Narrow Results

Let $H$ be a Hopf algebra. The concept of a symmetric universal ${ℛ}$-matrix of $H$ is introduced (see Definition 2.1). Subsequently, we prove that symmetry phenomenon exists on some well-known quasitriangular Hopf algebras, including some infinite families of small quantum groups. Through an examination of symmetric universal ${ℛ}$-matrices, we present a simple method to determine universal ${ℛ}$-matrices of some Hopf algebras. Furthermore, as part of our applications, we demonstrate that the universal ${ℛ}$-matrices of $K(8n,\sigma ,\tau )$ (refer to Sec. 2 for their definition) are symmetric, where $K(8n,\sigma ,\tau )$ are families of abelian extensions which include the well-known eight-dimensional Kac algebra. Subsequently, we demonstrate how symmetry can be utilized to significantly simplify the determination of universal ${ℛ}$-matrices for $K(8n,\sigma ,\tau )$, ultimately yielding the complete set of universal ${ℛ}$-matrices for this Hopf algebra.

In this paper, we establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra $U_{r,s}({\mathrm{\text{C}}}_n^{(1)})$ and the $R$-matrix realization á la Faddeev, Reshetikhin and Takhtajan.

In this paper a list of R-matrices on a certain coupled Lie algebra is obtained. With one of these R-matrices, we construct infinitely many bi-Hamiltonian structures for each of the two-component BKP and the Toda lattice hierarchies. We also show that, when such two hierarchies are reduced to their subhierarchies, these bi-Hamiltonian structures are reduced correspondingly.

We construct a family of $q$ deformations of $\mathrm{\text{E}}(2)$ group for nonzero complex parameters $\left|q\right|<1$ as locally compact braided quantum groups over the circle group $𝕋$ viewed as a quasitriangular quantum group with respect to the unitary $R$-matrix $\mathrm{\text{R}}(m,n):={(q/\bar{q})}^{mn}$ for all $m,n\in \mathbb{Z}$. For real $0<\left|q\right|<1$, the deformation coincides with Woronowicz’s ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups. As an application, we study the braided analogue of the contraction procedure between ${\mathrm{\text{SU}}}_{\mathrm{\text{q}}}(2)$ and ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü–Wigner group contraction. Consequently, we obtain the bosonization of braided ${\mathrm{\text{E}}}_{\mathrm{\text{q}}}(2)$ groups by contracting ${\mathrm{\text{U}}}_{\mathrm{\text{q}}}(2)$ groups.

The R-matrix method is applied to the Continuum Discretized Coupled Channel (CDCC) approximation. The variational basis is chosen as Lagrange functions, which are shown to be efficient and accurate. We apply the general formalism to the d + ⁵⁸Ni elastic scattering at E_d = 80 MeV. Future developments are briefly discussed.

The Siegert states are approached in the framework of Bloch–Lane–Robson theory of quantum collisions. Both the bound and the quasi-stationary Siegert states are subject of the equation relating the channel R-matrix element to the logarithmic derivative. The Siegert state dependence on the decay channel parameters results in channel renormalization of reduced widths, especially near threshold. The neutron subthreshold and the electron Rydberg states are examples of subthreshold Siegert states. The Siegert approach results in Heisenberg’s S-matrix formula for the bound state and the subthreshold resonance. The Siegert state residue is a spectroscopic asymptotic normalization constant. The decay width of the electron Rydberg channel resonance is, up to a factor, the electron strength function.

The quantum phase space described by Heisenberg algebra possesses undeformed Hopf algebroid structure. The κ-deformed phase space with noncommutative coordinates is realized in terms of undeformed quantum phase space. There are infinitely many such realizations related by similarity transformations. For a given realization, we construct corresponding coproducts of commutative coordinates and momenta (bialgebroid structure). The κ-deformed phase space has twisted Hopf algebroid structure. General method for the construction of twist operator (satisfying cocycle and normalization condition) corresponding to deformed coalgebra structure is presented. Specially, twist for natural realization (classical basis) of κ-Minkowski space–time is presented. The cocycle condition, κ-Poincaré algebra and R-matrix are discussed. Twist operators in arbitrary realizations are constructed from the twist in the given realization using similarity transformations. Some examples are presented. The important physical applications of twists, realizations, R-matrix and Hopf algebroid structure are discussed.

