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The alternating PBW basis for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
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abstract
The positive part $U^+_q$ of $U_q(\widehat{\mathfrak{sl}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. We introduce a PBW basis for $U^+_q$, said to be alternating. Each element of this PBW basis commutes with exactly one of $A$, $B$, $qAB-q^{-1}BA$. This gives three types of PBW basis elements; the elements of each type mutually commute. We interpret the alternating PBW basis in terms of a $q$-shuffle algebra associated with affine $\mathfrak{sl}_2$. We show how the alternating PBW basis is related to the PBW basis for $U^+_q$ found by Damiani in 1993.
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Freidel-Maillet type equations on fused K-matrices over the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$
Fused K-matrices with explicit Catalan-word entries satisfy Freidel-Maillet type equations for all dimensions 2j+1.
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