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56
Cohomological quotients and smashing localizations
 Amer. J. Math
"... Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the coh ..."
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Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic Ktheory and demonstrates the relevance of the telescope
The Realization of InputOutput Maps Using Bialgebras
, 1991
"... We use the theory of bialgebras to provide the algebraic background for state space realization theorems for inputoutput maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space realizations of nonlinear systems and more recent resul ..."
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Cited by 11 (7 self)
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We use the theory of bialgebras to provide the algebraic background for state space realization theorems for inputoutput maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space realizations of nonlinear systems and more recent results involving analysis related to families of trees. If H is a bialgebra, we say that p H # is di#erentially produced by the algebra R with the augmentation # if there is right Hmodule algebra structure on R and there exists f h).
Conway's Problem and the commutation of languages
 Bulletin of EATCS
, 2001
"... We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The se ..."
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Cited by 8 (5 self)
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We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those languages commuting with a given code – the conjecture is that the commutation with codes may be characterized as in free monoids. We present here simple proofs for the known results on these two problems. 1
Smarandache Rings
 MAGNA An international journal
, 2002
"... w. b. vasantha kandasamy smarandache rings a 5 a 3 a 8 a 7 a 16 a 14 a 18 a 1 a 0 a 2 american research press 2002 a 6 a 9 a 10 a 13 a 17 a 11 a 4 a 15 a 12 ..."
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w. b. vasantha kandasamy smarandache rings a 5 a 3 a 8 a 7 a 16 a 14 a 18 a 1 a 0 a 2 american research press 2002 a 6 a 9 a 10 a 13 a 17 a 11 a 4 a 15 a 12
ASSOCIATIVE CONFORMAL ALGEBRAS WITH FINITE FAITHFUL REPRESENTATION
, 2004
"... Abstract. We describe irreducible conformal subalgebras of CendN and build the structure theory of associative conformal algebras with finite faithful representation. 1. ..."
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Cited by 5 (5 self)
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Abstract. We describe irreducible conformal subalgebras of CendN and build the structure theory of associative conformal algebras with finite faithful representation. 1.
DIVISION ALGEBRAS AND NONCOMMENSURABLE ISOSPECTRAL MANIFOLDS
, 2005
"... Abstract. A. Reid [R] showed that if Γ1 and Γ2 are arithmetic lattices in G = PGL2(R) or in PGL2(C) which give rise to isospectral manifolds, then Γ1 and Γ2 are commensurable (after conjugation). We show that for d ≥ 3 and S = PGLd(R)/POd(R), or S = PGLd(C)/PUd(C), the situation is quite different: ..."
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Cited by 5 (1 self)
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Abstract. A. Reid [R] showed that if Γ1 and Γ2 are arithmetic lattices in G = PGL2(R) or in PGL2(C) which give rise to isospectral manifolds, then Γ1 and Γ2 are commensurable (after conjugation). We show that for d ≥ 3 and S = PGLd(R)/POd(R), or S = PGLd(C)/PUd(C), the situation is quite different: there are arbitrarily large finite families of isospectral noncommensurable compact manifolds covered by S. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants. 1.
Finitely generated subnormal subgroups of GLn(D) are central
 Journal of Algebra
, 2000
"... Let D be an infinite division algebra of finite dimension over its centre. Assume that N is a subnormal subgroup of GLn(D) with n ¡ that if N is finitely generated, then N is central. 1. It is shown Let D be an infinite division algebra of degree m over its centre Z(D) = F. Denote by D ¢ the commut ..."
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Let D be an infinite division algebra of finite dimension over its centre. Assume that N is a subnormal subgroup of GLn(D) with n ¡ that if N is finitely generated, then N is central. 1. It is shown Let D be an infinite division algebra of degree m over its centre Z(D) = F. Denote by D ¢ the commutator subgroup of the multiplicative group D = D − {0}. The aim of this note is to investigate the structure of finitely generated subnormal subgroups of GLn(D) with n £ 1. Assume that n £ 2 and N is a normal subgroup of GLn(D). It is shown in [1] that if N is finitely generated, then N is central. A similar result for finitely generated normal subgroups is obtained for the case n = 1 in [2]. Here we shall generalize some of the main results appeared in [1] and [2] to subnormal subgroups of GLn(D) with n £ 1. To be more precise, assume that N is a subnormal subgroup of GLn(D) with n £ 1. It is proved that if N is finitely generated, then N is central. Using this, it is also shown that GLn(D)/Z(GLn(D)) contains no nontrivial finitely generated subnormal subgroups. Furthermore, given an infinite subnormal subgroup N of GLn(D), it is proved that N contains no finitely generated maximal subgroups. Therefore, GLn(D) contains no finitely generated maximal subgroups. The reader may consult [5], [6], [7], and the references thereof for more recent results on multiplicative subgroups of GLn(D). For convenience we shall deal with the case n = 1 separately. Our key result is the following Theorem 1. Let D be a finite dimensional division algebra with centre F. Then any finitely generated subnormal subgroup of D is central. Proof. We first claim that if D contains a noncentral finitely generated subgroup N, which is subnormal, then F is finitely generated over its prime subfield.
Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generaliza ..."
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We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language  the largest set commuting with a language , of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago  asking whether or not the centralizer of any rational language is rational  in the case of periodic, binary, and ternary sets of words, as well as for rational ccodes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an ccode X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.
ISOMORPHISMS BETWEEN LEAVITT ALGEBRAS AND THEIR MATRIX RINGS
, 2008
"... Abstract. Let K be any field, let Ln denote the Leavitt algebra of type (1, n − 1) having coefficients in K, and let Md(Ln) denote the ring of d × d matrices over Ln. In our main result, we show that Md(Ln) ∼ = Ln if and only if d and n − 1 are coprime. We use this isomorphism to answer a question ..."
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Cited by 4 (3 self)
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Abstract. Let K be any field, let Ln denote the Leavitt algebra of type (1, n − 1) having coefficients in K, and let Md(Ln) denote the ring of d × d matrices over Ln. In our main result, we show that Md(Ln) ∼ = Ln if and only if d and n − 1 are coprime. We use this isomorphism to answer a question posed in [14] regarding isomorphisms between various C*algebras. Furthermore, our result demonstrates that data about the K0 structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple Kalgebras.
Associative algebras related to conformal algebras
 Appl. Categ. Structures
"... Abstract. In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TCalgebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we generally mean what is known as Hpseudoalgebra over t ..."
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Abstract. In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TCalgebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we generally mean what is known as Hpseudoalgebra over the polynomial Hopf algebra H = k[T1,..., Tn]. Some recent results in structure theory of conformal algebras are applied to get a description of TCalgebras. 1.