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Register a new userSome algebras having relations like those for the 4-dimensional Sklyanin algebras
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The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted A(E,tau), that depend on a quartic elliptic curve E in P^3 and a translation automorphism tau of E. They are graded algebras generated by four degree-one elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates apart from the minor difference that they are not commutative. They are elliptic analogues of the enveloping algebra of sl(2,C) and the quantized enveloping algebras U_q(gl_2). Recently, Cho, Hong, and Lau, conjectured that a certain 2-parameter family of algebras arising in their work on homological mirror symmetry consists of 4-dimensional Sklyanin algebras. This paper shows their conjecture is false in the generality they make it. On the positive side, we show their algebras exhibit features that are similar to, and differ from, analogous features of the 4-dimensional Sklyanin algebras in interesting ways. We show that most of the Cho-Hong-Lau algebras determine, and are determined by the graph of a bijection between two 20-point subsets of the projective space P^3. The paper also examines a 3-parameter family of 4-generator 6-relator algebras admitting presentations analogous to those of the 4-dimensional Sklyanin algebras. This class includes the 4-dimensional Sklyanin algebras and most of the Cho-Hong-Lau algebras.
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In 1982 E.K. Sklyanin defined a family of graded algebras $A(E,tau)$, depending on an elliptic curve $E$ and a point $tau in E$ that is not 4-torsion. The present paper is concerned with the structure of $A$ when $tau$ is a point of finite order, $n$ say. It is proved that every simple $A$-module has dimension $le n$ and that almost all have dimension precisely $n$. There are enough finite dimensional simple modules to separate elements of $A$; that is, if $0 e a in A$, then there exists a simple module $S$ such that $a.S e 0.$ Consequently $A$ satisfies a polynomial identity of degree $2n$ (and none of lower degree). Combined with results of Levasseur and Stafford it follows that $A$ is a finite module over its center. Therefore one may associate to $A$ a coherent sheaf, ${mathcal A}$ say, of finite ${mathcal O}_S$ algebras where $S$ is the projective 3-fold determined by the center of $A$. We determine where ${mathcal A}$ is Azumaya, and prove that the division algebra ${rm Fract}({mathcal A})$ has rational center. Thus, for each $E$ and each $tau in E$ of order $n e 0,2,4$ one obtains a division algebra of degree $s$ over the rational function field of ${mathbb P}^3$, where $s=n$ if $n$ is odd, and $s={{1} over {2}} n$ if $n$ is even. The main technical tool in the paper is the notion of a fat point introduced by M. Artin. A key preliminary result is the classification of the fat points: these are parametrized by a rational 3-fold.
In this paper, we compute all possible differential structures of a $3$-dimensional DG Sklyanin algebra $mathcal{A}$, which is a connected cochain DG algebra whose underlying graded algebra $mathcal{A}^{#}$ is a $3$-dimensional Sklyanin algebra $S_{a,b,c}$. We show that there are three major cases depending on the parameters $a,b,c$ of the underlying Sklyanin algebra $S_{a,b,c}$: (1) either $a^2 eq b^2$ or $c eq 0$, then $partial_{mathcal{A}}=0$; (2) $a=-b$ and $c=0$, then the $3$-dimensional DG Sklyanin algebra is actually a DG polynomial algebra; and (3) $a=b$ and $c=0$, then the DG Sklyanin algebra is uniquely determined by a $3times 3$ matrix $M$. It is worthy to point out that case (2) has been systematically studied in cite{MGYC} and case (3) is just the DG algebra $mathcal{A}_{mathcal{O}_{-1}(k^3)}(M)$ in cite{MWZ}. We solve the problem on how to judge whether a given $3$-dimensional DG Sklyanin algebra is Calabi-Yau.
The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperbergs 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaevs invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.
S-Heun operators on linear and $q$-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The Continuous Hahn and Big $q$-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Millers structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and $q$-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators
In this paper we introduce a trace-like invariant for the irreducible representations of a finite dimensional complex Hopf algebra H. We do so by considering the trace of the map induced by the antipode S on the endomorphisms End(V) of a self-dual module V. We also compute the values of this trace for the representations of two non-semisimple Hopf algebras: u_q(sl_2) and D(H_n(q)), the Drinfeld double of the Taft algebra.
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