相关论文: Integral homology of loop groups via Langlands dua…

We study a Hopf algebra $H$, which is finitely generated and projective over a commutative ring $k$, as a $P$-Frobenius algebra. We define modular functions in this setting, and provide a complete proof of Radford's formula for the fourth…

Let $S$ be the affine plane $\C^2$ together with an appropriate $\mathbb T = \C^*$ action. Let $\hil{m,m+1}$ be the incidence Hilbert scheme. Parallel to \cite{LQ}, we construct an infinite dimensional Lie algebra that acts on the direct…

Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected (but different from) group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to…

Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and Korkmaz one can identify the homotopy-type of the classifying space of the stable non-orientable mapping class group $N_\infty$ (after plus-construction). At odd primes p, the…

A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…

There is considerable current interest in applications of generalised Lie algebras graded by an abelian group $\Gamma$ with a commutative factor $\omega$. This calls for a systematic development of the theory of such algebraic structures.…

We describe a map from the equivariant twisted K-homology of a compact, connected, simply connected Lie group $G$ to the Verlinde ring. Our map is described at the level of `D-cycles' for the geometric twisted K-homology of $G$, and is…

We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was…

Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this…

Let i be a homomorphism of the multiplicative group into a connected reductive algebraic group over C. Let G^i be the centralizer of the image i. Let LG be the Lie algebra of G and let L_nG (n integer) be the summands in the direct sum…

We introduce the notion of iHopf algebra, a new associative algebra structure defined on a Hopf algebra equipped with a Hopf pairing. The iHopf algebra on a Borel quantum group endowed with a $\tau$-twisted Hopf pairing is shown to be a…

A description of a ring of functions on the base of a universal formal deformation for several moduli problems is given. The answer is given in terms of a homology group of a certain dg Lie algebra canonically (up to an essentially unique…

Let $G$ be a Lie group with a maximal torus $T$. Combining Schubert calculus in the flag manifold $G/T$ with the Serre spectral sequence of the fibration $G\rightarrow G/T$, we construct the integral cohomology ring $H^{\ast}(G)$ uniformly…

We develop a general homological approach to presentations of connected graded associative algebras, and apply it to the loop homology of moment-angle complexes $Z_K$ that correspond to flag simplicial complexes $K$. For arbitrary…

We study the categorical homology of Zappa-Sz\'ep products of small categories, which include all self-similar actions. We prove that the categorical homology coincides with the homology of a double complex, and so can be computed via a…

The total homology of the loop space of the configuration space of ordered distinct n points in R^m has a structure of a Hopf algebra defined by the 4-term relations if m>2. We describe a relation of between the cohomology of this loop…

A new highly symmetrical model of the compact Lie algebra $\mathfrak{g}^c_2$ is provided as a twisted ring group for the group $\mathbb{Z}_2^3$ and the ring $\mathbb{R}\oplus\mathbb{R}$. The model is self-contained and can be used without…

We prove several results concerning quasi-bialgebra morphisms $\mathcal{D}^\omega(G)\to\mathcal{D}^\eta(H)$ of twisted group doubles. We take a particular focus on the isomorphisms which are simultaneously isomorphisms…

Let $A$ and $H$ be two Hopf algebras. We shall classify up to an isomorphism that stabilizes $A$ all Hopf algebras $E$ that factorize through $A$ and $H$ by a cohomological type object ${\mathcal H}^{2} (A, H)$. Equivalently, we classify up…

For a complex reductive group $G$, we prove a homological mirror symmetry between the wrapped Fukaya category of the affine Toda system for $G$ and coherent sheaves on the regular centralizer group scheme for the Langlands dual group…