On homology with coefficients and generalized inductions

Homology of Lie Algebras of Orthogonal and Symplectic Generalized Jacobi Matrices

In this note, we compute the homology with trivial coefficients of Lie algebras of generalized Jacobi matrices of type $B, C$ and $D$ over an associative unital $k$-algebra with $k$ being a field of characteristic $0$.

带adjoint表示张量积的李代数上同调与相对多项式次数

Lie algebra homology with coefficients tensor products of the adjoint representation in relative polynomial degree 2

本文计算了自由李代数在adjoint表示张量积系数下的相对次数2的上同调，证实了Gadish和Hainaut的猜想。

Generalized Artin and Brauer induction for compact Lie groups

Let G be a compact Lie group. We present two induction theorems for certain generalized G-equivariant cohomology theories. The theory applies to G-equivariant K-theory K_G, and to the Borel cohomology associated to any complex oriented cohomology theory. The coefficient ring of K_G is the representation ring R(G) of G. When G is a finite group the induction theorems for K_G coincide with the classical Artin and Brauer induction theorems for R(G).

Categorifying induction formulae via divergent series

We show how to get explicit induction formulae for finite group representations, and more generally for rational Green functors, by summing a divergent series over Dwyer's subgroup and centralizer decomposition spaces. This results in formulae with rational coefficients. The former space yields a well-known induction formula, the latter yields a new one. As essentially immediate corollaries of the existing literature, we get similar formulae in group cohomology and stable splittings of classifying spaces.

Homology of the braid group with coefficients in the reduced Burau representation

The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the $n$-punctured disc. In this note, we calculate $H_*(B_n;V_n)$ as a module over the Laurent polynomial ring $\mathbb{Q}[t,t^{-1}]$.

具有任意系数和结合光滑性的导出表示方案

Derived representation schemes with arbitrary coefficients and associative smoothness

研究通过任意有限维代数系数的表示同调探讨结合光滑性，证明有限生成形式光滑代数高阶同调消失，任意系数检验更强，对有限维代数而言其表示同调完全刻画形式光滑性，为相关理念提供证据。

A Topological Proof for a Version of Artin's Induction Theorem

We define a Euler characteristic $χ(X,G)$ for a finite cell complex $X$ with a finite group $G$ acting cellularly on it. Then, each $K_{i}(X)$ (a complex vector space with basis the $i$-cells of $X$) is a representation of $G$, and we define $χ(X,G)$ to be the alternating sum of the representations $K_{i}(X)$, as elements of the representation ring $R(G)$ of $G$. By adapting the ordinary proof that the alternating sum of the dimensions of the chain complexes is equal to the alternating sum of the dimensions of the homology groups, we prove that there is another definition of $χ(X,G)$ with the …

对合代数的表示同调与超八面体同调

Representation homology of involutive algebras and hyperoctaherdral homology

本文研究对合代数的表示同调，证明特征0域上对合代数的超八面体同调与其一维对合表示同调同构，进而计算相关代数及群代数的超八面体同调。

The geometric realization of regular path complexes via (co-)homology

The aim of this paper is to give the geometric realization of regular path complexes via (co)homology groups with coefficients in a ring $R$. Concretely, for each regular path complex $P$, we associate it with a singular $Δ$-complex $S(P)$ and show that the (co)homology groups of $P$ are isomorphic to those of $S(P)$ with coefficients in $R$. As a direct result we recognize path (co)homology as Hochschild (co)homology in case that $R$ is commutative and $P$ regular finite. Analogues of the Eilenberg-Zilber theorem and Künneth formula are also showed for the Cartesian product and the join of tw…

Sur l'homologie des groupes orthogonaux et symplectiques à coefficients tordus

We compute the stable homology of orthogonal and symplectic groups over a finite field k with coefficients coming from an usual endofunctor F of k-vector spaces (exterior, symmetric, divided powers...), that is, for all natural integer i, we compute the colimits of the vector spaces $H_i(O_{n,n}(k) ; F(k^{2n}))$ and $H_i(Sp_{2n}(k) ; F(k^{2n}))$. In this situation, the stabilization is a classical result of Charney. We give a formal framework to connect stable homology of some families of groups and homology of suitable small categories thanks to a spectral sequence which collapses in several …