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4 The of this quantity care for new notions of whose papers algebras function is usual to have So are known as producing operations. they dial- bras. the 1st motivation to introduce such constructions a algebraic was once challenge in It grew to become out later that a few of them dendriform algebraic K-theory. (the comparable are to within the of ren- dialgebras) heavily Hopf algebras occuring conception malization of A. Connes and D. Kreimer. also are concerning the They heavily thought of Gerstenhaber homotopy algebra. allow us to first describe the incentive from The algebraic K-theory. al- braic of a aren't just like the yet K-groups ring periodic topological K-groups, of of a few them exhibits the life of a computation periodicity phenomenon. for example the zero are of four for > 2. The teams K,,(Z) Q periodic interval n are built at the linear GL. If algebraic K-groups basic we - team it its additive that's the Lie then where by way of counterpart, algebra gl, analogue of is : it's denoted HC. It algebraic K-theory computable cyclic homology, turns that for out this the is definitely understood. concept periodicity phenomenon It takes the shape of a designated : lengthy series - - - -+ -.+ -4 HCn_1 HH,,, -+ HC,, --+ HCn-2 HCn+1 the place HHstands for Hochschild In different homology. phrases, cyclic homology isn't really however the obstruction to is it truly is periodic (in basic) periodicity identified, Hochschild homology.

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