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Yuki Arano - One of the best experts on this subject based on the ideXlab platform.

Jose Manuel Gomez - One of the best experts on this subject based on the ideXlab platform.

  • twisted equivariant k theory of Compact Lie Group actions with maximal rank isotropy
    Journal of Mathematical Physics, 2018
    Co-Authors: Alejandro Adem, Jose Cantarero, Jose Manuel Gomez
    Abstract:

    We consider twisted equivariant K-theory for actions of a Compact Lie Group G on a space X where all the isotropy subGroups are connected and of maximal rank. We show that the associated rational spectral sequence a la Segal has a simple E2-term expressible as invariants under the Weyl Group of G. Specifically, if T is a maximal torus of G, they are invariants of the π1(XT)-equivariant Bredon cohomology of the universal cover of XT with suitable coefficients. In the case of the inertia stack ΛY, this term can be expressed using the cohomology of YT and algebraic invariants associated with the Lie Group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when Y = {*}.We consider twisted equivariant K-theory for actions of a Compact Lie Group G on a space X where all the isotropy subGroups are connected and of maximal rank. We show that the associated rational spectral sequence a la Segal has a simple E2-term expressible as invariants under the Weyl Group of G. Specifically, if T is a maximal torus of G, they are invariants of the π1(XT)-equivariant Bredon cohomology of the universal cover of XT with suitable coefficients. In the case of the inertia stack ΛY, this term can be expressed using the cohomology of YT and algebraic invariants associated with the Lie Group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when Y = {*}.

  • equivariant k theory of Compact Lie Group actions with maximal rank isotropy
    Journal of Topology, 2012
    Co-Authors: Alejandro Adem, Jose Manuel Gomez
    Abstract:

    We consider twisted equivariant K--theory for actions of a Compact Lie Group $G$ on a space $X$ where all the isotropy subGroups are connected and of maximal rank. We show that the associated rational spectral sequence \`a la Segal has a simple $E_2$--term expressible as invariants under the Weyl Group of $G$. Namely, if $T$ is a maximal torus of $G$, they are invariants of the $\pi_1(X^T)$-equivariant Bredon cohomology of the universal cover of $X^T$ with suitable coefficients. In the case of the inertia stack $\Lambda Y$ this term can be expressed using the cohomology of $Y^T$ and algebraic invariants associated to the Lie Group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when $Y=\{*\}$.

  • twisted equivariant k theory of Compact Lie Group actions with maximal rank isotropy
    arXiv: Algebraic Topology, 2012
    Co-Authors: Alejandro Adem, Jose Cantarero, Jose Manuel Gomez
    Abstract:

    Let G denote a Compact connected Lie Group with torsion-free fundamental Group acting on a Compact space X such that all the isotropy subGroups are connected subGroups of maximal rank. Let $T\subset G$ be a maximal torus with Weyl Group W. If the fixed-point set $X^T$ has the homotopy type of a finite W-CW complex, we prove that the rationalized complex equivariant K-theory of X is a free module over the representation ring of G. Given additional conditions on the W-action on the fixed-point set $X^T$ we show that the equivariant K-theory of X is free over R(G). We use this to provide computations for a number of examples, including the ordered n-tuples of commuting elements in G with the conjugation action.

Chun-nip Lee - One of the best experts on this subject based on the ideXlab platform.

  • Stable splittings of classifying spaces of Compact Lie Group
    Mathematische Zeitschrift, 1997
    Co-Authors: Chun-nip Lee
    Abstract:

