12 problems

Let ${^Cimath\mathscr{Y}}$ be regarded as a right coideal subalgebra of the Yangian ${\mathscr{Y}}$. For generating series … let $\xi_i(u)$ be the corresponding Yangian Cartan gene…

Irreducibility conjecture. The representation $L(s,s,3/2)\otimes V(1/2)$ is an irreducible representation of $Y^+(3)$ for all $s\in\mathbb{Z}_{+}+1/2$.

Let the twisted affine Yangian be the extension of $TY_\hbar(\mathfrak{so}(2n))$ introduced in the paper, and let the affine Yangian associated with $\widehat{\mathfrak{sl}}(2n)$ b…

Let $TY_\hbar(\widehat{\mathfrak{so}}(2n))$ be the associative algebra generated by … with the defining relations specified in the paper. Let the affine Yangian associated with…

AII shifted twisted Yangian conjecture. A version of Theorems A and B should hold true for $beta=-1$ when $y_n^+(sigma)$ is replaced by the as-yet-undefined shifted twisted Yangian…

A split twisted Yangian is an algebra defined here by Drinfeld-type current generators and relations; the corresponding RTT presentation is the realization of the same algebra by R…

Let $A=(c_{ij})$ be a Cartan matrix of type ADE, and let ${\mathscr{Y}}(A)$ be the algebra defined by the generators and relations obtained from the relations in Theorem 2 for the…

Let $z_1,\ldots,z_k$ be real parameters satisfying … Let $L(z_1,\ldots,z_k)$ and $L'(z_1,\ldots,z_k,\cdot)$ denote the modules used in the paper's restriction and multiplicity-spac…

For $k>0$, let $Y_3^+(k)$ be the $k$-shifted twisted Yangian for $\mathfrak{so}_3(\mathbb C)$, namely the subalgebra of $Y_3^+$ generated by … The preceding corollary gives…

Let $k>0$, let $n$ be a positive integer, and set … Let $e\in\mathfrak g$ be a nilpotent element with Jordan type $(n+2k,n,n)$, let $U(\mathfrak g,e)$ be the associated finite $W$-…

Quantisation conjecture for $\beta_{n,1}$. In the twisted Yangian case for $m=2$, the quantum conjugation relations should extend to all levels, and the element $\beta_{n,1}$ shoul…

General block-size braid-group action conjecture. The braid-group action for general $m\times m$ matrix blocks should have the same form as and, with