Showing papers on "Centralizer and normalizer published in 2009"
TL;DR: In this paper, effective equidistribution for large closed orbits of semisimple groups on homogeneous spaces was proved with polynomial rate for the special case where the acting group should have a finite centralizer in the ambient group.
Abstract: We prove effective equidistribution, with polynomial rate, for large closed orbits of semisimple groups on homogeneous spaces, under certain technical restrictions (notably, the acting group should have finite centralizer in the ambient group). The proofs make extensive use of spectral gaps, and also of a closing lemma for such actions.
78 citations
TL;DR: In this article, the exact value of the essential dimension of the maximal torus in the projective linear group PGLn for every n ≥ 1 has been derived, where p is a prime, k is a field of characteristic 6 = p and N is the normalizer of the torus.
Abstract: Let p be a prime, k be a field of characteristic 6= p and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential dimension edk(N ; p) of N at p for every n ≥ 1.
41 citations
TL;DR: For partially commutative metabelian groups, the centralizers of elements of commutator subgroups are described and canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved as mentioned in this paper.
Abstract: For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory.
41 citations
TL;DR: In this article, it was shown that the space of C 1 diffeomorphisms of a compact manifold contains a residual subset of diffEomorphisms whose centralizers are trivial.
Abstract: Answering a question of Smale, we prove that the space of C 1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.
35 citations
TL;DR: In this paper, an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero is given.
Abstract: We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.
34 citations
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TL;DR: In this article, a generalized flag manifold is defined as a homogeneous space of the form G/K, where G is the centralizer of a torus in a compact connected semisimple Lie group.
Abstract: A generalized flag manifold is a homogeneous space of the form $G/K$, where $K$ is the centralizer of a torus in a compact connected semisimple Lie group $G$. We classify all flag manifolds with four isotropy summands and we study their geometry. We present new $G$-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics.
33 citations
TL;DR: In this article, it was shown that OutcðG Þ¼ 1 for the members G of certain classes of metabelian groups have the normalizer property, from which the Herman-Li result follows, and the main result implies that, for an arbitrary group G, the group ZyðUÞ of hypercentral units of U is contained in ZðU ÞG.
Abstract: For a group G, let U be the group of units of the integral group ring ZG. The group G is said to have the normalizer property if NUðG Þ¼ ZðUÞG. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being non-trivial. Groups G for which class- preserving automorphisms are inner automorphisms, OutcðG Þ¼ 1, have the normalizer prop- erty. Recently, Herman and Li have shown that OutcðG Þ¼ 1 for a finite Blackburn group G. We show that OutcðG Þ¼ 1 for the members G of certain classes of metabelian groups, from which the Herman-Li result follows. Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group G, the group ZyðUÞ of hypercentral units of U is contained in ZðUÞG.
31 citations
TL;DR: In this paper, the authors define the 0-Hecke-group algebra as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations, which is closely related to the monoid algebras of respectively nondecreasing functions and parking functions.
31 citations
Journal Article•
TL;DR: In this article, it was shown that there is a natural set of maps jb : Ib? I which enjoy affine, H-equivariant, map any apartment of Ib into an apartment of I and are compatible with the Lie algebra filtrations of g and h.
Abstract: Let Fo be a non-archimedean locally compact field of residual characteristic not 2. Let G be a classical group over Fo (with no quaternionic algebra involved) which is not of type An for n > 1. Let b be an element of the Lie algebra g of G that we assume semisimple for simplicity. Let H be the centralizer of b in G and h its Lie algebra. Let I and Ib denote the (enlarged) Bruhat-Tits buildings of G and H respectively. We prove that there is a natural set of maps jb : Ib ? I which enjoy the following properties: they are affine, H-equivariant, map any apartment of Ib into an apartment of I and are compatible with the Lie algebra filtrations of g and h. In a particular case, where this set is reduced to one element, we prove that jb is characterized by the last property in the list. We also prove a similar characterization result for the general linear group.
