Completely X-symmetric S-matrices corresponding to theta functions

We extend to the sl(N ) case the results that we previously obtained on the construction of W q,p algebras from the elliptic algebra A q,p ( sl(2) c ). The elliptic algebra A q,p ( sl(N ) c ) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N (N − 1)/2 integers, defining q-deformations of the W N algebra, are constructed. The operators t(z) also close an exchange algebra when (−pIt becomes Abelian when in addition p = q N h where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W N algebras depending on the parity of h, characterizing the exchange structures at p = q N h as new W q,p (sl(N )) algebras. RésuméNousétendons au cas sl(N ) les résultats que nous avons obtenus précédemment concernant la construction des algèbres W q,pà partir de l'algèbre elliptique A q,p ( sl (2)

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“…The N-elliptic R-matrix in End(C N ) ⊗ End(C N ), associated to the Z N -vertex model, is defined as follows [33,34]:…”

Section: Notations and Basic Definitions 21 Definition Of The N -Ellmentioning

confidence: 99%

Deformed ${\cal W}_{N}$ Algebras from Elliptic sl ( N ) Algebras

Avan

Frappat

Rossi

et al. 1999

Communications in Mathematical Physics

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“…As conjectured by Belavin [Bel80] and later proved by Cherednik [Che82], Chudnovsky and Chudnovsky [CC81] and Tracy [Tra85], when = 1 these operators satisfy the equation…”

Section: 41mentioning

confidence: 75%

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

Chirvăsitu

Kanda

Smith

2021

Forum of Mathematics, Sigma

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The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

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“…One defines A q,p ( gl(N ) c ) by imposing the following constraints on L ij (z): [17][18][19] defined in Appendix B and R * 12 (z) = R 12 (z, q, p * = pq −2c ), see (B.14). It is useful to introduce the following two matrices:…”

Section: The Elliptic Quantum Algebramentioning

confidence: 99%

“…The starting point for the definition of the elliptic quantum algebras of vertex type is the N -elliptic R-matrix in End(C N ) ⊗ End(C N ) associated to the Z N -vertex model [17][18][19], given by…”

Section: A Jacobi Theta Functionsmentioning

confidence: 99%

“…Definition of the N -elliptic R-matrixThe starting point for the definition of the elliptic quantum algebras of vertex type is the N -elliptic R-matrix in End(C N ) ⊗ End(C N ) associated to the Z N -vertex model[17][18][19], given byZ(z, q, p) = z 2/N −2 1 κ(z 2 ) ζ, τ ) (α 1 ,α 2 )∈Z N ×Z N W (α 1 ,α 2 ) (ξ, ζ, τ ) I (α 1 ,α 2 ) ⊗ I −1 (α 1 ,α 2 ) ,(B.1) where the variables z, q, p are related to the variables ξ, ζ, τ by z = e iπξ , q = e iπζ , p = e 2iπτ .…”

mentioning

confidence: 99%

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Elliptic deformation of $\mathcal{W}_N$-algebras

2019

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We construct q-deformations of quantum W N algebras with elliptic structure functions. Their spin k + 1 generators are built from 2k products of the Lax matrix generators of A q,p ( gl(N ) c )). The closure of the algebras is insured by a critical surface condition relating the parameters p, q and the central charge c. Further abelianity conditions are determined, either as c = −N or as a second condition on p, q, c. When abelianity is achieved, a Poisson bracket can be defined, that we determine explicitly. One connects these structures with previously built classical q-deformed W N algebras and quantum W q,p (sl N ).

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Completely X-symmetric S-matrices corresponding to theta functions

Cited by 25 publications

References 7 publications

Deformed ${\cal W}_{N}$ Algebras from Elliptic sl ( N ) Algebras

Deformed ${\cal W}_{N}$ Algebras from Elliptic sl ( N ) Algebras

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

Elliptic deformation of $\mathcal{W}_N$-algebras

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