Let $G$ be an affine algebraic reductive group over $\mathbb{C}$ whose identity component is a torus $T$, and let $K:=G/T$. We realize $\Rep(G)$ as an equivariantization $\Rep(T)^K$ and describe its finite indecomposable semisimple module categories in terms of equivariant module-category data over $\Rep(T)$. For such an equivariantization, rank one is characterized by transitivity of the induced action on the simple objects of the underlying $\Rep(T)$-module category, and nondegeneracy of a projective cocycle on a point stabilizer. We use this criterion to parametrize fiber functors on $\Rep(G)$, prove that every such fiber functor is classical, hence arises from a Hopf $2$-cocycle on $\mathscr{O}(G)$, and classify (minimal) Hopf $2$-cocycle on $\mathscr{O}(G)$. For commutative direct products $G=T\times K$, we also give the canonical K\"unneth decomposition of the group of gauge classes, including the mixed component, and compare it with our classification.
Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with $H_1(X ; \mathbb{Z}) = 0$. Let $p$ be a prime and suppose that $G = \mathbb{Z}_p$ acts smoothly on $X$ (if $p=2$ we also require an assumption on how $G$ acts on $H^2(X ; \mathbb{Z})$). Using equivariant Yang--Mills theory, Hambleton--Lee and...
We prove an equivariant Landweber exact functor theorem for abelian compact Lie groups $G$. For a graded module $N$ over the $G$-equivariant Lazard ring $L_G$, we give necessary and sufficient algebraic conditions on $N$ for the functor \[ X\longmapsto (MU_G)_*(X)\otimes_{L_G} N \] to define a homology theory on the ca...
Let $L$ be a finite-dimensional pointed Hopf algebra over an algebraically closed field. We establish an intrinsic characterization of index-two extensions: every Hopf algebra $H$ containing $L$ as a Hopf subalgebra and satisfying $H_0=L_0$ and $\dim H=2\dim L$ is a normalized quadratic extension of $L$. For a fixed su...
Let $X$ be a smooth projective surface over $\mathbb{C}$ with an effective action of a finite group $G$, and let $L$ be a $G$-linearized line bundle. We develop a residue-theoretic and birational approach to computing the equivariant Euler characteristic $\chi_G(X,L)$. Every such $L$ admits an equivariant divisor prese...
Let $G=\mathrm{SL}_n(\mathbf C)$ and let $P\subset G$ be a standard parabolic subgroup with Levi factor $L$. For a $G$-dominant weight $\lambda$, consider the vector bundle on $T^*(G/P)$ obtained by pulling back the vector bundle on $G/P$ associated to the irreducible $L$-module $V_L(\lambda)^*$. We express its cohomol...
We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\infty,1)$-categorical group completion of topological groupoids of fi...
David Aretz, L. Stehouwer· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
