Quantised Loop Algebra Uq(sl2[z±])

• Quantised Loop Algebra Uq(sl2[z±]) is a q-deformation of the universal enveloping algebra of the loop algebra sl2, featuring dual algebraic presentations and a rich representation theory.
• The algebra admits both Drinfeld–Jimbo and Drinfeld new/current presentations, offering formulations through Chevalley–Serre generators and current operators integrated within a Hopf algebra structure.
• Applications span categorification via Hall algebras and quantum integrable systems through q-oscillator representations, explicit tensor product decomposition, and geometric realization via shuffle algebras.

The quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$, also known as the quantum loop algebra of type $A_1$, is a $q$-deformation of the universal enveloping algebra of the loop algebra $\mathfrak{sl}_2[z,z^{-1}]$. This structure appears as the rank-one case within the broader theory of quantum affine algebras, possesses diverse presentations (Drinfeld–Jimbo/Chevalley–Serre and Drinfeld “new”/current), and governs fundamental aspects of representation theory, Hall algebras, and mathematical physics.

1. Algebraic Presentations

The quantised loop algebra admits two principal presentations, each capturing distinct features:

Drinfeld–Jimbo (Chevalley–Serre) Presentation:

The generators are $e_i$, $f_i$, $K_i^{\pm1}$ ($i=0,1$), with Cartan matrix $$A = \begin{pmatrix}2 & -2 \ -2 & 2 \end{pmatrix}$$, and $K_0 K_1 = 1$. The defining relations are: $$\begin{aligned} &K_iK_j=K_jK_i,\quad K_iK_i^{-1}=1,\ &K_i\,e_j\,K_i^{-1}=q^{a_{ij}} e_j,\quad K_i\,f_j\,K_i^{-1}=q^{-a_{ij}} f_j,\ &[e_i,f_j]=\delta_{ij}\,\frac{K_i-K_i^{-1}}{q-q^{-1}},\ \end{aligned}$$ with $q$-Serre relations

$$e_i^3 e_j - [3]_q e_i^2 e_j e_i + [3]_q e_i e_j e_i^2 - e_j e_i^3 = 0$$

and similarly for the $f_i$ ($i\neq j$), where $[3]_q = q^2 + 1 + q^{-2}$ (Nirov et al., 2016).

Drinfeld “New” (Current) Presentation:

Generators are $x_k^{\pm}$ ($k\in\mathbb{Z}$), $h_r$ ($r\in\mathbb{Z}\setminus\{0\}$), central $c$, and $K^{\pm1}$, subject to relations: $$\begin{aligned} &K x_k^\pm K^{-1} = q^{\pm2} x_k^\pm, \ &[h_r, x_k^\pm] = \pm \frac{[2r]_q}{r} x_{k+r}^\pm, \ [x_k^+, x_\ell^-] = \frac{\psi^+_{k+\ell} - \psi^-_{k+\ell}}{q - q^{-1}}, \end{aligned}$$ with

$$\psi^+(z) = c\,\exp\left((q-q^{-1})\sum_{r>0} h_r z^{-r}\right),\quad \psi^-(z) = c^{-1}\,\exp\left(-(q-q^{-1})\sum_{r>0} h_{-r} z^{r}\right).$$

The generating series $x^\pm(z) = \sum_{k\in\mathbb{Z}} x_k^{\pm} z^{-k}$ encode the currents (Young, 2012).

2. Defining Relations and Hopf Algebra Structure

The defining relations in the Drinfeld current presentation are:

• Cartan–Cartan: $$\phi^{\epsilon}(z)\phi^{\epsilon}(w) = \phi^{\epsilon}(w)\phi^{\epsilon}(z)$$, and

$$\phi^+(z)\phi^-(w) = \frac{q^2 z - w}{z - q^2 w} \phi^-(w)\phi^+(z)$$

for $\epsilon = \pm$ (Dou et al., 2010).

• Cartan–raising/lowering:

$$\phi^+(z)x^\pm(w)\phi^+(z)^{-1} = \frac{q^{\pm 2}z - w}{z - q^{\pm2} w} x^\pm(w)$$

and analogous formulas for $\phi^-(z)$ (Dou et al., 2010, Neguţ et al., 2021).

• Currents commutator:

$$[x^+(z), x^-(w)] = \frac{1}{q - q^{-1}} \left( \delta(z/w)\phi^+(w) - \delta(z/w)\phi^-(z) \right),$$

with $\delta(z) = \sum_{k\in\mathbb{Z}} z^k$ (Neguţ et al., 2021).

The Hopf algebra structure is given on generating series as: $$\begin{aligned} &\Delta(x^+(z)) = x^+(z) \otimes 1 + \phi^-(C^{1/2}z) \otimes x^+(Cz),\ &\Delta(x^-(z)) = 1 \otimes x^-(z) + x^-(Cz) \otimes \phi^+(C^{-1/2}z),\ &\Delta(\phi^{\pm}(z)) = \phi^{\pm}(C^{\pm1/2}z) \otimes \phi^{\pm}(C^{\mp1/2}z),\ &\Delta(C) = C \otimes C,\quad \Delta(d) = d \otimes 1 + 1 \otimes d. \end{aligned}$$ Counit and antipode are determined by standard axioms, with the antipode,

$$S(x^+(z)) = -\phi^-(z)^{-1} x^+(z),\quad S(x^-(z)) = -x^-(z) \phi^+(z)^{-1} [1002.1316, 2102.11269].$$

3. PBW Bases and Shuffle Algebra Realizations

The quantum loop algebra admits a PBW-type basis indexed by sequences of loop variables. In the rank-one case, Lyndon words correspond to single symbols $1^{(d)}$ of “vertical degree” $d$, with total order $1^{(d_1)} < 1^{(d_2)} \iff d_1 > d_2$. Every basis monomial in the positive part has the shape $x^+(d_1)x^+(d_2)\cdots x^+(d_k)$ for $d_1 \geq d_2 \geq \cdots \geq d_k$ (Neguţ et al., 2021).

