Loday–Quillen–Tsygan Theorem for Coalgebras

The original Loday–Quillen–Tsygan Theorem (LQT) is proven by Loday and Quillen [13] and independently by Tsygan [20]. It states that the ordinary Lie homology (here referred as Chevalley–Eilenberg–Lie homology) of the Lie algebra of the infinite matrices gl(A) over an unital associative algebra A is generated by the cyclic homology of A as an exterior algebra. Although Lie algebras have been studied extensively, their non-commutative counterparts, Leibniz algebras and their homology, are defined only recently by Loday [12]. In this setting, LQT is extended by Cuvier [5] proving Leibniz homology (here referred as Chevalley– Eilenberg–Leibniz homology) of gl(A) is generated by the Hochschild homology of A as a tensor algebra. There is also a slightly different proof of Cuvier’s result by Oudom [17] by using a specific filtration on the Chevalley–Eilenberg–Leibniz complex of gl(A) and a spectral sequence. All of these proofs and the proof we present here for the coalgebras rely heavily on Weyl’s invariant theory [11, Chapter 9]. We would like to mention that in [1] Aboughazi and Ogle gave an alternative proof of LQT which did not use Weyl’s invariant theory. There is also a general LQT type result for algebras over operads by Fresse [6]. Coassociative coalgebras and their homologies received some attention recently in the context of Hopf and bialgebra cyclic (co)homology [2, 3, 7, 9]. On the other hand, Lie coalgebras have not received much attention in their own right [15, 18, 4] even though they appear as auxillary structures in important results such as Hinich’s explanation [8] of Tamarkin’s proof of Kontsevic Formality Theorem [19]. The only reference to Leibniz coalgebras we found was in Livernet [10]. In this paper we prove that LQT generalizes to the case of coalgebras. Specifically, we show that the Chevalley–Eilenberg–Lie homology of the Lie coalgebra of matrices gl(C) over a coassociative coalgebra C is generated by the cyclic homology of the underlying coalgebra C as an exterior algebra. Here is the plan of this paper. In Section 2 we give a self-contained account of coassociative, Leibniz and Lie coalgebras and their comodules. In Section 3 and Section 4 we develop the bar and Hochschild homology theories for coassociative coalgebras. In Section 5 we define several homology theories for Leibniz coalgebras including Chevalley–Eilenberg–Leibniz homology and symmetric Chevalley–Eilenberg–Leibniz homology which we call as Chevalley–Eilenberg–Lie homology. In Section 6 we connect all of these homology theories together for the Lie coalgebra Lie(C) of a coassociative coalgebra C. Section 7 contains several results about matrix coalgebras. Finally, in Section 8 we prove Loday–Quillen–Tsygan Theorem for coassociative counital coalgebras. Acknowledgements. We would like to thank Henri Moscovici for his helpful remarks.