Quantized coordinate rings of the unipotent radicals of the standard Borel subgroups in \(\mathrm{SL}_{n+1}\). (Q403804)

The author finds noncommutative analogues of the coordinate rings of the unipotent radicals of the standard Borel subgroups in \(\mathrm{SL}_{n+1}\). Two subalgebras of the quantized coordinate ring of the standard Borel in \(\mathrm{SL}_{n+1}\) are defined, both of which can be considered quantizations of the unipotent radical. Presentations are given for these algebras and they are proven to be isomorphic. It is then shown that these algebras also arise as coinvariants of a natural comodule algebra action using the Hopf algebra structure of \(O_q(\mathrm{SL}_{n+1})\). Finally, using a dual pairing, it is shown that these algebras are isomorphic to \(U_q^\pm(\mathfrak{sl}_{n+1})\).