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Mathematics, Representation Theory, math.RT
Abstract:
We construct a family of PBWD (Poincar\'e-Birkhoff-Witt-Drinfeld) bases for
the quantum loop algebras $U_v(L\mathfrak{sl}_n),
U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ in the new Drinfeld
realizations. In the 2-parameter case, this proves Theorem 3.11 of
[Hu-Rosso-Zhang] (stated in loc. cit. without a proof), while in the super case
it proves a conjecture of [Zhang]. The main ingredient in our proofs is the
interplay between those quantum loop algebras and the corresponding shuffle
algebras, which are trigonometric counterparts of the elliptic shuffle algebras
of Feigin-Odesskii. Our approach is similar to that of [Enriquez] in the formal
setting, but the key novelty is an explicit shuffle algebra realization of the
corresponding algebras, which is of independent interest. This also allows us
to strengthen the above results by constructing a family of PBWD bases for the
RTT forms of those quantum loop algebras as well as for the Lusztig form of
$U_v(L\mathfrak{sl}_n)$. The rational counterparts provide shuffle algebra
realizations of the type $A$ (super) Yangians and their Drinfeld-Gavarini dual
subalgebras.
