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110
Representation Theory of Artin Algebras
 Studies in Advanced Mathematics
, 1994
"... The representation theory of artin algebras, as we understand it today, is a relatively new area of mathematics, as most of the main developments have occurred ..."
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Cited by 644 (10 self)
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The representation theory of artin algebras, as we understand it today, is a relatively new area of mathematics, as most of the main developments have occurred
Finite Quantum Groupoids and Their Applications
"... Abstract. We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3manifolds. Contents ..."
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Cited by 56 (5 self)
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Abstract. We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3manifolds. Contents
Perfect SpaceTime Codes for Any Number of Antennas
"... In a recent paper, perfect (n × n) spacetime codes were introduced as the class of linear dispersion spacetime codes having full rate, nonvanishing determinant, a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions and uniform average transmitted en ..."
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Cited by 37 (3 self)
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In a recent paper, perfect (n × n) spacetime codes were introduced as the class of linear dispersion spacetime codes having full rate, nonvanishing determinant, a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions and uniform average transmitted energy per antenna. Consequence of these conditions include optimality of perfect codes with respect to the ZhengTse DiversityMultiplexing Gain tradeoff (DMT), as well as excellent lowSNR performance. Yet perfect spacetime codes have been constructed only for 2, 3, 4 and 6 transmit antennas. In this paper, we construct perfect codes for all channel dimensions, present some additional attributes of this class of spacetime codes and extend the notion of a perfect code to the rectangular case.
The classification problem for torsionfree abelian groups of finite rank
 J. Amer. Math. Soc
, 2001
"... In 1937, Baer [5] introduced the notion of the type of an element in a torsionfree abelian group and showed that this notion provided a complete invariant for the classification problem for torsionfree abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] ..."
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Cited by 32 (10 self)
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In 1937, Baer [5] introduced the notion of the type of an element in a torsionfree abelian group and showed that this notion provided a complete invariant for the classification problem for torsionfree abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory
Dynamical quantum groups at roots of 1
 DMITRI NIKSHYCH AND LEONID VAINERMAN
"... Given a dynamical twist for a finitedimensional Hopf algebra, we construct two weak Hopf algebras, using methods of P. Xu and of P. Etingof and A. Varchenko, and we show that they are dual to each other. We generalize the theory of dynamical quantum groups to the case when the quantum parameter q i ..."
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Cited by 28 (7 self)
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Given a dynamical twist for a finitedimensional Hopf algebra, we construct two weak Hopf algebras, using methods of P. Xu and of P. Etingof and A. Varchenko, and we show that they are dual to each other. We generalize the theory of dynamical quantum groups to the case when the quantum parameter q is a root of unity. These objects turn out to be selfdual—which is a fundamentally new property, not satisfied by the usual DrinfeldJimbo quantum groups.
Perfect spacetime codes with minimum and nonminimum delay for any number of antennas
 IEEE Trans. Inform. Theory
, 2005
"... Abstract — Perfect spacetime codes were first introduced by Oggier et. al. to be the spacetime codes that have full rate, full diversitygain, nonvanishing determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping of the constellation. These d ..."
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Cited by 26 (8 self)
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Abstract — Perfect spacetime codes were first introduced by Oggier et. al. to be the spacetime codes that have full rate, full diversitygain, nonvanishing determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping of the constellation. These defining conditions jointly correspond to optimality with respect to the ZhengTse DMG tradeoff, independent of channel statistics, as well as to near optimality in maximizing mutual information. All the above traits endow the code with error performance that is currently unmatched. Yet perfect spacetime codes have been constructed only for 2, 3,4 and 6 transmit antennas. We construct minimum and nonminimum delay perfect codes for all channel dimensions. A. Definition of Perfect Codes I.
Galois module structure of pthpower classes of extensions of degree p
 Israel J. Math
"... Abstract. In the mid1960s Borevič and Faddeev initiated the study of the Galois module structure of groups of pthpower classes of cyclic extensions K/F of pthpower degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois m ..."
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Cited by 25 (16 self)
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Abstract. In the mid1960s Borevič and Faddeev initiated the study of the Galois module structure of groups of pthpower classes of cyclic extensions K/F of pthpower degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F. In 1947 ˇ Safarevič initiated the study of Galois groups of maximal pextensions of fields with the case of local fields [12], and this study has grown into what is both an elegant theory as well as an efficient tool in the arithmetic of fields. From the very beginning it became clear that the groups of pthpower classes of the various field extensions of a base field encode basic information about the structure of the Galois groups of maximal pextensions. (See [7] and [13].) Such groups of pthpower classes arise naturally in studies in arithmetic algebraic geometry, for example in the study of elliptic curves. In 1960 Faddeev began to study the Galois module structure of pthpower classes of cyclic pextensions, again in the case of local fields, and during the mid1960s he and Borevič established the structure of these Galois modules using basic arithmetic invariants attached to Galois extensions. (See [6] and [4].) In 2003 two of the authors ascertained the Galois module structure of pthpower classes in the case of cyclic extensions of degree p over all base fields F containing a primitive pth root of unity [9]. Very recently, this work paved the way for the determination of the entire Galois cohomology as a Galois module in
Coordinate algebras of extended affine Lie algebras of type A1, J. of Algebra 234
, 2000
"... Abstract. The cores of extended affine Lie algebras of reduced types were classified except for type A1. In this paper we determine the coordinate algebra of extended affine Lie algebras of type A1. It turns out that such an algebra is a unital Zngraded Jordan algebra of a certain type, called a Jo ..."
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Cited by 22 (7 self)
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Abstract. The cores of extended affine Lie algebras of reduced types were classified except for type A1. In this paper we determine the coordinate algebra of extended affine Lie algebras of type A1. It turns out that such an algebra is a unital Zngraded Jordan algebra of a certain type, called a Jordan torus. We classify Jordan tori and get five types of Jordan tori.
On twisting of finitedimensional Hopf algebras
 J. Algebra
"... Abstract. In this paper we study the properties of Drinfeld’s twisting for finitedimensional Hopf algebras. We determine how the integral of the dual to a unimodular Hopf algebra H changes under twisting of H. We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimpl ..."
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Cited by 18 (5 self)
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Abstract. In this paper we study the properties of Drinfeld’s twisting for finitedimensional Hopf algebras. We determine how the integral of the dual to a unimodular Hopf algebra H changes under twisting of H. We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple quasitriangular finitedimensional Hopf algebras are stable under twisting. We also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular Hopf algebra by two different twists on two sides, and describe the representation theory of its dual. Next, we define the notion of a nondegenerate twist for a Hopf algebra H, and set up a bijection between such twists for H and H ∗ if both are semisimple. This bijection generalizes to the noncommutative case the procedure of inverting a nondegenerate bilinear form on a vector space. Finally, we apply these results to classification of twists in group algebras and of cosemisimple triangular finitedimensional Hopf algebras in positive characteristic, generalizing the previously known classification in characteristic zero. 1.