2016
DOI: 10.1093/imrn/rnw016
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Chern–Simons Forms and Higher Character Maps of Lie Representations

Abstract: Abstract. This paper is a sequel to [4], where we study the derived representation scheme DRep g (a) parametrizing the representations of a Lie algebra a in a reductive Lie algebra g. In [4], we constructed two canonical maps Trg(a) : HC (r)W called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. In this paper, we give an explicit formula for the Drinfeld trace in terms of Chern-Simons forms. As a consequence, we show that, if a is an abelian Lie algebra, the composite map Φg(a) •… Show more

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Cited by 5 publications
(8 citation statements)
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“…Our next goal is to describe certain natural trace maps with values in representation homology. These maps were originally constructed in [3, 4] as (derived) characters of finite‐dimensional Lie representations. We will give a topological interpretation of these characters in terms of free loop spaces.…”
Section: Spaces With Polynomial Representation Homology and The Stron...mentioning
confidence: 99%
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“…Our next goal is to describe certain natural trace maps with values in representation homology. These maps were originally constructed in [3, 4] as (derived) characters of finite‐dimensional Lie representations. We will give a topological interpretation of these characters in terms of free loop spaces.…”
Section: Spaces With Polynomial Representation Homology and The Stron...mentioning
confidence: 99%
“…For P0.16em0.16emIm+1(g)$P\,\in \,I^{m+1}({\mathfrak {g}})$, the Chern‐Simons form TP(θ)0.16em0.16emB2m+1$TP(\theta )\,\in \,\mathcal {B}^{2m+1}$ satisfies δ(TPfalse(θfalse))0.16em=0.16emP(normalΩm+1)$\delta (TP(\theta ))\,=\,P(\Omega ^{m+1})$, where normalΩ0.16em0.16emB2frakturg$\Omega \,\in \,\mathcal {B}^2 \otimes {\mathfrak {g}}$ is the curvature of θ$\theta$. Since Ωm+1=0$\Omega ^{m+1}=0$ by [4, Proposition A.2], TP(θ)0.16em0.16emB2m+1$TP(\theta )\,\in \,\mathcal {B}^{2m+1}$ is a cocycle. It follows that s2mTP(θ)$s^{2m}TP(\theta )$ defines a map of complexes 1false(m+1false)!smTP(θ)0.16em:0.16emC*(gfalse(trueA¯false);C)…”
Section: Spaces With Polynomial Representation Homology and The Stron...mentioning
confidence: 99%
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