An approach to describing nonlinear Lax type integrable dynamical systems of modern mathematical and theoretical physics, based on the Marsden–Weinstein reduction method on canonically symplectic manifolds with group symmetry, is proposed. Its natural relationship with the well-known Adler–Kostant–Souriau–Berezin–Kirillov method and the associated R-matrix approach is analyzed.

A new generalized exactly solvable spatially one-dimensional quantum superradiance model, describing a charged fermionic medium interacting with external electromagnetic field, is suggested. The Lax type operator spectral problem is presented, the related R-structure is calculated. The Hamilton operator renormalization procedure subject to a physically stable vacuum is described, the quantum excitations and quantum solitons, related with the thermodynamical equilibrity of the model, are discussed.

K-shell photoionization (PI) of Li, Be${}^{+}$ and B${}^{2+}$ from ground state $1s^{2}2s^2{\mathrm{\text{S}}}^e$ have been studied by using the $R$-matrix method with pseudostates. The K-shell PI process is featured with the contributions from the core-excited metastable states or dominated by the Auger states ²P^o. The resonant parameters of the Auger states ²P^o and the PI cross-sections have been calculated and compared with the available experimental and theoretical works. Our results agree very well with that of the published works. It is worth noting that compared with previous theoretical calculations, our results of B${}^{2+}$ show better agreements with the latest high-resolution advanced light source measurements [A. Müller et al., J. Phys. B43 (2010) 135602].

This paper gives a connection between well-chosen reductions of the Links–Gould invariants of oriented links and powers of the Alexander–Conway polynomial. This connection is obtained by showing the representations of the braid groups we derive the specialized Links–Gould polynomials from can be seen as exterior powers of a direct sum of Burau representations.

The Gukov–Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of “strange identities” for positive braid knots.

We prove that the construction of Vassiliev invariants by expanding the link polynomials P_g,V(q, q⁻¹) at the point q=1 is equivalent to the construction of Vassiliev invariants from Chern-Simons perturbation theory. In both constructions a simple Lie algebra g and an irreducible representation V of g should be specified.

We give an example of a Vassiliev invariant of order six which cannot be obtained by these constructions if we restrict ourselves to simple Lie algebras and do not allow semisimple ones.

The explicit description of primitive elements in the Kontsevich Hopf algebra is given.

The reaction ${}^{13}$C${(\alpha ,n)}^{16}$O is an important astrophysical process producing neutrons for $s$-process nucleosynthesis of nuclei heavier than iron in low-mass AGB stars. The reaction rate of ${}^{13}$C${(\alpha ,n)}^{16}$O at relevant energies is dominated by a ${1/2}^{+}$ state in ${}^{17}$O near the $\alpha$-threshold. The adopted value for the excitation energy of the state is −3keV below the threshold at 6359keV. However, recent investigations have indicated that the state is, instead of a sub-threshold state, an above threshold resonance state. The observation is also corroborated by neutron scattering studies from ${}^{16}$O. The location of the state and its implication on the low energy behavior of the astrophysical $S$-factor are of definite interest. The aim of this work is to ascertain the energy location of ${1/2}^{+}$ state in the excitation spectrum of ${}^{17}$O and to estimate its neutron and $\alpha$-partial widths. Subsequently, we look into the effect of the resultant set of parameters of the state on the $S$-factor and reaction rate at important astrophysical energies. A multilevel, multichannel $R$-matrix analysis has been performed for the existing ${}^{16}$O$(n,n)$, ${}^{13}$C$(\alpha ,n)$, ${}^{13}$C$(\alpha ,\alpha )$ and ${}^{16}$O$(n,\alpha )$ data. The excitation energy of the $1/2+$ state in ${}^{17}$O is found to be at 6371.9keV, about 12.9keV above the threshold. The resulting $S$-factor around the Gamow energy of 190keV $(T_9\sim 0.1)$ is $(1.91\pm 0.27)\times 10^6$$\mathrm{\text{MeV}}\cdot \mathrm{\text{b}}$. Relatively, higher $S$-factor value yields a larger reaction rate for ${}^{13}$C($\alpha ,n$)${}^{16}$O at the required temperature window. Consequently, larger number of neutrons relevant to $S$-process nucleosynthesis will be produced from ${}^{13}$C${(\alpha ,n)}^{16}$O reaction.

