Rigorous identification of the Yang-Baxter (RLL) algebra of the Maulik-Okounkov R-matrix with the affine Yangian of $\widehat{\mathfrak{gl}}(1)$ by factorization over its infinite-dimensional center

The affine Yangian $Y(\widehat{\mathfrak{gl}}(1))$ of $\mathfrak{gl}(1)$, introduced by Tsymbaliuk as a three-parameter current algebra whose Fock representations realize the $\mathcal{W}_{1+\infty}$ symmetry of two-dimensional conformal field theory, is obtained from the Yang-Baxter (RLL) algebra generated by the Maulik-Okounkov R-matrix acting on tensor products of free-boson Fock modules by factorization over an infinite-dimensional center; in conformal-field-theory language this R-matrix is the Liouville reflection operator. Litvinov and Vilkoviskiy (2020) showed at physics level of rigor that the two algebras do not coincide literally: the current realization of the RLL algebra satisfies the quadratic and cubic Serre relations of $Y(\widehat{\mathfrak{gl}}(1))$ only up to local terms, compensated by an infinite-dimensional center built from singular vectors of $\mathcal{W}_n$ algebras, and argued that $Y(\widehat{\mathfrak{gl}}(1))$ is obtained from the RLL algebra by factorization over this center, in the same way the Yangians of $\mathfrak{gl}(n)$ and $\mathfrak{sl}(n)$ differ by central elements. The problem is to turn this physics-level identification into a theorem: construct the quotient map explicitly, determine its kernel completely, and prove that the quadratic and Serre relations together with the center exhaust all relations of the RLL algebra. Later literature verifies parts of the dictionary at low levels, postpones the same question in the $\mathfrak{gl}(2)$ and $\mathfrak{gl}(1|1)$ analogues, or settles only adjacent problems, so the identification remains open.

Background

The integrable structure of two-dimensional conformal field theory can be described in two superficially different algebraic languages, and relating them precisely is the subject of this problem.

On one side stands the affine Yangian $Y(\widehat{\mathfrak{gl}}(1))$ constructed by Tsymbaliuk: a deformation of the universal enveloping algebra of the loop (current) algebra of $\mathfrak{gl}(1)$, generated by currents $e(z)$, $f(z)$, $\psi(z)$ subject to quadratic relations and cubic Serre relations (higher-order constraints of the kind familiar from quantum groups), and depending on three parameters $\varepsilon_1$, $\varepsilon_2$, $\varepsilon_3$ with $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$. It is the rational degeneration of the quantum toroidal algebra of $\mathfrak{gl}(1)$ (the trigonometric, $q$-deformed version of the same algebra), and its Fock modules — the standard representations of the Heisenberg algebra of a single free boson, acting on polynomials in infinitely many variables — realize the $\mathcal{W}_{1+\infty}$ symmetry algebra of conformal field theory (the Virasoro algebra extended by chiral currents of every integer spin).

On the other side stands the R-matrix of Maulik and Okounkov, constructed geometrically in their theory connecting quantum groups with quantum cohomology (via equivariant cohomology of Hilbert schemes of points on the plane). In rank one and in conformal-field-theory terms it coincides with the Liouville reflection operator: up to normalization it is the unique operator on a tensor product of two Fock modules that conjugates a product of two screening operators (integrals of exponentials of the free boson) into the same product with shifted spectral parameters, and it satisfies the Yang-Baxter equation, the cubic factorization condition $R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)$ guaranteeing consistency of the associated transfer matrices. Expanding $R(u)$ over a basis of an auxiliary Fock module yields matrix elements $L_{\lambda,\mu}(u)$, and the relations among them define the Yang-Baxter (RLL) algebra $\mathrm{YB}(\widehat{\mathfrak{gl}}(1))$, in the same way RLL-type presentations define ordinary Yangians.

