Mathematics

Let C be category over a commutative ring k , its Hochschild-Mitchell homology and cohomology are denoted respectively H H ∗ (C) and H H ∗ (C). Let G be a group acting on C , and C[G] be the skew category. We provide decompositions of the (co)homology of C[G] along the conjugacy classes of G . For Hochschild homology of a k -algebra, this corresponds to the decomposition obtained by M. Lorenz. If the coinvariants and invariants functors are exact, we obtain isomorphisms (H H ∗ (C)) G ≃H H {1} ∗ (C[G]) and (H H ∗ (C)) G ≃H H ∗ {1} (C[G]), where {1} is the trivial conjugacy class of G . We first obtain these isomorphisms in case the action of G is free on the objects of C . Then we introduce an auxiliary category M G (C) with an action of G which is free on its objects, related to the infinite matrix algebra considered by J. Cornick. This category enables us to show that the isomorphisms hold in general, and in particular for the Hochschild (co)homology of a k -algebra with an action of G by automorphisms. We infer that (H H ∗ (C)) G is a canonical direct summand of H H ∗ (C[G]) . This provides a frame for monomorphisms obtained previously, and which have been described in low degrees.

For an integer n>2 we define a polylogarithm, which is a holomorphic function on the universal abelian cover of C-{0,1} defined modulo (2 pi i)^n/(n-1)!. We use the formal properties of its functional relations to define groups lifting Goncharov's Bloch groups of a field F, and show that they fit into a complex lifting Goncharov's Bloch complex. When F=C we show that the imaginary part (when n is even) or real part (when n is odd) of the holomorphic polylogarithm agrees with Goncharov's real valued polylogarithm on the first cohomology group of the lifted Bloch complex. When n=2, this group is Neumann's extended Bloch group. Goncharov's complex conjecturally computes the rational motivic cohomology of F, and one may speculate whether the lifted complex computes the integral motivic cohomology. Finally, we construct a lift of Goncharov's regulator on the 5th homology of SL(3,C) to a complex regulator. This use polylogarithm relations arising from the cluster ensemble structure on the Grassmannians Gr(3,6) and Gr(3,7).

Enumerative geometry studies the intersection theory of virtual fundamental classes in the homology of moduli spaces. These usually depend on auxiliary parameters and are then related by wall-crossing formulas. We construct a homological graded Lie bracket on the homology of moduli spaces which can be used to express universal wall-crossing formulas. For this we develop a new topological theory of pushforward operations for principal bundles with orientations in twisted K-theory. We prove that all rational pushforward operations are obtained from a single new projective Euler operation, which also has applications to Chern classes in twisted K-theory.

Nichols algebras are an important tool for the classification of Hopf algebras. Within those with finite GK dimension, we study homological invariants of the super Jordan plane, that is, the Nichols algebra A=B(V(−1,2)) . These invariants are Hochschild homology, the Hochschild cohomology algebra, the Lie structure of the first cohomology space - which is a Lie subalgebra of the Virasoro algebra - and its representations H n (A,A) and also the Yoneda algebra. We prove that the algebra A is K 2 . Moreover, we prove that the Yoneda algebra of the bosonization of A is also finitely generated, but not K 2 .

Let H n be the n -th group homology functor (with integer coeffcients) and let { G i } i∈N be any tower of groups such that all maps G i+1 → G i are surjective. In this work we study kernel and cokernel of the following natural map: H n ( lim ← − G i )→ lim ← − H n ( G i ) For n=1 Barnea and Shelah [BS] proved that this map is surjective and its kernel is a cotorsion group for any such tower { G i } i∈N . We show that for n=2 the kernel can be non-cotorsion group even in the case when all G i are abelian and after it we study these kernels and cokernels for towers of abelian groups in more detail.

Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C*-algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum-Connes conjecture. The construction is based on the triangulated category approach to the Baum-Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.

For an infinite field F , we study the cokernel of the map of homology groups H n+1 ( GL n−1 (F),k)→ H n+1 ( GL n (F),k) , where k is a field such that (n−2)!∈ k × , and the kernel of the natural map H n ( GL n−1 (F),Z[ 1 (n−2)! ])→ H n ( GL n (F),Z[ 1 (n−2)! ]) . We give conjectural estimates of these cokernel and kernel and prove our conjectures for n≤4 .

For every n≥1 , we calculate the Hochschild homology of the quantum monoids M q (n) , and the quantum groups G L q (n) and S L q (n) with coefficients in a 1-dimensional module coming from a modular pair in involution.

We introduce a graded homology theory for graded étale groupoids. For Z -graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras.

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005, including a Lie-theoretic version of the Deligne conjecture.