Limits of Gaudin Algebras,Quantization of Bending Flows,Jucys–Murphy Elements and Gelfand–Tsetlin Bases

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra \({U(\mathfrak {g})}\) of a semisimple Lie algebra \({\mathfrak {g}}\). This family is parameterized by collections of pairwise distinct complex numbers z ₁, . . . , z _n . We obtain some new commutative subalgebras in \({U(\mathfrak {g})^{\otimes n}}\) as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the Hamiltonians of bending flows and to the Gelfand–Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases.     

Non-Negative Integral Level Affine Lie Algebra Tensor Categories and Their Associativity Isomorphisms

For a finite-dimensional simple Lie algebra \({\mathfrak{g}}\), we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra \({{\widehat{\mathfrak{g}}}}\) at a fixed level \({\ell\in\mathbb{N}}\) with a certain tensor category of finite-dimensional \({\mathfrak{g}}\)-modules. More precisely, the category of level ? standard \({{\widehat{\mathfrak{g}}}}\)-modules is the module category for the simple vertex operator algebra \({L_{\widehat{\mathfrak{g}}}(\ell, 0)}\), and as is well known, this category is equivalent as an abelian category to \({\mathbf{D}(\mathfrak{g},\ell)}\), the category of finite-dimensional modules for the Zhu’s algebra \({A{(L_{\widehat{\mathfrak{g}}}(\ell, 0))}}\), which is a quotient of \({U(\mathfrak{g})}\). Our main result is a direct construction using Knizhnik–Zamolodchikov equations of the associativity isomorphisms in \({\mathbf{D}(\mathfrak{g},\ell)}\) induced from the associativity isomorphisms constructed by Huang and Lepowsky in \({{L_{\widehat{\mathfrak{g}}}(\ell, 0) - \mathbf{mod}}}\). This construction shows that \({\mathbf{D}(\mathfrak{g},\ell)}\) is closely related to the Drinfeld category of \({U(\mathfrak{g})}\)[[h]]-modules used by Kazhdan and Lusztig to identify categories of \({{\widehat{\mathfrak{g}}}}\)-modules at irrational and most negative rational levels with categories of quantum group modules.     

Parabolic Presentations of the Super Yangian {Y(\mathfrak{gl}_{M|N})} Associated with Arbitrary 01-Sequences

Let μ be an arbitrary composition of M + N and let \({\mathfrak{s}}\) be an arbitrary \({0^{M}1^{N}}\)- sequence. A new presentation, depending on \({\mu \rm and \mathfrak{s}}\), of the super Yangian Y_M|N associated to the general linear Lie superalgebra \({\mathfrak{gl}_{M|N}}\) is obtained.     

Dorey’s Rule and the q-Characters of Simply-Laced Quantum Affine Algebras

Let \({U_q(\widehat{\mathfrak g})}\) be the quantum affine algebra associated to a simply-laced simple Lie algebra \({\mathfrak{g}}\) . We examine the relationship between Dorey’s rule, which is a geometrical statement about Coxeter orbits of \({\mathfrak{g}}\) -weights, and the structure of q-characters of fundamental representations V _i,a of \({U_q(\widehat{\mathfrak g})}\) . In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product \({V_{i,a}\otimes V_{j,b}\otimes V_{k,c}}\) .     

Entanglement Rates and the Stability of the Area Law for the Entanglement Entropy

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed \({\mathfrak{gl}(1|1)}\) spin-chain and its continuum limit—the \({c=-2}\) symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones–Temperley–Lieb algebra JTL _N , goes over in the continuum limit to a bigger algebra than \({\boldsymbol{\mathcal{V}}}\), the product of the left and right Virasoro algebras. This algebra, \({\mathcal{S}}\)—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field \({S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}\), with a symmetric form \({S_{\alpha\beta}}\) and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL _N in the \({\mathfrak{gl}(1|1)}\) spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of \({\mathfrak{sp}_{N-2}}\). The semi-simple part of JTL _N is represented by \({U \mathfrak{sp}_{N-2}}\), providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL _N image represented in the spin-chain. On the continuum side, simple modules over \({\mathcal{S}}\) are identified with “fundamental” representations of \({\mathfrak{sp}_\infty}\).     

