[Paper Review] Cohomological Hall algebras and affine quantum groups

Yaping Yang, Gufang Zhao|arXiv (Cornell University)|Apr 7, 2016

Algebraic structures and combinatorial models7 references3 citations

TL;DR

This paper constructs a comultiplication on the preprojective cohomological Hall algebra (CoHA) associated to a quiver $Q$ and a formal group $ \mathbb{G}$, turning it into a bialgebra, and defines its Drinfeld double, which realizes a quantum affine algebra quantizing $U(\mathfrak{g}_Q[[u]])$. When $\mathbb{G}$ is the additive group, the Drinfeld double is isomorphic to the Drinfeld-Jimbo Yangian $Y_\hbar(\mathfrak{g}_Q)$, providing a new algebraic realization of affine quantum groups via CoHA and shuffle algebras.

ABSTRACT

We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in an earlier work for any quiver $Q$ and any one-parameter formal group $\mathbb{G}$. In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra $\mathfrak{g}_Q$ associated to $Q$, whose quantization comes from the formal group $\mathbb{G}$. We prove, when the group $\mathbb{G}$ is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian.

Motivation & Objective

• To construct a comultiplication on the preprojective cohomological Hall algebra (CoHA) for any quiver $Q$ and formal group $\mathbb{G}$, making it a bialgebra.
• To define the Drinfeld double of this bialgebra, which quantizes the universal enveloping algebra $U(\mathfrak{g}_Q[[u]])$ of the symmetric Kac-Moody algebra associated to $Q$.
• To establish a uniform algebraic construction of both known and new affine-type quantum groups via the CoHA framework.
• To show that when $\mathbb{G}$ is the additive group, the Drinfeld double of the CoHA is isomorphic to the Drinfeld-Jimbo Yangian $Y_\hbar(\mathfrak{g}_Q)$.

Proposed method

• The construction uses the spherical extension $\mathcal{P}^{\mathfrak{s},\mathfrak{e}}(Q,A)$ of the CoHA, which is a shuffle algebra with two equivariant parameters $t_1, t_2$.
• A comultiplication $\Delta: \underline{\mathcal{P}}^{\mathfrak{s},\mathfrak{e}} \to \underline{\mathcal{P}}^{\mathfrak{s},\mathfrak{e}} \widehat{\otimes} \underline{\mathcal{P}}^{\mathfrak{s},\mathfrak{e}}$ is defined via a residue pairing on an ad\'ele version of the algebra.
• The Drinfeld double is constructed using a non-degenerate bialgebra pairing, with multiplication and comultiplication defined by the same formulas as in the original bialgebra.
• The construction relies on the shuffle algebra structure and the geometry of the moduli of representations of the preprojective algebra.
• The reduced Drinfeld double $\overline{D}(\underline{\mathcal{P}}^{\mathfrak{s},\mathfrak{e}})$ is defined by quotienting out torsion and imposing a symmetry condition $H_k^+(u) = H_k^-(-u)$.
• The isomorphism with the Yangian is established by mapping generators $x_k^+(u), x_k^-(u), \xi_k(u)$ to $E_k(u), F_k(u), H_k(u)$ in the double, and verifying the Yangian relations via residue calculations.

Experimental results

Research questions

• RQ1 Can a comultiplication be defined on the preprojective cohomological Hall algebra for any quiver $Q$ and formal group $\mathbb{G}$, making it a bialgebra?
• RQ2 Does the Drinfeld double of this bialgebra realize a quantization of $U(\mathfrak{g}_Q[[u]])$ for the symmetric Kac-Moody algebra $\mathfrak{g}_Q$?
• RQ3 Is the Drinfeld double isomorphic to the Drinfeld-Jimbo Yangian when $\mathbb{G}$ is the additive group?
• RQ4 Can this construction recover known quantum groups such as the elliptic quantum group of [Fed94] when $\mathbb{G}$ is an elliptic curve?
• RQ5 Is the map from the Yangian to the reduced Drinfeld double an isomorphism for quivers of finite and affine Dynkin type?

Key findings

• The preprojective cohomological Hall algebra $\mathcal{P}(Q,A)$ admits a comultiplication $\Delta$ making it a bialgebra, defined via a residue pairing on an ad\'ele version of the algebra.
• The Drinfeld double $D(\underline{\mathcal{P}}^{\mathfrak{s},\mathfrak{e}})$ is a quantization of $U(\mathfrak{g}_Q[[u]])$, with the pairing defined using residues on a completed tensor product.
• When $\mathbb{G}$ is the additive group, the reduced Drinfeld double $\overline{D}(\underline{\mathcal{P}}^{\mathfrak{s},\mathfrak{e}})$ at $t_1 = t_2 = \hbar/2$ is isomorphic to the Drinfeld-Jimbo Yangian $Y_\hbar(\mathfrak{g}_Q)$ for quivers without edge-loops.
• For quivers of finite type, the map from $Y_\hbar(\mathfrak{g}_Q)$ to $\overline{D}(\underline{\mathcal{P}}^{\mathfrak{s},\mathfrak{e}})$ is an isomorphism, as confirmed by matching triangular decompositions.
• The construction generalizes to formal groups not arising from algebraic groups, yielding new affine quantum groups, and recovers the elliptic quantum group of [Fed94] when $\mathbb{G}$ is an elliptic curve.
• The double Yangian $DY_\hbar(\mathfrak{g})$ is expected to embed into a reduced version of the Drinfeld double, suggesting a broader realization of double quantum groups via this CoHA framework.