TY - JOUR

T1 - The ring structure for equivariant twisted K-theory

AU - Tu, Jean Louis

AU - Xu, Ping

N1 - Funding Information:
1) Research partially supported by NSF grants DMS03-06665 and DMS-0605725 & NSA grant H98230-06-1-0047.

PY - 2009/10

Y1 - 2009/10

N2 - We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map for any crossed module N → and prove that any element in the image is ∞-multiplicative. As a consequence, we prove, under some mild conditions, for a crossed module N → and any , that the equivariant twisted K-theory group admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group , defined as the K-theory group of a certain groupoid C*-algebra, is endowed with a canonical ring structure , where . The relation with Freed-Hopkins-Teleman theorem Loop groups and twisted K-theory, III, math.AT/0312155 still needs to be explored.

AB - We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map for any crossed module N → and prove that any element in the image is ∞-multiplicative. As a consequence, we prove, under some mild conditions, for a crossed module N → and any , that the equivariant twisted K-theory group admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group , defined as the K-theory group of a certain groupoid C*-algebra, is endowed with a canonical ring structure , where . The relation with Freed-Hopkins-Teleman theorem Loop groups and twisted K-theory, III, math.AT/0312155 still needs to be explored.

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U2 - 10.1515/CRELLE.2009.077

DO - 10.1515/CRELLE.2009.077

M3 - Article

AN - SCOPUS:71049117762

SP - 97

EP - 148

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 635

ER -