The ${}^{12}$C(${}^{20}$Ne, ${}^{16}$O)${}^{16}$O reaction has been used for the first time to determine ANC of ${}^{16}$O levels. The ${}^{12}$C(${}^{20}$Ne, ${}^{16}$O)${}^{16}$O $\alpha$-transfer angular distributions are measured at $E_{\mathrm{\text{lab}}}=150$MeV and used to determine the ANC of the ground, 6.05MeV ($0^{+}$), 6.13MeV ($3^{-}$), 6.92MeV ($2^{+}$) and 7.12MeV ($1^{-}$) states of ${}^{16}$O. The ANC values are $C^2$$(0^{+})=(22.2\pm 7.1)\times 10^4$${\mathrm{\text{fm}}}^{-1}$, $C^2$$(0^{+})=(1.40\pm 0.54)\times 10^6{\mathrm{\text{fm}}}^{-1}$, $C^2$$(3^{-})=(2.30\pm 0.92)\times 10^4{\mathrm{\text{fm}}}^{-1}$, $C^2$$(2^{+})=(1.07\pm 0.32)\times 10^{10}{\mathrm{\text{fm}}}^{-1}$ and $C^2$$(1^{-})=(32.29\pm 10.14)\times 10^{28}{\mathrm{\text{fm}}}^{-1}$. Astrophysical S-factors at 300keV are extracted $\{S_{E1}^{300}=(79.18\pm 26.60$) keV b and $S_{E2}^{300}=(52.51\pm 24.3)$ keV b$\}$ using these ANC values by R-matrix theory.

The isotope ${}^{16}\mathrm{\text{O}}$ is one of the most abundant light isotopes, and therefore the process ${}^{16}\mathrm{\text{O}}{(n,\gamma )}^{17}\mathrm{\text{O}}$ reaction plays a critical role in stellar nucleosynthesis. Reaction cross-sections and rates for neutron capture by ${}^{16}\mathrm{\text{O}}$ at astrophysical energies have been calculated with special attention to low-lying, narrow resonances while implementing $R$-matrix formalism during integration.

We prove that the singularities of the $R$-matrix $R(k)$ of the minimal quantization of the adjoint representation of the Yangian $Y(𝔤)$ of a finite dimensional simple Lie algebra $𝔤$ are the opposite of the roots of the monic polynomial $p(k)$ entering in the OPE expansions of quantum fields of conformal weight $3/2$ of the universal minimal affine $W$-algebra at level $k$ attached to $𝔤$.

Using the skew-Hopf pairing, we obtain ${ℛ}$-matrix for the two-parameter quantum algebra $U_{v,t}$. We further construct a strict monoidal functor ${𝒯}$ from the tangle category $(\mathrm{\text{OTa}},\otimes ,\varnothing )$ to the category $(\mathrm{\text{Mod}},\otimes ,\mathbb{Q}(v,t))$ of $U_{v,t}$-modules. As a consequence, the quantum knot invariant of the tangle $L$ of type $(n,n)$ is obtained by the action of ${𝒯}$ on the closure $\tilde{L}$ of $L$.

In this paper we continue our studies of Hitchin systems on singular curves (started in [1]). We consider a rather general class of curves which can be obtained from the projective line by gluing two subschemes together (i.e. their affine part is: Spec {f ∈ ℂ[z] : f(A(∊)) = f(B(∊)); ∊^N = 0}, where A(∊), B(∊) are arbitrary polynomials). The most simple examples are the generalized cusp curves which are projectivizations of Spec {f ∈ ℂ[z] : f′(0) = f″(0) = ⋯ f^N-1(0) = 0}. We describe the geometry of such curves; in particular we calculate their genus (for some curves the calculation appears to be related with the iteration of polynomials A(∊), B(∊) defining the subschemes). We obtain the explicit description of moduli space of vector bundles, the dualizing sheaf, Higgs field and other ingredients of the Hitchin integrable systems; these results may deserve the independent interest. We prove the integrability of Hitchin systems on such curves. To do this we develop r-matrix formalism for the functions on the truncated loop group GL_n(ℂ[z]), z^N = 0. We also show how to obtain the Hitchin integrable systems on such curves as hamiltonian reduction from the more simple system on some finite-dimensional space.