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Register a new userHomology and cohomology via the partial group algebra
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We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$ with coefficients in an irreducible (resp. indecomposable) $mathbb{K}_{par}G$-module with the ordinary homology and cohomology groups of $G$ with in general non-trivial coefficients. Furthermore, we compare the standard cohomological dimension $cd_{ mathbb{K}}(G)$ (over a field $mathbb{K}$) with the partial cohomological dimension $cd_{ mathbb{K}}^{par}(G)$ (over $mathbb{K}$) and show that $cd_{ mathbb{K}}^{par}(G) geq cd_{ mathbb{K}}(G)$ and that there is equality for $G = mathbb{Z}$.
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Inner relations are derived between partial augmentations of certain elements (units or idempotents) in group rings.
Given a semilattice $X$ we study the algebraic properties of the semigroup $upsilon(X)$ of upfamilies on $X$. The semigroup $upsilon(X)$ contains the Stone-Cech extension $beta(X)$, the superextension $lambda(X)$, and the space of filters $phi(X)$ on $X$ as closed subsemigroups. We prove that $upsilon(X)$ is a semilattice iff $lambda(X)$ is a semilattice iff $phi(X)$ is a semilattice iff the semilattice $X$ is finite and linearly ordered. We prove that the semigroup $beta(X)$ is a band if and only if $X$ has no infinite antichains, and the semigroup $lambda(X)$ is commutative if and only if $X$ is a bush with finite branches.
A weakly complete vector space over $mathbb{K}=mathbb{R}$ or $mathbb{K}=mathbb{C}$ is isomorphic to $mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is illustrated by the fact that the underlying vector space of the Lie algebra of any pro-Lie group is weakly complete. In this study, weakly complete real or complex associative algebras are studied because they are necessarily projective limits of finite dimensional algebras. The group of units $A^{-1}$ of a weakly complete algebra $A$ is a pro-Lie group with the associated topological Lie algebra $A_{rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $expcolon Ato A^{-1}$ as the exponential function of $A^{-1}$. With each topological group, a weakly complete group algebra $mathbb{K}[G]$ is associated functorially so that the functor $Gmapsto mathbb{K}[G]$ is left adjoint to $Amapsto A^{-1}$. The group algebra $mathbb{K}[G]$ is a weakly complete Hopf algebra. If $G$ is compact, the $mathbb{R}[G]$ contains $G$ as the set of grouplike elements. The category of all real Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is shown to be equivalent to the category of compact groups. The group algebra $A=mathbb{R}[G]$ of a compact group $G$ contains a copy of the Lie algebra $mathcal{L}(G)$ in $A_{rm Lie}$; it also contains a copy of the Radon measure algebra $M(G,mathbb{R})$. The dual of the group algebra $mathbb{R}[G]$ is the Hopf algebra ${mathcal R}(G,mathbb{R})$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis of a duality ${mathcal R}(G,mathbb{R})leftrightarrow mathbb{R}[G]$ and thus yields a new aspect of Tannaka duality.
We prove that the cup product of $Delta$-decomposable quasimorphisms with any bounded cohomology class of arbitrary positive degree is trivial. As a corollary we obtain that this is also the case for Brooks quasimorphisms (in particular on selfoverlapping words) and Rolli quasimorphisms.
We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the $p$-toral rank of $mathrm{HH}^1(A,A)$. We also prove a more general result concerning Iwanaga-Gorenstein algebras, using a more general notion of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory. These results are proven by first establishing the stable invariance of the $B_infty$-structure of the Hochschild cochain complex. In the appendix we explain how the $p$-power operation on Hochschild cohomology can be seen as an artifact of this $B_infty$-structure. In particular, we establish well-definedness of the $p$-power operation, following some -- originally topological -- methods due to May, Cohen and Turchin, using the language of operads.
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