arXiv subjects
Publications and source records attributed to D. Algethami.
We give a construction of electrical quantum group and its superpartner as coideal subalgebras in $U_q(\mathfrak{gl}(N))$ and, respectively, in $U^u_q(\mathfrak{gl}(1|N-1))$, where the latter is an "ungraded" (non-standard) quantum general linear supergroup. We derive them as quantum isotropy subgroups of triangular twisted K-matrices found in one of our recent papers.
We list classical spherical subalgebras in basic matrix Lie superalgebras which are quantizable to coideal subalgebras in the standard quantum supergroups, for any choice of Borel subalgebra. We classify the corresponding Satake-type diagrams and prove that each of them defines a family of proper spherical subalgebras.
We classify constant numerical solutions of the $\Z_2$-graded twisted reflection equation for the fundamental $GL$-type R-matrix and an arbitrary parity sequence, without assuming evenness or invertibility. The classification is based on a four-index locality principle established for this type of reflection equation. We determine the singular and invertible solutions and recover previously conjectured matrices for the minimal symmetric grading. Even invertible solutions comply with the recent classification of graded Satake diagrams. The classical stabilizer of a full-support lower-triangular non-even family is a Lie superalgebra whose even part is an electrical Lie algebra.
We introduce quantum super-spherical pairs as coideal subalgebras in general linear and orthosymplectic quantum supergroups. These subalgebras play a role of isotropy subgroups for matrices solving $\mathbb{Z}_2$-graded reflection equation. They generalize quantum (pseudo)-symmetric pairs of Letzter-Kolb-Regelskis-Vlaar.
We list solutions of the graded reflection equation associated with the fundamental vector representation of the quantum supergroup of GL-type.