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Cyclic homology and lambda operations. (English) Zbl 0719.19002

Algebraic \(K\)-theory: Connections with geometry and topology, Proc. Meet., Lake Louise/Can. 1987, NATO ASI Ser., Ser. C 279, 209-224 (1989).
[For the entire collection see Zbl 0685.00007.]
Let \(A\) be a commutative algebra with unit over a field \(K\) of characteristic zero. We denote by \(\mathfrak{gl}_ n(A)\) the Lie algebra of \(A\)-endomorphisms of the free \(A\)-module \(A^ n\), and we define \(\mathfrak{gl}(A)\) to be the direct limit \(\lim_{n\to \infty}\mathfrak{gl}_ n(A)\). Using bases, the \(k\)-th exterior power functor \(\Lambda^ n\) induces a Lie algebra homomorphism \(\Lambda^ k_ n: \mathfrak{gl}_ n(A)\to\mathfrak{gl}(A)\). Defining \(\lambda^ k_ n\) by the usual formula \(\lambda^ k_ n=\oplus^{k}_{i=0}(-1)^ i\binom{n-1+i}{i} \Lambda_ n^{k- i}\) and passing to homology, we obtain a mapping \((\lambda^ k_ n)_*: H_*(\mathfrak{gl}_ n(A);K)\to H_*(\mathfrak{gl}(A);K)\) which induces a mapping \(\lambda^ k: H_*(\mathfrak{gl}(A);K)\to H_*(\mathfrak{gl}(A);K)\). The primitive part of the coalgebra \(H_*(\mathfrak{gl}(A);K)\) is known to be the cyclic homology \(HC_{*-1}(A)\). It can be shown that \(\lambda^ k\) induces a mapping (denoted by the same symbol) \(\lambda^ k: HC_*(A)\to HC_*(A)\), and that \(HC_*(A)\) endowed with \(\lambda^ k\) \((k=0,1,2,\ldots)\) is a \(\lambda\)-algebra with trivial multiplication.
The authors present a simple explicit formula for \(\lambda^ k\) on \(HC_*(A)\). They also prove that their formula is equivalent to the Feigin-Tsygan formula for the Adams operations \(\psi^ k\) [see B. L. Feĭgin and B. L. Tsygan, “Additive K-theory”, Lect. Notes Math. 1289, 67–129 (1987; Zbl 0635.18008)].

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
17B55 Homological methods in Lie (super)algebras
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
18G60 Other (co)homology theories (MSC2010)
16T05 Hopf algebras and their applications