27 citations
TL;DR: In this paper, the essential dimension of a connected linear algebraic group is bounded from below by rank(H) 0, where rank denotes the rank of the maximal torus in the centralizer CG(H).
Abstract: Let G be a connected linear algebraic group defined over an algebraically closed field k and H be a finite abelian subgroup of G whose order does not divide char(k). We show that the essential dimension of G is bounded from below by rank(H) - rank?CG(H)0, where rank?CG(H)0 denotes the rank of the maximal torus in the centralizer CG(H). This inequality, conjectured by J.-P. Serre, generalizes previous results of Reichstein�Youssin (where char(k) is assumed to be 0 and CG(H) to be finite) and Chernousov�Serre (where H is assumed to be a 2-group).
26 citations
TL;DR: In this article, the exterior centralizer of an element x of a group G is introduced in order to improve some known results on the non-abelian tensor product of two groups.
Abstract: The notion of the exterior centralizer \({C_G^{^\wedge}(x)}\) of an element x of a group G is introduced in the present paper in order to improve some known results on the non-abelian tensor product of two groups. We study the structure of G by looking at that of \({C_G^{^\wedge}(x)}\) and we find some bounds for the Schur multiplier M(G) of G.
TL;DR: In this paper, it was shown that if the centralizer of a Banach space X is infinite-dimensional, then every nonempty relatively weakly open subset of the closed unit ball of X has diameter equal to 2.
Abstract: We prove that, if the centralizer of a Banach space X is infinite-dimensional, then every nonempty relatively weakly open subset of the closed unit ball of X has diameter equal to 2. This result, together with a suitable refinement also proven in the paper, contains (and improves in some cases) previously known facts for C
*-algebras, JB
*-triples, spaces of vector valued continuous functions, and spaces of operators.
TL;DR: In this article, it was shown that a group of finite $c$-dimension is soluble of derived length in terms of roughly εk, and that the rank of its quotient by the Hirsch-Plotkin radical is bounded in ε k.
Abstract: The $c$-dimension of a group is the maximum length of a chain of
nested centralizers. It is proved that a periodic locally soluble
group of finite $c$-dimension $k$ is soluble of derived length
bounded in terms of~$k$, and the rank of its quotient by the
Hirsch--Plotkin radical is bounded in terms of~$k$. Corollary: a
pseudo-(finite soluble) group of finite $c$-dimension $k$ is
soluble of derived length bounded in terms of~$k$.
TL;DR: In this paper, an algorithm to construct a maximal order Λ in a finite-dimensional semisimple rational algebra A is presented, where the discriminants of the simple components of Λ allow one to read off the Wedderburn structure of A.
TL;DR: In this paper, the centralizer of a homeomorphism of a disk D that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers is proved.
Abstract: We prove a global fixed point theorem for the centralizer of a homeomorphism of the two-dimensional disk D that has attractor–repeller dynamics on the boundary with at least two attractors and two repellers. As one application we give an elementary proof of Morita’s Theorem, that the mapping class group of a closed surface S of genus g does not lift to the group of C 2 diffeomorphisms of S and we improve the lower bound for g from 5 to 3 .
TL;DR: In this article, it was shown that the center of CyCy and the centre of CYCY are smooth group schemes over a field KK whose characteristic is very good for GG, and that the automorphism of Lie(C)Lie (C) determined by the differential of σσ at zero is a scalar multiple of the identity.
09 Sep 2009
TL;DR: In this article, the authors studied the centralizers of reductive algebraic groups and split Kac-Moody groups, outside of characteristic 2, and proved that the centralizer of a reductive automorphism is finitely generated.
