scholarly journals The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)

2009 ◽  
Vol 146 (2) ◽  
pp. 379-393 ◽  
Author(s):  
TYLER LAWSON
Keyword(s):  
Unitary Matrices ◽  
Bott Periodicity ◽  
Representation Ring ◽  
K Theory

AbstractWe show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its “deformation representation ring,” analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices.

On the relation of connective K-theory to homology

1970 ◽  
Vol 68 (3) ◽  
pp. 637-639 ◽  
Author(s):  
Larry Smith
Keyword(s):  
Exact Sequence ◽  
Natural Transformation ◽  
Homology Theory ◽  
Finite Complex ◽  
Ring Spectrum ◽  
Bott Periodicity ◽  
Ring Spectra ◽  
K Theory ◽  
Image Position ◽  
Natural Way

Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectrainducing a morphismof homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequencewhere Z is viewed as a Z[t] module via the augmentationand, is induced by η*in the natural way.

scholarly journals Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory

2013 ◽  
Vol 13 (1) ◽  
pp. 83-113 ◽  
Author(s):  
El-Kaïoum M. Moutuou
Keyword(s):  
Vector Bundles ◽  
Isomorphism Theorem ◽  
Locally Compact ◽  
Bott Periodicity ◽  
C Algebras ◽  
Thom Isomorphism ◽  
K Theory ◽  
Thom Isomorphism Theorem

AbstractWe develop equivariant KK–theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K–theory.

scholarly journals Symmetries of Kirchberg Algebras

2003 ◽  
Vol 46 (4) ◽  
pp. 509-528 ◽  
Author(s):  
David J. Benson ◽  
Alex Kumjian ◽  
N. Christopher Phillips
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Direct Sum ◽  
Representation Ring ◽  
Direct Sum Decomposition ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
The Way

AbstractLet G0 and G1 be countable abelian groups. Let γi be an automorphism of Gi of order two. Then there exists a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with [1A] = 0 in K0(A), and an automorphism α ∈ Aut(A) of order two, such that K0(A) ≅ G0, such that K1(A) ≅ G1, and such that α* : Ki(A) → Ki(A) is γi. As a consequence, we prove that every -graded countable module over the representation ring R() of is isomorphic to the equivariant K-theory K (A) for some action of on a unital Kirchberg algebra A.Along the way, we prove that every not necessarily finitely generated []-module which is free as a -module has a direct sum decomposition with only three kinds of summands, namely [] itself and on which the nontrivial element of acts either trivially or by multiplication by −1.

An Introduction to K-Theory for C*-Algebras

2000 ◽  
Author(s):  
M. Rørdam ◽  
F. Larsen ◽  
N. Laustsen
Keyword(s):  
C Algebras ◽  
K Theory

On a question of swan in algebraic k-theory

1973 ◽  
Vol 6 (1) ◽  
pp. 85-94 ◽  
Author(s):  
Pramod K. Sharma ◽  
Jan R. Strooker
Keyword(s):  
K Theory

scholarly journals DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN
Keyword(s):  
General Scheme ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Additional Structure ◽  
Unitary Matrices ◽  
Decomposition Scheme ◽  
Single Qubit ◽  
Matlab Program

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

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