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

For a Hopf algebra B, we endow the Heisenberg double \({\mathcal{H}(B^*)}\) with the structure of a module algebra over the Drinfeld double \({\mathcal{D}(B)}\). Based on this property, we propose that \({\mathcal{H}(B^*)}\) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan–Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) quantum group that is Kazhdan–Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair \({(\mathcal{D}(B),\mathcal{H}(B^*))}\) is “truncated” to \({(\overline{\mathcal{U}}_{\mathfrak{q}} s\ell2,\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2))}\), where \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)}\) is a \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) module algebra that turns out to have the form \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)=\mathbb{C}_{\mathfrak{q}}[z,\partial]\otimes\mathbb{C}[\lambda]/(\lambda^{2p}-1)}\), where \({\mathbb{C}_{\mathfrak{q}}[z,\partial]}\) is the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\)-module algebra with the relations z ^p  = 0, ? ^p  = 0, and \({\partial z = \mathfrak{q}-\mathfrak{q}^{-1} + \mathfrak{q}^{-2} z\partial}\).     

Automorphisms and Derivations of the Insertion–Elimination Algebra and Related Graded Lie Algebras

This paper addresses several structural aspects of the insertion–elimination algebra \({\mathfrak{g}}\), a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of \({\mathfrak{g}}\), the automorphism group of \({\mathfrak{g}}\), the derivation Lie algebra of \({\mathfrak{g}}\), and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.     

Conformal Mappings and Dispersionless Toda Hierarchy

Let \({\mathfrak{D}}\) be the space consists of pairs (f, g), where f is a univalent function on the unit disc with f(0) = 0, g is a univalent function on the exterior of the unit disc with g(∞) = ∞ and f′(0)g′(∞) = 1. In this article, we define the time variables \({t_n, n\in \mathbb{Z}}\), on \({\mathfrak{D}}\) which are holomorphic with respect to the natural complex structure on \({\mathfrak{D}}\) and can serve as local complex coordinates for \({\mathfrak{D}}\) . We show that the evolutions of the pair (f, g) with respect to these time coordinates are governed by the dispersionless Toda hierarchy flows. An explicit tau function is constructed for the dispersionless Toda hierarchy. By restricting \({\mathfrak{D}}\) to the subspace Σ consists of pairs where \({f(w)=1/\overline{g(1/\bar{w})}}\), we obtain the integrable hierarchy of conformal mappings considered by Wiegmann and Zabrodin [31]. Since every C ¹ homeomorphism γ of the unit circle corresponds uniquely to an element (f, g) of \({\mathfrak{D}}\) under the conformal welding \({\gamma=g^{-1}\circ f}\), the space Homeo _C (S ¹) can be naturally identified as a subspace of \({\mathfrak{D}}\) characterized by f(S ¹) = g(S ¹). We show that we can naturally define complexified vector fields \({\partial_n, n\in \mathbb{Z}}\) on Homeo _C (S ¹) so that the evolutions of (f, g) on Homeo _C (S ¹) with respect to ? _n satisfy the dispersionless Toda hierarchy. Finally, we show that there is a similar integrable structure for the Riemann mappings (f ^?1, g ^?1). Moreover, in the latter case, the time variables are Fourier coefficients of γ and 1/γ ^?1.     

Hölder Regularity of the Solutions of the Cohomological Equation for Roth Type Interval Exchange Maps

There is a general method for constructing a soliton hierarchy from a splitting \({{L_{\pm}}}\) of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra \({\mathcal{L}_{+}}\). Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup \({L_-}\) a solution \({u_{f}}\) of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function \({\tau_{f}}\) for each element \({f \in L_-}\). In this paper, we give integral formulas for variations of \({\ln\tau_{f}}\) and second partials of \({\ln\tau_{f}}\), discuss whether we can recover solutions \({u_{f}}\) from \({\tau_{f}}\), and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the \({GL(n,\mathbb{C})}\)-hierarchy.     

On a Classical Limit of <Emphasis Type="Italic">q</Emphasis>-Deformed Whittaker Functions

Previously, we derive a representation of q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function as a sum over Gelfand–Zetlin patterns. This representation provides an analog of the Shintani–Casselman–Shalika formula for q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker functions. In this note, we provide a derivation of the Givental integral representation of the classical \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function as a limit q → 1 of the sum over the Gelfand–Zetlin patterns representation of the q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function. Thus, Givental representation provides an analog the Shintani–Casselman–Shalika formula for classical Whittaker functions.     

