The Experts below are selected from a list of 252 Experts worldwide ranked by ideXlab platform
Markus Linckelmann - One of the best experts on this subject based on the ideXlab platform.
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block algebras with hh1 a simple lie algebra
Quarterly Journal of Mathematics, 2018Co-Authors: Markus Linckelmann, Lleonard Rubio Y DegrassiAbstract:The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH1(B) is a simple Lie algebra and such that B has a unique Isomorphism Class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH1(B) is in that case isomorphic to the Witt algebra HH1(kP). In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single Isomorphism Class of simple modules.
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on blocks of defect two and one simple module and lie algebra structure of hh1
Journal of Pure and Applied Algebra, 2017Co-Authors: David J Benson, Radha Kessar, Markus LinckelmannAbstract:Let k be a field of odd prime characteristic p. We calculate the Lie algebra structure of the first Hochschild cohomology of a Class of quantum complete intersections over k. As a consequence, we prove that if B is a defect 2-block of a finite group algebra $kG$ whose Brauer correspondent C has a unique Isomorphism Class of simple modules, then a basic algebra of B is a local algebra which can be generated by at most 2√I elements, where I is the inertial index of B, and where we assume that k is a splitting field for B and C.
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block algebras with hh1 a simple lie algebra
arXiv: Representation Theory, 2016Co-Authors: Markus Linckelmann, Lleonard Rubio Y DegrassiAbstract:We show that if $B$ is a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ such that $\HH^1(B)$ is a simple Lie algebra and such that $B$ has a unique Isomorphism Class of simple modules, then $B$ is nilpotent with an elementary abelian defect group $P$ of order at least $3$, and $\HH^1(B)$ is in that case isomorphic to the Jacobson-Witt algebra $\HH^1(kP)$. In particular, no other simple modular Lie algebras arise as $\HH^1(B)$ of a block $B$ with a single Isomorphism Class of simple modules.
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on blocks of defect two and one simple module and lie algebra structure of hh 1
arXiv: Representation Theory, 2016Co-Authors: David J Benson, Radha Kessar, Markus LinckelmannAbstract:Let $k$ be a field of odd prime characteristic $p$. We calculate the Lie algebra structure of the first Hochschild cohomology of a Class of quantum complete intersections over $k$. As a consequence, we prove that if $B$ is a defect $2$-block of a finite group algebra $kG$ whose Brauer correspondent $C$ has a unique Isomorphism Class of simple modules, then a basic algebra of $B$ is a local algebra which can be generated by at most $2\sqrt I$ elements, where $I$ is the inertial index of $B$, and where we assume that $k$ is a splitting field for $B$ and $C$.
Jean Saint-raymond - One of the best experts on this subject based on the ideXlab platform.
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Descriptive complexity of some Isomorphism Classes of Banach spaces
Journal of Functional Analysis, 2018Co-Authors: Gilles Godefroy, Jean Saint-raymondAbstract:Abstract We present a topological frame in which it is possible to estimate the complexity of some Borel families of separable Banach spaces. We use this frame for evaluating the complexity of the Isomorphism Class of the Hilbert space l 2 , of certain asymptotically Hilbertian spaces, and of the l p spaces under the condition 1 p ∞ . The complexity appears to increase with the distance to the Hilbert space. Commented open problems conclude the article.
Lleonard Rubio Y Degrassi - One of the best experts on this subject based on the ideXlab platform.
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block algebras with hh1 a simple lie algebra
Quarterly Journal of Mathematics, 2018Co-Authors: Markus Linckelmann, Lleonard Rubio Y DegrassiAbstract:The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH1(B) is a simple Lie algebra and such that B has a unique Isomorphism Class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH1(B) is in that case isomorphic to the Witt algebra HH1(kP). In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single Isomorphism Class of simple modules.
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block algebras with hh1 a simple lie algebra
arXiv: Representation Theory, 2016Co-Authors: Markus Linckelmann, Lleonard Rubio Y DegrassiAbstract:We show that if $B$ is a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ such that $\HH^1(B)$ is a simple Lie algebra and such that $B$ has a unique Isomorphism Class of simple modules, then $B$ is nilpotent with an elementary abelian defect group $P$ of order at least $3$, and $\HH^1(B)$ is in that case isomorphic to the Jacobson-Witt algebra $\HH^1(kP)$. In particular, no other simple modular Lie algebras arise as $\HH^1(B)$ of a block $B$ with a single Isomorphism Class of simple modules.
Will Brian - One of the best experts on this subject based on the ideXlab platform.
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The Isomorphism Class of the shift map
Topology and its Applications, 2020Co-Authors: Will BrianAbstract:Abstract The shift map, denoted σ, is the self-homeomorphism of induced by the successor function n ↦ n + 1 on ω. We prove that the Isomorphism Classes of σ and σ − 1 cannot be separated by a Borel set in H ( ω ⁎ ) , the space of all self-homeomorphisms of ω ⁎ equipped with the compact-open topology. Van Douwen proved it is consistent for σ and σ − 1 not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while σ and σ − 1 may not be isomorphic, there is no simple topological property that distinguishes them. As a relatively straightforward consequence of the main theorem, we deduce that OCA + MA implies the set of continuous images of σ fails to be Borel in H ( ω ⁎ ) . (Here a “continuous image” of σ is meant in the sense of topological dynamics: any h ∈ H ( ω ⁎ ) such that q ∘ σ = h ∘ q for some continuous surjection q : ω ⁎ → ω ⁎ .) This contrasts starkly with a recent theorem of the author showing that under CH , the continuous images of σ form a closed subset of H ( ω ⁎ ) .
