12 problems

Let ${\mathcal M}$ be the space of quasi-triangular solutions to the classical Yang–Baxter equation, and let $r_{CG}(i,n)$ be the Cremmer–Gervais $r$-matrix indexed by positive cop…

Let $\mathfrak{sp}_{2n}$ be the symplectic Lie algebra, and consider a parabolic subalgebra of type (b) with block parameters $d=(d_1,\ldots,d_r,d_{r+1},d_r,\ldots,d_1)$. Let…

For every $n\geqslant3$, let ${\mathcal O}(\boldsymbol{\lambda})\subset{\mathfrak{sl}}_n$ be the unique nilpotent orbit with … where ${\mathfrak b}$ is a Borel subalgebra and…

Let ${\mathfrak g}$ be a simple Lie algebra. A Lie subalgebra is Frobenius when its index is zero; let ${\EuScript M}_{\sf Fr}$ denote the corresponding maximal dimension invariant…

Let $\mathfrak g$ be a reductive Lie algebra, let $\mathfrak p$ be a parabolic subalgebra of $\mathfrak g$, and let $\mathfrak p^{\sf nil}$ denote its nilpotent radical. Write…

Let ${\mathfrak g}$ be a simple Lie algebra. Let $\mathfrak{Ab}({\mathfrak g})$ be the finite set of abelian ideals and $\mathfrak{Par}({\mathfrak g})$ the finite set of standard p…

Let ${\mathfrak g}$ be a simple Lie algebra, let $\mathfrak{Ab}({\mathfrak g})$ be the set of its abelian ideals, and let $\mathfrak{Par}({\mathfrak g})$ be the set of its standard…

Let $\mathfrak{g}$ be a complex simple Lie algebra, let $\mathfrak{p}$ be a parabolic subalgebra of cominuscule type, and let $\mathfrak{l}$ be its Levi factor. Let $w_0$ be the lo…

For a composition $\mathbf{a}=(a_1,\dots,a_r)\in\mathcal{C}_n$, let $p(\mathbf{a})=r$ be its number of parts, and let $F_{n,p}$ denote the number of Frobenius standard parabolic su…

Let $\mathfrak p$ be a parabolic subalgebra with nilradical $\mathfrak m$, and let $N$ be the corresponding unipotent subgroup. Let $S$ be the distinguished-root set, let $\Phi$ be…

Orbit-dimension conjecture. The maximal dimension of an $N$-orbit in $\mathfrak m$ is equal to

Invariant-field conjecture. The field of invariants $\mathcal F^N$ is the field of rational functions of the polynomials $M_\xi$, $\xi\in S$, and $L_q$, $q\in Q$.