On the Extension of Stringlike Localised Sectors in 2+1 Dimensions

In the framework of algebraic quantum field theory, we study the category \({\Delta_{{\rm BF}}^{\mathfrak{A}}}\) of stringlike localised representations of a net of observables \({\mathcal{O} \mapsto \mathfrak{A}(\mathcal{O})}\) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of \({\Delta_{{\rm BF}}^{\mathfrak{A}}}\) with respect to the braiding. This implies that \({\Delta_{{\rm BF}}^{\mathfrak{A}}}\) cannot be modular when non-trivial DHR sectors exist. Modular tensor categories, however, are important for topological quantum computing. For this reason, we discuss a method to remove this obstruction to modularity.Indeed, the obstruction can be removed by passing from the observable net \({\mathfrak{A}(\mathcal{O})}\) to the Doplicher-Roberts field net \({\mathfrak{F}(\mathcal{O})}\). It is then shown that sectors of \({\mathfrak{A}}\) can be extended to sectors of the field net that commute with the action of the corresponding symmetry group. Moreover, all such sectors are extensions of sectors of \({\mathfrak{A}}\). Finally, the category \({\Delta_{{\rm BF}}^{\mathfrak{F}}}\) of sectors of \({\mathfrak{F}}\) is studied by investigating the relation with the categorical crossed product of \({\Delta_{{\rm BF}}^{\mathfrak{A}}}\) by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category \({\Delta_{{\rm BF}}^{\mathfrak{F}}}\).     

Harmonic Pinnacles in the Discrete Gaussian Model

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.     

Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case

We study the ODE/IM correspondence for ODE associated to \({\widehat{\mathfrak{g}}}\)-valued connections, for a simply-laced Lie algebra \({\mathfrak{g}}\). We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called \({\Psi}\)-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.     

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

A quantum system (with Hilbert space \({\mathcal {H}_{1}}\)) entangled with its environment (with Hilbert space \({\mathcal {H}_{2}}\)) is usually not attributed to a wave function but only to a reduced density matrix \({\rho_{1}}\). Nevertheless, there is a precise way of attributing to it a random wave function \({\psi_{1}}\), called its conditional wave function, whose probability distribution \({\mu_{1}}\) depends on the entangled wave function \({\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}\) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of \({\mathcal {H}_{2}}\) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about \({\mu_{1}}\), e.g., that if the environment is sufficiently large then for every orthonormal basis of \({\mathcal {H}_{2}}\), most entangled states \({\psi}\) with given reduced density matrix \({\rho_{1}}\) are such that \({\mu_{1}}\) is close to one of the so-called GAP (Gaussian adjusted projected) measures, \({GAP(\rho_{1})}\). We also show that, for most entangled states \({\psi}\) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval \({[E, E+ \delta E]}\)) and most orthonormal bases of \({\mathcal {H}_{2}}\), \({\mu_{1}}\) is close to \({GAP(\rm {tr}_{2} \rho_{mc})}\) with \({\rho_{mc}}\) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then \({\mu_{1}}\) is close to \({GAP(\rho_\beta)}\) with \({\rho_\beta}\) the canonical density matrix on \({\mathcal {H}_{1}}\) at inverse temperature \({\beta=\beta(E)}\). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.     

A Note on Expansiveness and Hyperbolicity for Generic Geodesic Flows

In this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥?2. We prove that there exists a C²-residual subset \(\mathscr{R}\) of metrics on a given compact Riemannian manifold such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if the closure of the set of periodic orbits of \({\varphi ^{t}_{g}}\) is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C²-residual \(\mathscr{R}\) such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if \({\varphi ^{t}_{g}}\) is an Anosov flow.     

On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

Let K be a field of characteristic zero. For \({n \in \mathbb{N}^{*}}\) , let \({\mathcal{T}^{\,\prime}_{n}}\) be the vector space of non-planar rooted trees with n vertices (Foissy in Bull Sci Math 126, no. 3, 193–239; no. 4, 249–288, 2002). Let \({\vartriangleright}\) be the left pre-Lie product of insertion of a tree inside another defined on infinitesimal characters of the graded Hopf algebra \({\mathcal{H}}\) introduced by Calaque, Ebrahimi-Fard and Manchon. Let \({\mathcal{T}^{\,\prime}=\oplus_{n\geq 2}\mathcal{T}^{\,\prime}_{n}}\) . In this work, we first prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) a pre-Lie algebra generated by the two ladders E ₁ and E ₂ where E ₁ is the ladder with one edge and E ₂ is the ladder with two edges. Second, we prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) is not a free pre-Lie algebra, and we exhibit a family of relations.     

