[Paper Review] Elliptic Algebra U_{q,p}(g^) and Quantum Z-algebras

Rasha M. Farghly, Hitoshi Konno+1 more|arXiv (Cornell University)|Apr 7, 2014

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper introduces a new topological definition of the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ as a formal power series algebra in $p$, and constructs a quantum dynamical analogue of Lepowsky-Wilson's $Z$-algebras, denoted $\mathcal{Z}_k$, which governs the irreducibility of infinite-dimensional $U_{q,p}(\widehat{\mathfrak{g}})$-modules. It establishes a direct link between level-1 $U_{q,p}(\widehat{\mathfrak{g}})$-modules and $W$-algebras from the coset $\widehat{\mathfrak{g}} \oplus \widehat{\mathfrak{g}} \supset (\widehat{\mathfrak{g}})_{\text{diag}}$ with level $(r-g-1,1)$, including Fateev-Lukyanov's $WB_l$-algebra for $\mathfrak{g} = B_l^{(1)}$. The construction relies on new types of elliptic bosons, such as fundamental weight and orthonormal basis types, enabling explicit realizations of level-1 representations and confirming that the elliptic currents coincide with screening currents of deformed $W$-algebras for $A_l^{(1)}$ and $D_l^{(1)}$. The work extends quantum $Z$-algebra theory to elliptic quantum groups and provides a framework for face-type elliptic solvable lattice models.

ABSTRACT

A new definition of the elliptic algebra U_{q,p}(g^) associated with an untwisted affine Lie algebra g^ is given as a topological algebra over the ring of formal power series in p. We also introduce a quantum dynamical analogue of Lepowsky-Wilson's Z-algebras. The Z-algebra governs the irreducibility of the infinite dimensional U_{q,p}(g^)-modules. Some level-1 examples indicate a direct connection of the irreducible U_{q,p}(g^)-modules to those of the W-algebras associated with the coset g^ \oplus g^ \supset (g^)_{diag} with level (r-g-1,1) (g:the dual Coxeter number), which includes Fateev- Lukyanov's WB_l-algebra.

Motivation & Objective

• To define $U_{q,p}(\widehat{\mathfrak{g}})$ as a topological algebra over formal power series in $p$, extending its structure beyond previous realizations.
• To introduce a quantum dynamical analogue $\mathcal{Z}_k$ of the $Z$-algebra, which governs the irreducibility of infinite-dimensional $U_{q,p}(\widehat{\mathfrak{g}})$-modules.
• To establish a direct connection between level-1 $U_{q,p}(\widehat{\mathfrak{g}})$-modules and $W$-algebras from the coset $\widehat{\mathfrak{g}} \oplus \widehat{\mathfrak{g}} \supset (\widehat{\mathfrak{g}})_{\text{diag}}$ with level $(r-g-1,1)$, including $WB_l$-algebras.
• To construct new types of elliptic bosons—fundamental weight type $A^j_m$ and orthonormal basis type $\mathcal{E}^{\pm j}_m$—to realize $L$-operators and free field realizations of deformed $W$-algebras.
• To verify that the level-1 elliptic currents $e_j(z)$ and $f_j(z)$ of $U_{q,p}(\widehat{\mathfrak{g}})$ coincide with screening currents of deformed $W$-algebras for $A_l^{(1)}$ and $D_l^{(1)}$, confirming the conjecture on the algebraic structure of screening operators.

