Drinfeld Presentation Research Articles

A Drinfeld type presentation of affine ıquantum groups I: Split ADE type

We establish a Drinfeld type new presentation for the ıquantum groups arising from quantum symmetric pairs of split affine ADE type, which includes the q-Onsager algebra as the rank 1 case. This presentation takes a form which can be viewed as a deformation of half an affine quantum group in the Drinfeld presentation.

Representations of Quantum Affine Algebras in their R-Matrix Realization

We use the isomorphisms between the R-matrix and Drinfeld presentations of the quantum affine algebras in types B, C and D produced in our previous work to describe finite-dimensional irreducible representations in the R-matrix realization.We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the R-matrix and Drinfeld presentations of the Yangians.

Isomorphism between the R-Matrix and Drinfeld Presentations of Quantum Affine Algebra: Types B and D

Following the approach of Ding and Frenkel [Comm. Math. Phys. 156 (1993), 277-300] for type $A$, we showed in our previous work [J. Math. Phys. 61 (2020), 031701, 41 pages] that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the quantum affine algebra yields the Drinfeld generators in all classical types. Complete details for type $C$ were given therein, while the present paper deals with types $B$ and $D$. The arguments for all classical types are quite similar so we mostly concentrate on necessary additional details specific to the underlying orthogonal Lie algebras.

Yangian doubles of classical types and their vertex representations

The Yangian double DYℏ(gN) is introduced for the classical types of gN=o2n+1, sp2n, and o2n. Via the Gauss decomposition of the generator matrix, the Yangian double is given the Drinfeld presentation. In addition, bosonization of level 1 realizations for the Yangian double DYℏ(gN) of non-simply laced types are explicitly constructed.

Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: Type C

An explicit isomorphism between the R-matrix and Drinfeld presentations of the quantum affine algebra in type A was given by Ding and Frenkel [Commun. Math. Phys. 156, 277–300 (1993)]. We show that this result can be extended to types B, C, and D and give a detailed construction for type C in this paper. In all classical types, the Gauss decomposition of the generator matrix in the R-matrix presentation yields the Drinfeld generators. To prove that the resulting map is an isomorphism, we follow the work of Frenkel and Mukhin [Sel. Math. 8, 537–635 (2002)] in type A and employ the universal R-matrix to construct the inverse map. A key role in our construction is played by a homomorphism theorem, which relates the quantum affine algebra of rank n − 1 in the R-matrix presentation with a subalgebra of the corresponding algebra of rank n of the same type.

Isomorphism Between the R-Matrix and Drinfeld Presentations of Yangian in Types B, C and D

It is well-known that the Gauss decomposition of the generator matrix in the R-matrix presentation of the Yangian in type A yields generators of its Drinfeld presentation. Defining relations between these generators are known in an explicit form, thus providing an isomorphism between the presentations. It has been an open problem since the pioneering work of Drinfeld to extend this result to the remaining types. We give a solution for the classical types B, C and D by constructing an explicit isomorphism between the R-matrix and Drinfeld presentations of the Yangian. It is based on an embedding theorem which allows us to consider the Yangian of rank $${n-1}$$ as a subalgebra of the Yangian of rank n of the same type.

Quantum Q systems: from cluster algebras to quantum current algebras

In this paper, we recall our renormalized quantum Q-system associated with representations of the Lie algebra $A_r$, and show that it can be viewed as a quotient of the quantum current algebra $U_q({\mathfrak n}[u,u^{-1}])\subset U_q(\widehat{\mathfrak sl}_2)$ in the Drinfeld presentation. Moreover, we find the interpretation of the conserved quantities in terms of Cartan currents at level 0, and the rest of the current algebra, in a non-standard polarization in terms of generators in the quantum cluster algebra.

Kirillov-Reshetikhin Conjecture: The General Case

We prove the Kirillov–Reshetikhin (KR) conjecture in the general case: for all twisted quantum affine algebras, we prove that the characters of KR modules solve the twisted Q-system [20] and we get explicit formulas for the character of their tensor products (the untwisted case was treated in [16, 33, 34]). The proof provides several new developments for the representation theory of twisted quantum affine algebras, in particular on the twisted Frenkel–Reshetikhin q-characters that we define (expected in [11, 12]) and on the parameterization of simple finite dimensional representations without Drinfeld presentation. We also prove the twisted T-system [30]. As an application, we get a proof of explicit (q)-characters formulas conjectured in various papers. We prove an isomorphism of Grothendieck rings between a twisted quantum affine algebra and the corresponding simply-laced quantum affine algebra.

The Kronecker quiver and bases of quantum affine [formula omitted

We compare various bases of the quantum group U ( s l ^ 2 ) in the context of the Kronecker quiver, and relate them to the Drinfeld presentation.

Parabolic Presentations of the Yangian

We derive some new presentations for the Yangian associated to the Lie algebra gl_n(C) that are adapted to parabolic subalgebras. At one extreme, the presentation is just the usual RTT presentation, whilst at the other extreme it is a variation on Drinfeld's presentation. All these presentations play an important role in our subsequent article "Shifted Yangians and finite W-algebras".