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Register a new userFeigin and Odesskiis elliptic algebras
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Alexandru Chirv\\u{a}situ L.
Publication date
2018
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English
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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 $Q_{n,k}(E,tau)$.
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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.
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.
We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a cycle, we show that the center is trivial, and hence the Calabi-Yau structure is unique. If the quiver is a cycle, we show that the algebra is a non-commutative crepant resolution of its center, the ring of functions on the corresponding multiplicative quiver variety with a type A surface singularity. We also prove that the
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