Abstract: Goal of the present work is the study of involutory automorphisms and their centralizers of reductive algebraic groups and of split Kac-Moody groups, outside of characteristic 2 The groups in question have in common that they admit a so-called twin BN-pair (B_+, B_-, N) and an associated so-called twin building C=(C_+, C_-, δ^*) Let G be such a group An involutory automorphism θ of G for which θ(B_+) is conjugate to B_- induces an almost isometry of the building C which interchanges the halves of the building and which we also denote by θ This now enables us to apply the rich structure theory of buildings An important tool for this is the so-called flip-flop system C_θ, consisting of all chambers c of C_+ for which the distance between c and θ(c) is maximal (with respect to the codistance of the twin building) Since C_θ is a subsystem of the building C_+, we can also regard it as a simplicial complex The centralizer G_θ of θ in G acts naturally on this complex In the present work we give criteria under which C_θ is a connected and pure simplicial complex For this we reduce the global question to a problem in rank 2 We then solve this rank 2 problem for several important cases Additional, we study the orbit structure of G_Simplizialkomplex on the building C and the flip-flop system C_θ As applications, we obtain for example a parametrization of the double coset space G_θ\G/B_+; a generlization of the Iwasawa decomposition; and a proof that for certain locally-finite Kac-Moody groups, the centralizer C_θ is finitely generated In closing we would like to remark that our results also apply to further groups with a root group datum as defined by Tits (like eg finite groups of Lie-type)
TL;DR: In this article, it was proved that every automorphism of an elementary adjoint Chevalley group of type A fixme l¯¯¯¯, D fixme l¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯�, D¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯� l¯¯¯¯ and E¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯�l¯¯¯¯ over a local commutative ring with 1/2 is a composition of a ring automomorphism and conjugation by some matrix from the normalizer of that Chevalery group in GL(V) (V is an adjoint representation space).
Abstract: It is proved that every automorphism of an elementary adjoint Chevalley group of type A
l
, D
l
, or E
l
over a local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the normalizer of that Chevalley group in GL(V) (V is an adjoint representation space)
TL;DR: This work implements hereditary substitutions in a functional programming language with sized heterogeneous inductive types $\Fhat$, arriving at an interpreter whose termination can be tracked by the type system of its host programming language.
Abstract: In the simply typed λ-calculus, a hereditary substitution replaces a free variable in a normal form r by another normal form s of type a, removing freshly created redexes on the fly. It can be defined by lexicographic induction on a and r, thus giving rise to a structurally recursive normalizer for the simply typed λ-calculus. We implement hereditary substitutions in a functional programming language with sized heterogeneous inductive types $\Fhat$ , arriving at an interpreter whose termination can be tracked by the type system of its host programming language.
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TL;DR: In this article, the homology of the based loop group Ωmega K with integer coefficients is shown to be a Hopf Hopf algebra of algebraic functions on Borel subgroups of the Langlands dual group scheme.
Abstract: Let K be a connected compact Lie group, and G be its complexification. The homology of the based loop group \Omega K with integer coefficients is naturally a \ZZ-Hopf algebra. After possibly inverting 2 or 3, we identify H_*(\Omega K,\ZZ) with the Hopf algebra of algebraic functions on B^\vee_e, where B^\vee is a Borel subgroup of the Langlands dual group scheme G^\vee of G and B^\vee_e is the centralizer in B^\vee of a regular nilpotent element e\in\Lie B^\vee. We also give a similar interpretation for the equivariant homology of \Omega K under the maximal torus action.
TL;DR: In this paper, a complex parametrization of a dense open subset of a generalized flag manifold is presented, where the authors give explicit expressions for the differential $d\pi$ of $\pi$ and for the Berezin symbols of $d \pi (g)$ ($g\in G$) and $d ǫ (X)$ (X\in \mathfrak{g}$).
Abstract: Let $M=G/H$ be a generalized flag manifold where $G$ is a compact, connected, simply-connected Lie group with Lie algebra $\mathfrak{g}$ and $H$ is the centralizer of a torus. Let $\pi$ be a unitary irreducible representation of $G$ which is holomorphically induced from a character of $H$. Using a complex parametrization of a dense open subset of $M$, we realize $\pi$ on a Hilbert space of holomorphic functions. We give explicit expressions for the differential $d\pi$ of $\pi$ and for the Berezin symbols of $\pi (g)$ ($g\in G$) and $d\pi (X)$ ($X\in \mathfrak{g}$). In particular, we recover some results of S. Berceanu and we partially generalize a result of K. H. Neeb.
