16 problems

Let $V_\ell$ denote the $U_q(gl_2(\mathbb{C}))$-module indexed by $\ell\geq 0$, and let $S_\sigma^n V_\ell$ be its $n$th braided symmetric power. Higher braided symmetric powers co…

Let $P_+$ denote the set of dominant weights, let $V_\lambda$ be the corresponding module, and let $S_\sigma(V_\lambda)$ and $\Lambda_\sigma(V_\lambda^*)$ be its braided symmetric…

Let $W$ be a Weyl group, let ${\cal C}$ be its set of reflections, and let $\Lambda_w$ be the left-invariant differential algebra associated with this differential structure. Let…

Let $\Omega_A$ and $\Omega_B$ be the differential algebras defined by the relations in equations (1), (12), (13) and (2), (15), respectively. Transmutation conjecture. The differen…

Let ${\cal N}$ be the algebra equipped with the braided differential operators defined by the matrix $D$, and let $Z({\cal N})$ denote its center. For each positive integer $k$, de…

Generating-relations conjecture. These relations generate $\mathfrak{I}(V)$ as an ideal.

Let $\mathcal{H}$ be the Hilbert space underlying a Wick algebra, let $T$ be a braided operator on $\mathcal{H}^{\mathcal{otimes}2}$, and let $R_m$ denote the corresponding Wick sy…

Let $V$ be an object in $\mathcal{O}_f$, and let $\overline{S_\sigma(V)}$ and $\overline{\Lambda_\sigma(V)}$ denote the classical limits of its braided symmetric and exterior algeb…

Let $V_{2k\omega_1}$ and $V_{2\omega_1}$ denote the indicated simple $sl_n$-modules, and let $S_\sigma(V)$ be the corresponding braided symmetric algebra. Let $V_q(2,k)$ denote the…

Let $A$ be a non-commutative algebra, let $gr(A)$ be the category of finitely graded $A$-modules, and let $qgr(A)$ be its quotient by the finite-dimensional modules; write…

Let $S_q(V)$ denote the braided symmetric algebra of a simple $U_q(sl_2)$-module $V$, and let its point modules be the corresponding non-commutative geometric objects. Zwicknagl's…

Let $G=S_n$ and let $X_n$ be the conjugacy class of 2-cycles, with $A=\mathbb{C}X_n$. In the braided category of $D(G)$-modules, form the braided symmetric algebra $BS(A)$ from the…

Let $B$ be the bosonic algebra, and consider the usual tensor product algebra formed from $p$ copies of $B$ together with a copy of a Clifford-like algebra. Green's conjecture. Gre…

Let $R$ be the root system of a finite Coxeter group, and let ${\cal B}_{S_n}$ denote the Nichols-Woronowicz algebra associated with the symmetric group $S_n$. Let ${\cal E}_n$ be…

Let $\mathfrak{g}$ be a simple Lie algebra, let $V^q$ be a simple $U_q(\mathfrak{g})$-module, and let $V$ be its classical limit. The degree-three braided symmetric and exterior po…

Let $\mathfrak{g}$ be a reductive Lie algebra, let $V^q$ be a finite-dimensional $U_q(\mathfrak{g})$-module, and let $V$ be its classical limit. The reduced symmetric algebra is…