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Register a new userElliptic R-matrices and Feigin and Odesskiis elliptic algebras
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The algebras $Q_{n,k}(E,tau)$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $n>kge 1$, a complex elliptic curve $E$, and a point $tauin E$. The main result in this paper is that $Q_{n,k}(E,tau)$ has the same Hilbert series as the polynomial ring on $n$ variables when $tau$ is not a torsion point. We also show that $Q_{n,k}(E,tau)$ is a Koszul algebra, hence of global dimension $n$ when $tau$ is not a torsion point, and, for all but countably many $tau$, it is Artin-Schelter regular. The proofs use the fact that the space of quadratic relations defining $Q_{n,k}(E,tau)$ is the image of an operator $R_{tau}(tau)$ that belongs to a family of operators $R_{tau}(z):mathbb{C}^notimesmathbb{C}^ntomathbb{C}^notimesmathbb{C}^n$, $zinmathbb{C}$, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.
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We study the elliptic algebras $Q_{n,k}(E,tau)$ introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers $n>kgeq 1$, an elliptic curve $E$, and a point $tauin E$. We consider and compare several different definitions of the algebras and provide proofs of various statements about them made by Feigin and Odesskii. For example, we show that $Q_{n,k}(E,0)$, and $Q_{n,n-1}(E,tau)$ are polynomial rings on $n$ variables. We also show that $Q_{n,k}(E,tau+zeta)$ is a twist of $Q_{n,k}(E,tau)$ when $zeta$ is an $n$-torsion point. This paper is the first of several we are writing about the algebras $Q_{n,k}(E,tau)$.
This paper examines an algebraic variety that controls an important part of the structure and representation theory of the algebra $Q_{n,k}(E,tau)$ introduced by Feigin and Odesskii. The $Q_{n,k}(E,tau)$s are a family of quadratic algebras depending on a pair of coprime integers $n>kge 1$, an elliptic curve $E$, and a point $tauin E$. It is already known that the structure and representation theory of $Q_{n,1}(E,tau)$ is controlled by the geometry associated to $E$ embedded as a degree $n$ normal curve in the projective space $mathbb P^{n-1}$, and by the way in which the translation automorphism $zmapsto z+tau$ interacts with that geometry. For $kge 2$ a similar phenomenon occurs: $(E,tau)$ is replaced by $(X_{n/k},sigma)$ where $X_{n/k}subseteqmathbb P^{n-1}$ is the characteristic variety of the title and $sigma$ is an automorphism of it that is determined by the negative continued fraction for $frac{n}{k}$. There is a surjective morphism $Phi:E^g to X_{n/k}$ where $g$ is the length of that continued fraction. The main result in this paper is that $X_{n/k}$ is a quotient of $E^g$ by the action of an explicit finite group. We also prove some assertions made by Feigin and Odesskii. The morphism $Phi$ is the natural one associated to a particular invertible sheaf $mathcal L_{n/k}$ on $E^g$. The generalized Fourier-Mukai transform associated to $mathcal L_{n/k}$ sends the set of isomorphism classes of degree-zero invertible $mathcal O_E$-modules to the set of isomorphism classes of indecomposable locally free $mathcal O_E$-modules of rank $k$ and degree $n$. Thus $X_{n/k}$ has an importance independent of the role it plays in relation to $Q_{n,k}(E,tau)$. The backward $sigma$-orbit of each point on $X_{n/k}$ determines a point module for $Q_{n,k}(E,tau)$.
Fix a pair of relatively prime integers $n>kge 1$, and a point $(eta , | , tau) in mathbb{C} times mathbb{H}$, where $mathbb{H}$ denotes the upper-half complex plane, and let ${{a ; ,b} choose {c , ; d}} in mathrm{SL}(2,mathbb{Z})$. We show that Feigin and Odesskiis elliptic algebras $Q_{n,k}(eta , | , tau)$ have the property $Q_{n,k} big( frac{eta}{ctau+d} ,bigvert , frac{atau+b}{ctau+d} big) cong Q_{n,k}(eta , | , tau)$. As a consequence, given a pair $(E,xi)$ consisting of a complex elliptic curve $E$ and a point $xi in E$, one may unambiguously define $Q_{n,k}(E,xi):=Q_{n,k}(eta , | , tau)$ where $tau in mathbb{H}$ is any point such that $mathbb{C}/mathbb{Z}+mathbb{Z}tau cong E$ and $eta in mathbb{C}$ is any point whose image in $E$ is $xi$. This justifies Feigin and Odesskiis notation $Q_{n,k}(E,xi)$ for their algebras.
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 $n>kge 1$, an elliptic curve $E$, and a point $tauin E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,tau)$ to the twisted homogeneous coordinate ring $B(X_{n/k},sigma,mathcal{L}_{n/k})$ 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 the homomorphism $Q_{n,k}(E,tau) to B(X_{n/k},sigma,mathcal{L}_{n/k})$ is surjective, that the relations for $B(X_{n/k},sigma,mathcal{L}_{n/k})$ are generated in degrees $le 3$, and the non-commutative 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 $B(X_{n/k},sigma,mathcal{L}_{n/k})$ 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.
The quiver Yangian, an infinite-dimensional algebra introduced recently in arXiv:2003.08909, is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $mathcal{N}=2$ supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.
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