TL;DR: In this article, it was shown that any clone without virtual constants is isomorphic to the centralizer clone of a unary universal algebra, and that adding one unary relation to these unary algebras produces algebraic systems representing any clone.
Abstract: We show that any clone without virtual constants is isomorphic to the centralizer clone of a unary universal algebra, and that adding one unary relation to these unary algebras produces algebraic systems representing any clone.
TL;DR: In this paper, normalizers of an infinite index irreducible inclusion N ⊆ M of II 1 factors were studied and the semigroup of one-sided normalizers was investigated.
TL;DR: In this article, it was shown that if Λ is a codimension-one hyperbolic attractor for a diffeomorphism f, where 2 ≤ r ≤ ∞, and f is not Anosov, then there is a neighborhood U of f in Diffr(M) and an open and dense set V of U such that any g ∈ V has a trivial centralizer on the basin of attraction for Λ.
Abstract: We show that if Λ is a codimension-one hyperbolic attractor for a Cr diffeomorphism f , where 2 ≤ r ≤ ∞, and f is not Anosov, then there is a neighborhood U of f in Diffr(M) and an open and dense set V of U such that any g ∈ V has a trivial centralizer on the basin of attraction for Λ.
TL;DR: In this paper, it was shown that the fundamental group of a compact Lie group can be computed in terms of the normalizer of the maximal torus, which is known to be surjective.
Abstract: The notion of p-compact group [10] is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of p-compact groups, one for each prime number p. A key feature of the theory of compact Lie groups is the relationship between centers and fundamental groups; these play off against one another, at least in the semisimple case, in that the center of the simply connected form is the fundamental group of the adjoint form. There are explicit ways to compute the center or fundamental group of a compact Lie group in terms of the normalizer of the maximal torus [1, 5.47]. For some time there has in fact been a corresponding formula for the center of a p-compact group [11, 7.5], but in general the fundamental group has eluded analysis. The purpose of the present paper is to remedy this deficit. For any space Y , we let H Zp i (Y ) denotes lim nHi(Y ;Z/p ). Suppose that X is a connected p-compact group, with maximal torus T and torus normalizer NT [10, §8]. It is known that the map π1(T ) → π1(X) is surjective [12, 6.11] [21, 5.6], or equivalently that the map H Zp 2 (BT ) → H Zp 2 (BX) is surjective. We prove the following statement.
TL;DR: In this article, it was shown that if G is a nonabelian group with ∇(G) ≅ ∇ (S4(q)) then G ≅ S4(qs), where q is a prime power.
Abstract: Let G be a nonabelian group, and associate the noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S4(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ∇(G) ≅ ∇(S4(q)) then G ≅ S4(q).
TL;DR: In this article, a new family of gauge invariant non-local order parameters Δ α A for non-abelian discrete gauge theories on a Euclidean lattice, which are in one-to-one correspondence with the excitation spectrum, is presented.
TL;DR: In this paper, the authors give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra, which is used to obtain an explicit set of generators.
Abstract: The quantum group analog of the normalizer of SU(1, 1) in is an important and nontrivial example of a noncompact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of this article is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra . It turns out that does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analog of the Casimir operator characterized by an affiliation relation to a von Neumann algebra. This is used to obtain an explicit set of generators. Having the dual quantum group the left regular corepresentation of the quantum group analog of the normalizer of SU(1, 1) in is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to -representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations. The detailed analysis of this example involves the analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately. This article is split into two parts: the first part gives almost all of the statements and the results, and the statements of this part are independent of the second part. The second part contains the proofs of all the statements.
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TL;DR: In this paper, it was shown that if [Q,P]=1, then the centralizer of P is the polynomial algebra k[P], and if [P,Q] = 1, then [P] = 0, then
Abstract: Let P,Q be elements of the Weyl algebra W. We prove that if [Q,P]=1, then the centralizer of P is the polynomial algebra k[P].
TL;DR: In this article, it was shown that Broue's abelian defect group conjecture is true for the non-principal block of SL ( 2, p n ) for a positive integer n.