Homology of quantum linear groups

Homology of Free Quantum Groups

We compute the Hochschild homology of the free orthogonal quantum group Ao(n). We show that it satisfies Poincaré duality and should be considered to be a 3-dimensional object. We then use recent results of R. Vergnioux to derive results about the l-homology of Ao(n) and estimates on the free entropy dimension of its set of generators. In particular, we show that the l Bettinumbers of Ao(n) all...

Equivariant Cyclic Homology for Quantum Groups

We define equivariant periodic cyclic homology for bornological quantum groups. Generalizing corresponding results from the group case, we show that the theory is homotopy invariant, stable and satisfies excision in both variables. Along the way we prove Radfords formula for the antipode of a bornological quantum group. Moreover we discuss anti-Yetter-Drinfeld modules and establish an analogue ...

Rigidity of the Low Dimensional Homology of Linear Groups

Homology of Linear Groups via Cycles in Bg×x

Homology theories for algebraic varieties are often constructed using simplicial sets of algebraic cycles. For example, Bloch’s higher Chow groups and motivic cohomology, as defined by Suslin and Voevodsky [17], are given in this fashion. In this paper, we construct homology groups Hi(X,G), where G is an algebraic group and X is a variety, by considering cycles on the simplicial scheme BG×X, an...

The Homology of Special Linear Groups over Polynomial Rings

We study the homology of SLn(F [t, t ]) by examining the action of the group on a suitable simplicial complex. The E–term of the resulting spectral sequence is computed and the differential, d, is calculated in some special cases to yield information about the low-dimensional homology groups of SLn(F [t, t ]). In particular, we show that if F is an infinite field, then H2(SLn(F [t, t ]), Z) = K...