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Register a new userA simple algorithm for extending the identities for quantum minors to the multiparametric case
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Authors
Zoran v{S}koda
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For any homogeneous identity between $q$-minors, we provide an identity between $P,Q$-minors.
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We survey a vast array of known results and techniques in the area of polynomial identities in pointed Hopf algebras. Some new results are proven in the setting of Hopf algebras that appeared in papers of D. Radford and N. Andruskiewitsch - H.-J. Schneider.
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.
We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfeld twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $mathbb{Z}_2^m$. This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.
We derive a formula for the trace of the antipode on endomorphism algebras of simple self-dual modules of nilpotent liftings of quantum planes. We show that the trace is equal to the quantum dimension of the module up to a nonzero scalar depending on the simple module.
The Structure Theorem for Hopf modules states that if a bialgebra $H$ is a Hopf algebra (i.e. it is endowed with a so-called antipode) then every Hopf module $M$ is of the form ${M}^{mathrm{co}{H}}otimes H$, where ${M}^{mathrm{co}{H}}$ denotes the space of coinvariant elements in $M$. Actually, it has been shown that this result characterizes Hopf algebras: $H$ is a Hopf algebra if and only if every Hopf module $M$ can be decomposed in such a way. The main aim of this paper is to extend this characterization to the framework of quasi-bialgebras by introducing the notion of preantipode and by proving a Structure Theorem for quasi-Hopf bimodules. We will also establish the uniqueness of the preantipode and the closure of the family of quasi-bialgebras with preantipode under gauge transformation. Then, we will prove that every Hopf and quasi-Hopf algebra (i.e. a quasi-bialgebra with quasi-antipode) admits a preantipode and we will show how some previous results, as the Structure Theorem for Hopf modules, the Hausser-Nill theorem and the Bulacu-Caenepeel theorem for quasi-Hopf algebras, can be deduced from our Structure Theorem. Furthermore, we will investigate the relationship between the preantipode and the quasi-antipode and we will study a number of cases in which the two notions are equivalent: ordinary bialgebras endowed with trivial reassociator, commutative quasi-bialgebras, finite-dimensional quasi-bialgebras.
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