Shuffle algebra realizations of type A super Yangians and quantum affine superalgebras for all Cartan data

Abstract

We introduce super Yangians of gl(V ), sl(V ) (in the new Drinfeld realization) associated to all Dynkin diagrams of gl(V ). We show that all of them are isomorphic to the super Yangians introduced by M. Nazarov in [Na], by identifying them with the corresponding RTT super Yangians. However, their "positive halves" are not pairwise isomorphic, and we obtain the shuffle algebra realizations for all of those in spirit of [T]. We adapt the latter to the trigonometric setup by obtaining the shuffle algebra realizations of the "positive halves" of type A quantum loop superalgebras associated to arbitrary Dynkin diagrams.

A Drinfeld-type presentation of the orthosymplectic Yangians

“…N |2m ) are implied by the Serre relations in the Lie superalgebra osp N |2m via the embedding (2.10), which are particular cases of (1.7) withr 1 = • • • = r k = s = 0.It was already shown in[30, Remark 2.61] how the super Serre relations in Theorem 6.1 follow from relations (1.8) with the use of the polynomials κ i r . The same argument applies to prove that all relations of the form (1.8) are implied by their particular case with r = s = 0 which holds in osp N |2m .We thus have an epimorphism from the algebra Y(osp N |2m ) defined in the Main Theorem to the Yangian Y(osp N |2m ), which takes the generators κ i r and ξ ± i r of Y(osp N |2m ) to the elements of Y(osp N |2m ) denoted by the same symbols.…”

“…Therefore, some sets of relations between the Gaussian generators follow from [13, Thm. 3] (via the change of parity); see also [30]. Furthermore, for the values of the indices i m + 1 the relations are implied by the Drinfeld presentation of X(o N ) given in [18] 1 .…”

“…The Gauss decomposition of the corresponding generator matrix T (u) was used in [13] to get Drinfeld-type presentations of Y(gl n|m ) and Y(sl n|m ). By changing the parity assumptions of [13] to their opposites (see also [30]), we find that the coefficients of the series h i (u) with i = 1, . .…”

“…Presentations of the Yangian for sl n|m analogous to [7] and [24] were given by Stukopin [29], while the construction of [4] was extended to the Yangian Y(gl n|m ) by Gow [13], where a Drinfeld-type presentation was devised together with an isomorphism with the R-matrix presentation. More general parabolic presentations of the Yangian Y(gl n|m ), corresponding to arbitrary Borel subalgebras in gl n|m were given by Peng [28]; see also Tsymbaliuk [30].…”

Yangian of the General Linear Lie Superalgebra

“…Still let us mention the work [3] which contains a direct proof of the centrality of the elements of Y(gl M |N ) from our second family. Let us also mention the work [9] which provides a generalization of Y(gl M |N ) to arbitrary parity sequences.…”