Transfer matrices built from $R(u)$ generate commuting Hamiltonians of the quantum Korteweg-de Vries hierarchy (the integrable hierarchy whose conserved quantities deform the Virasoro generators) and of the intermediate long wave system (a one-dimensional model of long surface waves with a nonlocal dispersion relation), and the affine Yangian program derives the Bethe ansatz equations for their spectra from the algebra. The linchpin is the precise relation between $\mathrm{YB}(\widehat{\mathfrak{gl}}(1))$ and $Y(\widehat{\mathfrak{gl}}(1))$. Litvinov and Vilkoviskiy established at physics level of rigor that the two algebras do not literally coincide: deriving the current realization of the RLL algebra, they found the quadratic and Serre relations satisfied up to local (contact-term) corrections, and exhibited an infinite-dimensional center (elements commuting with the whole algebra) — elements $D_{m,n}(u)$ attached to singular vectors (vectors annihilated by all positive oscillator modes) of the $\mathcal{W}_n$ algebras at the degenerate Kac values $u_{m,n}$ of the parameter, where two screening operators fuse. They argued that quotienting $\mathrm{YB}(\widehat{\mathfrak{gl}}(1))$ by the ideal generated by this center yields exactly $Y(\widehat{\mathfrak{gl}}(1))$, the analogue of the classical fact that the Yangians of $\mathfrak{gl}(n)$ and $\mathfrak{sl}(n)$ differ by central elements. What is missing is the theorem: an explicit quotient map, a complete description of the kernel, and a completeness proof of Poincaré-Birkhoff-Witt type (a theorem that ordered monomials in the currents form a basis) showing that the quadratic relations, the Serre relations, and the center exhaust all relations of the RLL algebra.

Problem Statement

Establish, with mathematical rigor, the precise algebraic relation between the Yang-Baxter algebra $\mathrm{YB}(\widehat{\mathfrak{gl}}(1))$ generated by the Maulik-Okounkov R-matrix in Fock representations and Tsymbaliuk's affine Yangian $Y(\widehat{\mathfrak{gl}}(1))$, in the regime of generic (non-resonant) parameters $\varepsilon_1$, $\varepsilon_2$ with $\varepsilon_3=-\varepsilon_1-\varepsilon_2$ (the generic regime treated in the source paper of Litvinov and Vilkoviskiy). Concretely:

(1) Fix an unambiguous definition of the RLL algebra as an abstract algebra; construct an explicit surjective homomorphism $\mathrm{YB}(\widehat{\mathfrak{gl}}(1))\to Y(\widehat{\mathfrak{gl}}(1))$ compatible with the current realization and determine its kernel — or prove that no such homomorphism exists.

(2) Prove or disprove that the kernel equals the ideal generated by the center exhibited by Litvinov and Vilkoviskiy, and complete the description of that center: show that the elements attached to singular vectors of the $\mathcal{W}_n$ algebras — the simplest being $D_{m,n}(u)=h_{s_{m,n}}(u)/(h(u-m\varepsilon_1)h(u-n\varepsilon_2))$, where $h(u)$ is the Cartan current of the affine Yangian and $h_{s_{m,n}}(u)$ is the modified current associated with the $\mathcal{W}_2$ singular vector labeled by the Kac pair $(m,n)$, in the notation of Litvinov and Vilkoviskiy — together with their products generate the entire center, or identify what is missing.

(3) Prove the completeness of the current realization: after factorization over the center, the currents obeying the quadratic relations and the cubic Serre relations, with higher currents obtained as iterated commutators, carry a Poincaré-Birkhoff-Witt-type basis, so that no additional independent relations occur at higher levels; equivalently, quadratic plus Serre relations plus center give a full presentation of the quotient. A disproof is equally acceptable: exhibit an explicit relation of the RLL algebra, at any finite level, that is independent of the quadratic, Serre, and center relations, or prove that the kernel differs from the ideal generated by the center.

The three components are one objective: an RLL presentation theorem for the affine Yangian of $\mathfrak{gl}(1)$. Each component may be settled in either direction: a proof that no surjective homomorphism exists settles component (1) and, the kernel then being undefined, resolves the objective in the negative; otherwise component (3) is an equivalent reformulation of the kernel-identification demand of component (2), so the objective is met once components (1) and (2) are settled, in either direction. Partial results (for example, a Poincaré-Birkhoff-Witt theorem for a subalgebra, or new central elements without the quotient map) do not by themselves resolve it.

References

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