \bar{B}\to X_{u}\ell \bar{\nu}_{\ell} and the determination of |V ub |

We study the impact of next-to-next-to-leading order (NNLO) QCD corrections on partial decay rates in $\bar{B}\to X_{u}\ell \bar{\nu}_{\ell}$ decays, at leading-order in the 1/m _b expansion for shape-function kinematics. These corrections are implemented within a modified form of the BLNP framework, which allows for arbitrary variations of the jet scale μ _i ～1.5 GeV. Our analysis includes a detailed comparison between resummed and fixed-order perturbation theory, and between the complete NNLO results and those obtained in the large-β ₀ approximation. For the default choice μ _i =1.5 GeV used in current extractions of |V _ub | within the BLNP framework, the NNLO corrections induce significant downward shifts in the central values of partial decay rates with cuts on the hadronic variable P ₊, the hadronic invariant mass, and the lepton energy. At the same time, perturbative uncertainties are reduced, especially those at the jet scale, which are the dominant ones at next-to-leading order (NLO). For higher values of μ _i and in fixed-order perturbation theory, the shifts between NLO and NNLO are more moderate. We combine our new results with known power-suppressed terms in order to illustrate the implications of our analysis on the determination of |V _ub | from inclusive decays.     

Free Boson Representation of DYħ(\widehat{sl}(M+1|N+1)) at Level One

We construct a realization of the central extension of super-Yangian double DYh(sl(M 1|N 1)) at level one in terms of free boson fields with a continuous parameter.     

Eleven-dimensional gauge theory for the M-algebra as an Abelian semigroup expansion of \mathfrak{osp} ( 32 | 1 )

A new Lagrangian realizing the symmetry of the M-algebra in eleven-dimensional space-time is presented. By means of the novel technique of Abelian semigroup expansion, a link between the M-algebra and the orthosymplectic algebra is established, and an M-algebra-invariant symmetric tensor of rank six is computed. This symmetric invariant tensor is a key ingredient in the construction of the new Lagrangian. The gauge-invariant Lagrangian is displayed in an explicitly Lorentz-invariant way by means of a subspace separation method based on the extended Cartan homotopy formula.     

Generalization of the $$\mathcal{U} $$ q (gl(N)) Algebra and Staggered Models

We develop a technique for the construction of integrable models with a ₂ grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang–Baxter equations are written down and their solution for the gl(N) case is found. We analyze in details the N = 2 case and find the corresponding quantum group behind this solution. It can be regarded as the quantum group , with a matrix deformation parameter q such that (q )² = q ². The symmetry behind these models can also be interpreted as the tensor product of the (–1)-Weyl algebra by an extension of _q (gl(N)) with a Cartan generator related to deformation parameter –1.     

Analysis of the Possible $D\bar{D}_{s0}^*(2317)$ and $D^*\bar{D}_{s1}^*(2460)$ Molecules with QCD Sum Rules *

In this article, we assume that there exist the pseudoscalar $D\bar{D}_{s0}^*(2317)$ and $D^*\bar{D}_{s1}^*(2460)$ molecular states $Z_{1,2}$ and construct the color singlet-singlet molecule-type interpolating currents to study their masses with the QCD sum rules. In calculations, we consider the contributions of the vacuum condensates up to dimension-10 and use the formula $\mu=\sqrt{M_{X/Y/Z}^{2}-(2{\mathbb{M}}_{c})^{2}}$ to determine the energy scales of the QCD spectral densities. The numerical results, $M_{Z_1}=4.61_{-0.08}^{+0.11}\,\text{GeV}$ and $M_{Z_2}=4.60_{-0.06}^{+0.07}\,\text{GeV}$, which lie above the $D\bar{D}_{s0}^*(2317)$ and $D^*\bar{D}_{s1}^*(2460)$ thresholds respectively, indicate that the $D\bar{D}_{s0}^*(2317)$ and $D^*\bar{D}_{s1}^*(2460)$ are difficult to form bound state molecular states, the $Z_{1,2}$ are probably resonance states.     

q-Difference Realization of Uq(sl(M|N)) and Its Application to Free Boson Realization of Uq $$(\widehat{{\text{sl}}}(2|1))$$

We present a q-difference realization of the quantum superalgebra U_q(sl(M|N)), which includes Grassmann even and odd coordinates and their derivatives. Based on this result, we obtain a free boson realization of the quantum affine superalgebra U_q of an arbitrary level k.