Yangian of the Strange Lie Superalgebra of Qn−1\boldsymbol{Q_{n-1}} Type, Drinfel'd Approach

The Yangian of the strange Lie superalgebras in Drinfel'd realization is defined. The current system generators and defining relations are described.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Solitary and Periodic Wave Solutions for Several Short Wave Model Equations

We study the periodic and solitary wave solutions to several short wave model equations arising from a so-called $\beta$-family equation for $\beta=1,2,4$. These are integrable cases which possess Lax pair and multi-soliton solutions. By phase plane analysis, either the loop or cuspon type solutions are predicted. Then, by introducing a hodograph, or reciprocal, transformation, a coupled system is derived for each $\beta$. Applying a travelling wave setting, we are able to find the periodic solutions exactly expressed in terms of Jacobi Elliptic functions. In the limiting cases of modulus k=1, they all converge to the known solitary waves

Mickelsson algebras and inverse Shapovalov form

Let $\mathcal{A}$ be an associative algebra containing the classical or quantum universal enveloping algebra $U$ of a semi-simple complex Lie algebra. Let $\mathcal{J}\subset \mathcal{A}$ designate the left ideal generated by positive root vectors in $U$. We construct the reduction algebra of the pair $(\mathcal{A},\mathcal{J})$ via the inverse Shapovalov form of $U$.Comment: 22 page

Mickelsson algebras via Hasse diagrams

Let $\mathcal{A}$ be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra $\mathfrak{g}$. We present a construction of the Mickelsson algebra $Z(\mathcal{A},\mathfrak{g})$ relative to the left ideal in $\mathcal{A}$ generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum $\mathfrak{g}$-modules. We give an explicit expression for a PBW basis in $Z(\mathcal{A},\mathfrak{g})$ in the case when $\mathcal{A}=U(\mathfrak{a})$ of a metric Lie algebra $\mathfrak{a}\supset \mathfrak{g}$. For $\mathcal{A}=U_q(\mathfrak{a})$ and $\mathfrak{g}$ the commutant of a Levi subalgebra in $\mathfrak{a}$, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to $\mathbb{C}[[\hbar]]$.Comment: 19 pages, no figure

Yangian of the Strange Lie Superalgebra of Qn₋₁ Type, Drinfel'd Approach

The Yangian of the strange Lie superalgebras in Drinfel'd realization is defined. The current system generators and defining relations are described

Solutions to graded reflection equation of GL-type

We list solutions of the graded reflection equation associated with the fundamental vector representation of the quantum supergroup of GL-type.Comment: arXiv admin note: text overlap with arXiv:math/020429