Epimorphism from the Drinfeld Yangian of the queer Lie superalgebra

Let YN(qn)Y_N(q_n) and YD(sqn)Y_D(sq_n) be the Yangians in the Drinfeld and RTT-type presentations, respectively, with generators xi,07(u)x^{7}_{i,0}(u), xi,17(u)x^{7}_{i,1}(u), hi,0(u)h_{i,0}(u), hi,1(u)h_{i,1}(u), and di,j(u)d_{i,j}(u) as in the paper. Let II denote the index set used there. The map is defined by

Φ:YN(qn)YD(sqn).\Phi:Y_N(q_n)\longrightarrow Y_D(sq_n).

The epimorphism conjecture. The map defined by

xi,0+(u)=Φ(xi+1,i(u)),iI,xi,0(u)=Φ(xi,i1+(u)),iI,xi,1+(u)=Φ(xi+1,i(u)),iI,xi,1(u)=Φ(xi,i+1+(u)),iI,hi,0(u)=Φ(di,i+(u)),iI,hi,1(u)=Φ((di,i(u))1)Φ(di+1,i1(u)),iI\begin{aligned} x^{+}_{i,0}(u)&=\Phi(x^-_{i+1,i}(u)), & i&\in I,\\ x^-_{i,0}(u)&=\Phi(x^+_{i,-i-1}(u)), & i&\in I,\\ x^{+}_{i,1}(u)&=\Phi(x^-_{i+1,-i}(u)), & i&\in I,\\ x^{-}_{i,1}(u)&=\Phi(x^+_{i,i+1}(u)), & i&\in I,\\ h_{i,0}(u)&=\Phi(d^+_{i,i}(u)), & i&\in I,\\ h_{i,1}(u)&=\Phi\bigl((d_{i,-i}(u))^{-1}\bigr)\Phi(d_{i+1,-i-1}(u)), & i&\in I \end{aligned}

is an epimorphism. The claim concerns the identification of the Drinfeld presentation with the RTT presentation through the displayed assignment; the source does not establish its status beyond stating the assertion.

Sources & referencesView supporting material

Primary source

Vladimir Stukopin, “Drinfeld Yangian of the queer Lie superalgebra. I”, arXiv:1807.00919 (2018).

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