The quantum toroidal superalgebra isomorphism for gl_{n|m}

From papers

Let V=Cn+mV=\mathbb{C}^{n+m} with the gradation defined by ϵi=(1)δi>n\epsilon_i=(-1)^{\delta_{i>n}}. The super RR-matrix determines shuffle algebras A+\mathcal{A}^+ and A\mathcal{A}^-, which combine with Ut(gl˙nm)U_t(\dot{\mathfrak{gl}}_{n|m}) into the Drinfeld double

A=A+Ut(gl˙nm)A.\mathcal{A}=\mathcal{A}^+\otimes U_t(\dot{\mathfrak{gl}}_{n|m})\otimes\mathcal{A}^-.

Quantum toroidal superalgebra isomorphism. There is an isomorphism

AUq,t(gl¨nm),\mathcal{A}\cong U_{q,t}(\ddot{\mathfrak{gl}}_{n|m}),

where the quantum toroidal superalgebra on the right-hand side is defined as in the cited reference. The super-case construction is presented as conjectural and the paper stops short of implementing the missing parts, so the claimed isomorphism remains open.

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Sources & referencesView supporting material

Primary source

Alexandr Garbali and Andrei Neguţ, “Shuffle algebras, lattice paths and quantum toroidal gl_n|m”, arXiv:2507.00538 (2026).

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