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Abstract In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $$K_0$$-group of its group $$C^\ast$$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum–Connes conjecture holds for a group, then Lott’s delocalized eta invariants take values in algebraic numbers. We also generalize Lott’s delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.
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JORDAN–KRONECKER INVARIANTS OF LIE ALGEBRA REPRESENTATIONS AND DEGREES OF INVARIANT POLYNOMIALS
Abstract For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ . Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ . Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.
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- Award ID(s):
- 2008021
- PAR ID:
- 10413236
- Date Published:
- Journal Name:
- Transformation Groups
- Volume:
- 28
- Issue:
- 2
- ISSN:
- 1083-4362
- Page Range / eLocation ID:
- 541 to 560
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00031-021-09661-0