The Exoticness and Realisability of Twisted Haagerup–Izumi Modular Data

The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data \({\mathcal{D}{\rm Hg}}\) fits into a family \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\) , where n ≥  0 and \({\omega\in \mathbb{Z}_{2n+1}}\) . We show \({\mathcal{D}^0 {\rm Hg}_{2n+1}}\) is related to the subfactors Izumi hypothetically associates to the cyclic groups \({\mathbb{Z}_{2n+1}}\) . Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions, etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of Izumi type \({\mathbb{Z}_7, \mathbb{Z}_9}\) and \({\mathbb{Z}_3^2}\) , and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type \({\mathbb{Z}_{11},\mathbb{Z}_{13},\mathbb{Z}_{15},\mathbb{Z}_{17},\mathbb{Z}_{19}}\) (previously, Izumi had shown uniqueness for \({\mathbb{Z}_3}\) and \({\mathbb{Z}_5}\)), and we identify their modular data. We explain how \({\mathcal{D}{\rm Hg}}\) (more generally \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\)) is a graft of the quantum double \({\mathcal{D} Sym(3)}\) (resp. the twisted double \({\mathcal{D}^\omega D_{2n+1}}\)) by affine so(13) (resp. so\({(4n^2+4n+5)}\)) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\) . For example we show there are exactly 2 possible character vectors (giving graded dimensions of all modules) for the Haagerup VOA at central charge c = 8. It seems unlikely that any of this twisted Haagerup-Izumi modular data can be regarded as exotic, in any reasonable sense.     

Constraining the electric charges of some astronomical bodies in Reissner–Nordström spacetimes and generic <Emphasis Type="Italic">r</Emphasis><Superscript>−2</Superscript>-type power-law potentials from orbital motions

We put independent model dynamical constraints on the net electric charge Q of some astronomical and astrophysical objects by assuming that their exterior spacetimes are described by the Reissner-Nordström, metric, which induces an additional potential \({U_{\rm RN} \propto Q^2 r^{-2}}\). From the current bounds \({\Delta \dot \varpi}\) on any anomalies in the secular perihelion rate \({\dot \varpi}\) of Mercury and the Earth–mercury ranging Δρ, we have \({\left|Q_{\odot}\right| \lesssim 1-0.4 \times 10^{18}\ {\rm C}}\). Such constraints are ~60–200 times tighter than those recently inferred in literature. For the Earth, the perigee precession of the Moon, determined with the Lunar Laser Ranging technique, and the intersatellite ranging Δρ for the GRACE mission yield \({\left|Q_{\oplus} \right| \lesssim 5-0.4 \times 10^{14}\ {\rm C}}\). The periastron rate of the double pulsar PSR J0737-3039A/B system allows to infer \({\left|Q_{\rm NS} \right| \lesssim 5\times 10^{19}\ {\rm C}}\). According to the perinigricon precession of the main sequence S2 star in Sgr A^*, the electric charge carried by the compact object hosted in the Galactic Center may be as large as \({\left|Q_{\bullet} \right| \lesssim 4\times 10^{27} \ {\rm C}}\). Our results extend to other hypothetical power-law interactions inducing extra-potentials \({U_{\rm pert} = \Psi r^{-2}}\) as well. It turns out that the terrestrial GRACE mission yields the tightest constraint on the parameter \({\Psi}\), assumed as a universal constant, amounting to \({|\Psi| \lesssim 5\times 10^{9}\ {\rm m^4\ s^{-2}}}\).     

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\). The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\)-module with central charge \({c \in \mathbb{C}}\), and its irreducible quotient \({\mathcal{V}_c}\), possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\). We show that for each integer \({n > 0}\), \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\)-algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\), respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\), respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\), and we prove this for \({n = 1}\). As an application, we show that invariant subalgebras of \({\beta\gamma}\)-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.