2 problems

Let $\mathbb{K}$ be a field and let $A=\bigoplus_{i\geq 0}A_i$ be a standard graded $\mathbb{K}$-algebra. For its $r$th Veronese algebra … denote by $\beta_{i,j}(A^{(r)})$ its grad…

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…