There is a faithful embedding of $U_q(L\mathfrak{sl}_2)^+$ into a trigonometric shuffle algebra $\mathcal{S}$, via $x^+(d) \mapsto [1^{(d)}]$, with convolution product determined by the $q^2$-commutation factors. The image is characterized as the “wheel-vanishing” subalgebra. The Enriquez homomorphism from $U_q(L\mathfrak{sl}_2)^+$ to $\mathcal{S}$ is an isomorphism for $A_1$ (Neguţ et al., 2021).

Algebraic Structure	Realization	Reference
Quantum loop algebra	Currents $x^\pm(z)$, $\phi^\pm(z)$	(Dou et al., 2010)
Shuffle algebra	Lyndon words/trigonometric shuffle product	(Neguţ et al., 2021)
Double Hall algebra	Sheaf classes on $\mathbb{P}^1$	(Dou et al., 2010)

4. Hall Algebra and Geometric Realizations

The double Hall algebra $DH(Coh(\mathbb{P}^1))$ of coherent sheaves over $\mathbb{P}^1$ is isomorphic to $\mathcal{U}_q(\mathfrak{sl}_2[z^\pm])$ in Drinfeld's current presentation. The correspondence is:

• $x_k^+ \leftrightarrow u_{\mathcal{O}(k)}$ (line bundles),
• $x_k^- \leftrightarrow -v u_{\mathcal{O}(-k)}$,
• $\phi^+_r \leftrightarrow T_r$ (torsion sheaf Hall elements),
• $K \leftrightarrow K_{[\mathcal{O}]}$ (Dou et al., 2010).

The combinatorial and extension-theoretic relations in the Hall algebra match precisely with the defining relations of $\mathcal{U}_q(\mathfrak{sl}_2[z^\pm])$, providing a geometric realization valuable for both structural and categorification results.

5. $\ell$-Weights, Classification, and Representation Theory

Modules for $\mathcal{U}_q(\mathfrak{sl}_2[z^\pm])$ decompose into $\ell$-weight spaces defined by the commutative action of the Cartan subalgebra. An $\ell$-weight is a rational function $Y(z)$ of the form

$$Y(z) = \kappa \prod_{a \in S} \frac{1 - a z q^{-1}}{1 - a z q},$$

for finite $S \subset \mathbb{C}^\times$; these functions control the decomposition of finite-dimensional modules and determine highest-$\ell$-weight representations (Young, 2012).

For each finite set $P \subset \mathbb{C}^\times$, there is a subcategory $\mathcal{C}_P$ of modules with poles of all $\ell$-weights in $P$. All irreducible finite-dimensional modules arise as highest $\ell$-weight representations with dominant monomials in $Y_a(z)$ for $a \in P$ (Young, 2012).

The auxiliary algebra $A_{sl_2,P}$ with generators $E_{a,m}$, $F_{a,m}$, $H_{a,m}$ acts with definite $\ell$-weight, and comes equipped with explicit commutation and “Serre-type” relations, such as

$$(a - b q^2)E_{a,m}E_{b,n} + a E_{a,m+1}E_{b,n} - b q^2 E_{a,m}E_{b,n+1} = (a q^2 - b)E_{b,n}E_{a,m} + a q^2 E_{b,n}E_{a,m+1} - b E_{b,n+1}E_{a,m}.$$

A homomorphism from the quantum loop algebra to this auxiliary algebra encodes the structure of modules in $\mathcal{C}_P$ (Young, 2012).

6. Verma Modules, $q$-Oscillator and Prefundamental Representations

Via the Jimbo homomorphism, Verma modules for $U_q(\mathfrak{gl}_2)$ induce explicit highest-weight modules for $U_q(L(\mathfrak{sl}_2))$. The positive Borel subalgebra $U_q^+(L(\mathfrak{sl}_2))$ admits a $q$-oscillator representation, realized via an algebra homomorphism

$\rho: U_q^+(L(\mathfrak{sl}_2)) \to {\rm Osc}_q,$

with $\rho(e_0) = b^\dagger q^N$, $\rho(e_1) = -(q-q^{-1})^{-1} b q^N$. Simple quotients of the $q$-oscillator module yield the prefundamental “factor modules” that serve as building blocks for universal $Q$-operators in quantum integrable systems (Nirov et al., 2016).

7. Applications and Further Directions

The algebraic and geometric structures of $\mathcal{U}_q(\mathfrak{sl}_2[z^\pm])$ have substantial implications across mathematical physics, especially in the theory of quantum integrable systems, the categorification program via Hall algebras, and in the theory of $Q$-operators. Shuffle algebra realizations and $\ell$-weight theory facilitate the precise classification of finite-dimensional representations, enable explicit tensor product decompositions, and support the development of categorifications and connection with quantum cluster algebras (Neguţ et al., 2021, Young, 2012).

A plausible implication is that the correspondence between Hall algebra realizations and quantum loop algebra presentations is essential for the ongoing interaction between geometric representation theory and quantum group theory, particularly in the development of categorified structures and in the computation of quantum invariants of moduli spaces.