The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $W_0, W_1$ and relations called the $q$-Dolan/Grady relations. Recently Baseilhac Kolb obtained a PBW basis for with elements denoted $\lbrace B_{n \delta+ \alpha_0} \rbrace_{n=0}^\infty, \lbrace \alpha_1} \delta} \rbrace_{n=1}^\infty $. In their recent study of current A_q$, Belliard conjecture that there exist W_{-k}\rbrace_{...

If q is an algebraic Lie algebra and Q is an algebraic group with Lie algebra q, then ind q equals the transcendence degree of the field of Q-invariant rational functions on q. If q is reductive, then q and q are isomorphic as q-modules and hence ind q = rk q. It is an important invariant-theoretic problem to study index and, more generally, the coadjoint representation for non-reductive Lie al...