    Let G be a Compact Lie gorup. In this paper, we study the stable splitting of BG completed at p into a wedge sum of indecomposable spectra. When G is finite, this question has been reduced to understanding the irreducible modular representations of the outer automorphism Group Out (Q) for various p-subGroups Q ⊆ G by work of [2], [14] and [18]. The principal tool used by these authors is a generalization of Segal’s Burnside ring conjecture which describes all stable maps between p-completions of the classifying spaces of p-Groups. One of the problems in going from finite Groups to Compact Lie Groups is that in the latter case one no longer has a convenient description of all stable maps between classifying spaces. Our solution to this difficulty is to pass from the ring of stable self-maps to the induced self-maps on Fp-homology. It is well-known that in terms of stable splittings, one does not lose any information by this process. There are two advantages to this approach. One is that a result of Henn [8] impLies that the ring of induced self-maps on the Fp-homology of BG is finite. Two is that one can now give a more explicit description of all the induced self-maps on Fp-homology for a large class of Compact Lie Groups which we call p-Roquette. The latter is exactly the class of Compact Lie Groups for which an appropriate density theorem is valid for a generalized Segal’s Burnside ring conjecture for Compact Lie Groups as shown by Minami [16] whose result built upon work of Feshbach on the original Segal’s Burnside ring conjecture for Compact Lie Groups [7]. In particular, every Compact Lie Group is p-Roquette if p is odd. With these reductions, one can follow a similar procedure for splitting BG∧ p as in the case when G is finite. Recall that a Compact Lie Group Q is said to be p-toral if it is an extension of a torus by a finite p-Group. The main result of this paper is that when G is p-Roquette, one can reduce the stable splitting of BG∧ p to the study

  • On stable summands of the classifying space of a Compact Lie Group
    Topology, 1994
    Co-Authors: Chun-nip Lee
    Abstract:

    LET G be a finite Group. Recent works of Benson and Feshbach [2] and Martin0 and Priddy [8,9] give a complete account of the question of stable splittings of the classifying space BG into indecomposable summands. This work is in part motivated by the work of Nishida [13] in which the notion of a dominant summand was first introduced. The purpose of this paper is to develop a framework for studying the analogous question for Compact Lie Groups. As in the case of finite Groups where a certain generalization of the affirmative solution to Segal’s Burnside ring conjecture was used, our work begin with a theorem of May, Snaith and Zelewski [lo] which describes the set of stable maps from BQ to BK where Q is finite and K is Compact Lie. If there were a similar description for Q Compact Lie, one could try to imitate the work of [2] and [8] and reduce the problem to one of representation theory. However, as shown by work of Feshbach [4] and later Bauer [l] on Segal’s Burnside ring conjecture for Compact Lie Groups, the situation when Q is not finite is much more intricate. The key ingredient in our investigation of the stable summands of BG when G is a Compact Lie Group is the utilization of certain Group-theoretic notions to analyze stable maps between classifying spaces of Compact Lie Groups. The end result is that we can define Group-theoretic invariants for stable summands of BG which are invariant under stable homotopy equivalence. This puts a restriction on what kind of stable summands can occur for a given Compact Lie Group G. As an application, we use this framework to study two special classes of stable summands. The first is a generalized notion of a dominant summand in BG applicable for Compact Lie Groups. Let ( )i denote completion at the prime p in the sense of Bousfield and Kan [3]. For technical reasons, we shall add a disjoint basepoint to each classifying space of G, denoting the result by BG, . Our main theorem in this case is _ THEOREM 3.8. Suppose Xi is a dominant summand of BGi+~ where Gi is a Compact Lie Group with p-Sylow subGroup Ni, i = 1,2. Zf X1 is stably homotopy equivalent to X2, then N1 is isomorphic to N1.

  • Stable splittings of the dual spectrum of the classifying space of a Compact Lie Group
    Transactions of the American Mathematical Society, 1992
    Co-Authors: Chun-nip Lee
    Abstract:

    For a Compact Lie Group G, there is a map from the G-equivariant fixed point spectrum of the zero sphere to the dual spectrum of the classifying space of G, DBG + . When G is finite, the affirmative solution to Segal's conjecture states that this map is an equivalence upon appropriate completion of the source. In the case of a Compact Lie Group, we obtain splitting results of DBG + via this map upon taking p-adic completions

Alejandro Adem - One of the best experts on this subject based on the ideXlab platform.

  • twisted equivariant k theory of Compact Lie Group actions with maximal rank isotropy
    Journal of Mathematical Physics, 2018
    Co-Authors: Alejandro Adem, Jose Cantarero, Jose Manuel Gomez
    Abstract:

