Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations
Hitoshi Konno
备注：
This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
For any affine Lie algebra ${\mathfrak g}$, we show that any finite dimensional representation of the universal dynamical $R$ matrix ${\cal R}(λ)$ of the elliptic quantum group ${\cal B}_{q,λ}({\mathfrak g})$ coincides with a corresponding connection matrix for the solutions of the $q$-KZ equation associated with $U_q({\mathfrak g})$. This provides a general connection between ${\cal B}_{q,λ}({\mathfrak g})$ and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of ${\cal R}(λ)$ for ${\mathfrak g}=A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $D_n^{(1)}$, and show that they coincide with the face weights derived by Jimbo, Miwa and Okado. We hence confirm the conjecture by Frenkel and Reshetikhin.
We study a general ansatz for an odd supersymmetric version of the Kronecker elliptic function, which satisfies the genus one Fay identity. The obtained result is used for construction of the odd supersymmetric analogue for the classical and quantum elliptic $R$-matrices. They are shown to satisfy the classical Yang-Baxter equation and the associative Yang-Baxter equation. The quantum Yang-Baxter is discussed as well. It acquires additional term in the case of supersymmetric $R$-matrices.
We construct quadratic quantum algebra based on the dynamical RLL-relation for the quantum $R$-matrix related to $SL(NM)$-bundles with nontrivial characteristic class over elliptic curve. This $R$-matrix generalizes simultaneously the elliptic nondynamical Baxter--Belavin and the dynamical Felder $R$-matrices,and the obtained quadratic relations generalize both -- the Sklyanin algebra and the relations in the Felder-Tarasov-Varchenko elliptic quantum group, which are reproduced in the particular cases $M=1$ and $N=1$ respectively.
We construct a flat connection on the elliptic configuration space associated to any complex semisimple Lie algebra g. This elliptic Casimir connection has logarithmic singularities, and takes values in the deformed double current algebra of g defined by Guay. It degenerates to the trigonometric Casimir connection of g constructed by the first author. By analogy with the rational and trigonometric cases, we conjecture that the monodromy of the elliptic Casimir connection is described by the quantum Weyl group operators of the quantum toroidal algebra of g.
Resonance Relations for Solutions of the Elliptic QKZB Equations, Fusion Rules, and Eigenvectors of Transfer Matrices of Restricted Interaction-round-a-face Models
We propose a new method to compute connection matrices of quantum Knizhnik-Zamolodchikov equations associated to integrable vertex models with super algebra and Hecke algebra symmetries. The scheme relies on decomposing the underlying spin representation of the affine Hecke algebra in principal series modules and invoking the known solution of the connection problem for quantum affine Knizhnik-Zamolodchikov equations associated to principal series modules. We apply the method to the spin representation underlying the $\mathcal{U}_q\bigl(\hat{\mathfrak{gl}}(2|1)\bigr)$ Perk-Schultz model. We sh…
We identify generating functionals that satisfy dynamical exchange relations with the Lax matrices defining the face-type elliptic quantum algebra ${\mathcal{B}}_{q,λ}(\hat{gl}_{2})_c$, when the central charge takes the two possible values $c =\pm 2$. These generating functionals are constructed as quadratic trace-like objects in terms of the Lax matrices. The obtained structures are characterized as "dynamical centers", i.e. the centrality property is deformed by dynamical shifts. For these values, the functionals define (genuine) abelian subalgebras of ${\mathcal{B}}_{q,λ}(\hat{gl}_{2})_c$.
We introduce an odd supersymmetric version of the Kronecker elliptic function. It satisfies the genus one Fay identity and supersymmetric version of the heat equation. As an application we construct an odd supersymmetric extensions of the elliptic $R$-matrices, which satisfy the classical and the associative Yang-Baxter equations.
In this paper we propose versions of the associative Yang-Baxter equation and higher order $R$-matrix identities which can be applied to quantum dynamical $R$-matrices. As is known quantum non-dynamical $R$-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum Yang-Baxter equation and a set of identities useful for different applications in integrable systems. The dynamical $R$-matrices satisfy the Gervais-Neveu-Felder (or dynamical Yang-Baxter) equation. Relation between the dynamical and non-dynamical cases is descr…
Invariants of links with flat connections in their complements.II. Holonomy R-matrices related to quantized universal enveloping algebras at roots of 1
The algebras $Q_{n,k}(E,τ)$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $n>k\ge 1$, a complex elliptic curve $E$, and a point $τ\in E$. The main result in this paper is that $Q_{n,k}(E,τ)$ has the same Hilbert series as the polynomial ring on $n$ variables when $τ$ is not a torsion point. We also show that $Q_{n,k}(E,τ)$ is a Koszul algebra, hence of global dimension $n$ when $τ$ is not a torsion point, and, for all but countably many $τ$, it is Artin-Schelter regular. The proof…
We give necessary and sufficient conditions for a weight system on multiloop chord diagrams to be obtainable from a metrized Lie algebra representation, in terms of a bound on the ranks of associated connection matrices. Here a multiloop chord diagram is a graph with directed and undirected edges so that at each vertex precisely one directed edge is entering and precisely one directed edge is leaving, and each vertex is incident with precisely one undirected edge. Weight systems on multiloop chord diagrams yield the Vassiliev invariants for knots and links. The $k$-th connection matrix of a fu…
We construct a twisted version of the genus one universal Knizhnik-Zamolodchikov-Bernard (KZB) connection introduced by Calaque-Enriquez-Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the moduli space of $Γ$-structured elliptic curves with marked points, where $Γ=\mathbb{Z}/M\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$, and $M,N\geq1$ are two integers. It restricts to a flat connection on $Γ$-twisted configuration spaces of points on elliptic curves, which can be used to construct a filtered-formality isomorphism for some interesting subg…