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The Isomorphism Class of the shift map
arXiv: General Topology, 2019Co-Authors: Will BrianAbstract:The \emph{shift map} $\sigma$ is the self-homeomorphism of $\omega^* = \beta\omega \setminus \omega$ induced by the successor function $n \mapsto n+1$ on $\omega$. We prove that the Isomorphism Classes of $\sigma$ and $\sigma^{-1}$ cannot be separated by a Borel set in $\mathcal H(\omega^*)$, the space of all self-homeomorphisms of $\omega^*$ equipped with the compact-open topology. Van Douwen proved it is consistent for $\sigma$ and $\sigma^{-1}$ not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while $\sigma$ and $\sigma^{-1}$ may not be isomorphic, there is no simple topological property that distinguishes them. As a relatively straightforward consequence of the main theorem, we deduce that $\mathsf{OCA}+\mathsf{MA}$ implies the set of continuous images of $\sigma$ fails to be Borel in $\mathcal H(\omega^*)$. (Here a ``continuous image'' of $\sigma$ is meant in the sense of topological dynamics: any $h \in \mathcal H(\omega^*)$ such that $q \circ \sigma = h \circ q$ for some continuous surjection $q: \omega^* \to \omega^*$.) This contrasts starkly with a recent theorem of the author showing that under $\mathsf{CH}$, the continuous images of $\sigma$ form a closed subset of $\mathcal H(\omega^*)$.
Daniel Barlet - One of the best experts on this subject based on the ideXlab platform.
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Holomorphic families of $[\lambda]-$primitive themes
2015Co-Authors: Daniel BarletAbstract:This article is the continuation of [B. 13-b] where we show how the Isomorphism Class of a $[\lambda]-$primitive theme with a given Bernstein polynomial may be characterized by a (small) finite number of complex parameters. We construct here a corresponding locally versal holomorphic deformation of $ [\lambda]-$primitive themes for each given Bernstein polynomial. Then we prove the universality of the corresponding ``canonical family'' in many cases. We also give some examples where no local universal family exists.
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Asymptotics of a vanishing period : General existence theorem and basic properties of frescos
2012Co-Authors: Daniel BarletAbstract:In this paper we introduce the word "fresco" to denote a \ $[\lambda]-$primitive monogenic geometric (a,b)-module. The study of this "basic object" (generalized Brieskorn module with one generator) which corresponds to the minimal filtered (regular) differential equation satisfied by a relative de Rham cohomology Class, began in [B.09] where the first structure theorems are proved. Then in [B.10] we introduced the notion of theme which corresponds in the \ $[\lambda]-$primitive case to frescos having a unique Jordan-H{ö}lder sequence. Themes correspond to asymptotic expansion of a given vanishing period, so to the image of a fresco in the module of asymptotic expansions. For a fixed relative de Rham cohomology Class (for instance given by a smooth differential form $d-$closed and $df-$closed) each choice of a vanishing cycle in the spectral eigenspace of the monodromy for the eigenvalue \ $exp(2i\pi.\lambda)$ \ produces a \ $[\lambda]-$primitive theme, which is a quotient of the fresco associated to the given relative de Rham Class itself. \\ The first part of this paper shows that, for any \ $[\lambda]-$primitive fresco there exists an unique Jordan-H{ö}lder sequence (called the principal J-H. sequence) with corresponding quotients giving the opposite of the roots of the Bernstein polynomial in a non decreasing order. Then we introduce and study the semi-simple part of a given fresco and we characterize the semi-simplicity of a fresco by the fact for any given order of the roots of its Bernstein polynomial we may find a J-H. sequence making them appear with this order. Then, using the parameter associated to a rank \ $2$ \ \ $[\lambda]-$primitive theme, we introduce inductiveley a numerical invariant, that we call the \ $\alpha-$invariant, which depends polynomially on the Isomorphism Class of a fresco (in a sens which has to be defined) and which allows to give an inductive way to produce a sub-quotient rank \ $2$ \ theme of a given \ $[\lambda]-$primitive fresco assuming non semi-simplicity.\\ In the last section we prove a general existence result which naturally associate a fresco to any relative de Rham cohomology Class of a proper holomorphic function of a complex manifold onto a disc. This is, of course, the motivation for the study of frescos.
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Changements de variable pour un th`eme.
2010Co-Authors: Daniel BarletAbstract:We study the behaviour of the notion of "thema", introduced in our previous article [B.09b], by a change of variable. We show not only that the fundamental invariants of such a thema, corresponding to the Bernstein polynomial, are stable by a change of variable, but also other numerical invariants called principal parameters. \\ We show on a rank 3 example that nevertheless the Isomorphism Class of a thema is not stable in general by a change of variable. We conclude in proving that a change of variable transforms an holomorphic family of thema in an holomorphic family. This implies that non principal parameters change holomorphically.
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Two finiteness theorem for $(a,b)$-modules
2008Co-Authors: Daniel BarletAbstract:1. For a proper holomorphic function \ $ f : X \to D$ \ of a complex manifold \ $X$ \ on a disc such that \ $\{ df = 0 \} \subset f^{-1}(0)$, we construct, in a functorial way, for each integer \ $p$, a geometric (a,b)-module \ $E^p$ \ associated to the (filtered) Gauss-Manin connexion of \ $f$.\\ This first theorem is an existence/finiteness result which shows that geometric (a,b)-modules may be used in global situations. 2. For any regular (a,b)-module \ $E$ \ we give an integer \ $N(E)$, explicitely given from simple invariants of \ $E$, such that the Isomorphism Class of \ $E\big/b^{N(E)}.E$ \ determines the Isomorphism Class of \ $E$.\\ This second result allows to cut asymptotic expansions (in powers of \ $b$) \ of elements of \ $E$ \ without loosing any information.