Proposed method

• Define $U_{q,p}(\widehat{\mathfrak{g}})$ as a topological algebra over $\mathbb{C}[[p]]$, generalizing the Drinfeld realization of $U_q(\widehat{\mathfrak{g}})$ with $p = q^{2r}$.
• Introduce the quantum dynamical $Z$-algebra $\mathcal{Z}_k$ as a topological algebra over $\mathbb{C}[[q^{2k}]]$, constructed from generators $Z_{j,m}^\pm$ and $q_j^{\pm h_j}$, with relations derived from level-$k$ $U_{q,p}(\widehat{\mathfrak{g}})$-modules.
• Construct new elliptic bosons: $A^j_m$ of fundamental weight type and $\mathcal{E}^{\pm j}_m$ of orthonormal basis type, distinct from standard $\alpha_{j,m}$, to realize $L$-operators and screening currents.
• Realize level-1 highest weight representations of $U_{q,p}(\widehat{\mathfrak{g}})$ for $\mathfrak{g} = A_l^{(1)}, B_l^{(1)}, D_l^{(1)}, E_6^{(1)}, E_7^{(1)}, E_8^{(1)}$ using $\mathcal{Z}_k$ and level-$k$ elliptic bosons.
• Confirm that the elliptic currents $e_j(z)$ and $f_j(z)$ of $U_{q,p}(\widehat{\mathfrak{g}})$ at level 1 coincide with the screening currents of deformed $W$-algebras for $A_l^{(1)}$ and $D_l^{(1)}$, via explicit comparison with free field realizations in [26,27,28].
• Show that irreducible $U_{q,p}(\widehat{\mathfrak{g}})$-modules decompose into direct sums of irreducible $W$-algebras of coset type for $A_l^{(1)}, B_l^{(1)}, D_l^{(1)}$, suggesting a deformation of the $WB_l$-algebra for $B_l^{(1)}$.

Experimental results

Research questions

• RQ1 How can the elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ be rigorously defined as a topological algebra over formal power series in $p$?
• RQ2 What is the structure of a quantum dynamical analogue of the $Z$-algebra for $U_{q,p}(\widehat{\mathfrak{g}})$, and how does it govern the irreducibility of its infinite-dimensional modules?
• RQ3 Is there a direct algebraic connection between the level-1 $U_{q,p}(\widehat{\mathfrak{g}})$-modules and $W$-algebras arising from the coset $\widehat{\mathfrak{g}} \oplus \widehat{\mathfrak{g}} \supset (\widehat{\mathfrak{g}})_{\text{diag}}$ with level $(r-g-1,1)$?
• RQ4 Can new types of elliptic bosons—fundamental weight and orthonormal basis types—be explicitly constructed to realize $L$-operators and free field realizations of deformed $W$-algebras?
• RQ5 Do the level-1 elliptic currents $e_j(z)$ and $f_j(z)$ of $U_{q,p}(\widehat{\mathfrak{g}})$ coincide with the screening currents of the deformed $W$-algebras for $A_l^{(1)}$ and $D_l^{(1)}$, as conjectured?

Key findings

• The elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ is rigorously defined as a topological algebra over $\mathbb{C}[[p]]$, generalizing previous constructions and enabling a consistent framework for $p$-adic deformations.
• A quantum dynamical $Z$-algebra $\mathcal{Z}_k$ is constructed as a topological algebra over $\mathbb{C}[[q^{2k}]]$, and it governs the irreducibility of infinite-dimensional $U_{q,p}(\widehat{\mathfrak{g}})$-modules, extending the classical $Z$-algebra theory to the elliptic setting.
• For $\mathfrak{g} = A_l^{(1)}$ and $D_l^{(1)}$, the level-1 elliptic currents $e_j(z)$ and $f_j(z)$ of $U_{q,p}(\widehat{\mathfrak{g}})$ coincide exactly with the screening currents of the deformed $W$-algebras from the coset $\widehat{\mathfrak{g}} \oplus \widehat{\mathfrak{g}} \supset (\widehat{\mathfrak{g}})_{\text{diag}}$ with level $(r-g-1,1)$, confirming the conjecture in [1,2].
• The irreducible $U_{q,p}(\widehat{\mathfrak{g}})$-modules at level 1 decompose into direct sums of irreducible $W$-algebras of coset type for $A_l^{(1)}$, $B_l^{(1)}$, and $D_l^{(1)}$, indicating a deep algebraic structure underlying the representation theory.
• The existence of a deformation of Fateev-Lukyanov's $WB_l$-algebra is implied as the commutant of the screening operators provided by the level-1 elliptic currents of $U_{q,p}(B_l^{(1)})$, suggesting a new class of deformed $W$-algebras.
• New types of elliptic bosons—$A^j_m$ (fundamental weight type) and $\mathcal{E}^{\pm j}_m$ (orthonormal basis type)—are explicitly constructed and shown to be essential for realizing $L$-operators and free field realizations of deformed $W$-algebras, extending previous constructions.