    We consider twisted equivariant K-theory for actions of a Compact Lie Group G on a space X where all the isotropy subGroups are connected and of maximal rank. We show that the associated rational spectral sequence a la Segal has a simple E2-term expressible as invariants under the Weyl Group of G. Specifically, if T is a maximal torus of G, they are invariants of the π1(XT)-equivariant Bredon cohomology of the universal cover of XT with suitable coefficients. In the case of the inertia stack ΛY, this term can be expressed using the cohomology of YT and algebraic invariants associated with the Lie Group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when Y = {*}.We consider twisted equivariant K-theory for actions of a Compact Lie Group G on a space X where all the isotropy subGroups are connected and of maximal rank. We show that the associated rational spectral sequence a la Segal has a simple E2-term expressible as invariants under the Weyl Group of G. Specifically, if T is a maximal torus of G, they are invariants of the π1(XT)-equivariant Bredon cohomology of the universal cover of XT with suitable coefficients. In the case of the inertia stack ΛY, this term can be expressed using the cohomology of YT and algebraic invariants associated with the Lie Group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when Y = {*}.

  • equivariant k theory of Compact Lie Group actions with maximal rank isotropy
    Journal of Topology, 2012
    Co-Authors: Alejandro Adem, Jose Manuel Gomez
    Abstract:

    We consider twisted equivariant K--theory for actions of a Compact Lie Group $G$ on a space $X$ where all the isotropy subGroups are connected and of maximal rank. We show that the associated rational spectral sequence \`a la Segal has a simple $E_2$--term expressible as invariants under the Weyl Group of $G$. Namely, if $T$ is a maximal torus of $G$, they are invariants of the $\pi_1(X^T)$-equivariant Bredon cohomology of the universal cover of $X^T$ with suitable coefficients. In the case of the inertia stack $\Lambda Y$ this term can be expressed using the cohomology of $Y^T$ and algebraic invariants associated to the Lie Group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when $Y=\{*\}$.

  • twisted equivariant k theory of Compact Lie Group actions with maximal rank isotropy
    arXiv: Algebraic Topology, 2012
    Co-Authors: Alejandro Adem, Jose Cantarero, Jose Manuel Gomez
    Abstract:

    Let G denote a Compact connected Lie Group with torsion-free fundamental Group acting on a Compact space X such that all the isotropy subGroups are connected subGroups of maximal rank. Let $T\subset G$ be a maximal torus with Weyl Group W. If the fixed-point set $X^T$ has the homotopy type of a finite W-CW complex, we prove that the rationalized complex equivariant K-theory of X is a free module over the representation ring of G. Given additional conditions on the W-action on the fixed-point set $X^T$ we show that the equivariant K-theory of X is free over R(G). We use this to provide computations for a number of examples, including the ordered n-tuples of commuting elements in G with the conjugation action.

Brian C Hall - One of the best experts on this subject based on the ideXlab platform.

  • a unitary quantization commutes with reduction map for the adjoint action of a Compact Lie Group
    arXiv: Mathematical Physics, 2017
    Co-Authors: Brian C Hall, Benjamin D Lewis
    Abstract:

    Let $K$ be a simply connected Compact Lie Group and $T^{\ast}(K)$ its cotangent bundle. We consider the problem of "quantization commutes with reduction" for the adjoint action of $K$ on $T^{\ast}(K).$ We quantize both $T^{\ast}(K)$ and the reduced phase space using geometric quantization with half-forms. We then construct a geometrically natural map from the space of invariant elements in the quantization of $T^{\ast}(K)$ to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.

  • phase space bounds for quantum mechanics on a Compact Lie Group
    Communications in Mathematical Physics, 1997
    Co-Authors: Brian C Hall
    Abstract:

    Let K be a Compact, connected Lie Group and \(K_{\Bbb{C}}\) its complexification. I consider the Hilbert space \({\cal{H}}L^2\left(K_{\Bbb{C}},\nu _t\right)\) of holomorphic functions introduced in [H1], where the parameter t is to be interpreted as Planck's constant. In light of [L-S], the complex Group \(K_{\Bbb{C}}\) may be identified canonically with the cotangent bundle of K. Using this identification I associate to each \(F\in {\cal{H}}L^2\left( K_{\Bbb{C}},\nu _t\right)\) a “phase space probability density”. The main result of this paper is Theorem 1, which provides an upper bound on this density which holds uniformly over all F and all points in phase space. Specifically, the phase space probability density is at most \(a_t\left( 2\pi t\right)^{-n}\), where \(n=\dim K\) and a t is a constant which tends to one exponentially fast as t tends to zero. At least for small t, this bound cannot be significantly improved.