Related papers: Noncommutative geometry and dual coalgebras

The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is…

A covariant functor from the category of generic complex algebraic curves to a category of the AF-algebras is constructed. The construction is based on a representation of the Teichmueller space of a curve by the measured foliations due to…

We classify the cohomology spaces $H^2(\mathfrak{g},K)$ for all filiform nilpotent Lie algebras of dimension $n\le 11$ over $K$ and for certain classes of algebras of dimension $n\ge 12$. The result is applied to the determination of affine…

The non-commutative algebraic analog of the moduli of vector and covector fields is built. The structure of moduli of derivations of non-commutative algebras are studied. The canonical coupling is introduced and the conditions for…

We compute the p-adic geometric pro-\'etale cohomology of the affine space (in any dimension). This cohomogy is non-zero, contrary to the \'etale cohomology, and can be described by means of differential forms.

We establish a structure theorem on the arc space of a $k$-scheme of finite type. More precisely, we show that the arc space is locally for the pro-smooth toplogy a product of an infinite dimensional affine space and of a non-noetherian…

Standard combinatorial construction, due to Kontsevich, associates to any $\ai$-algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We propose an…

This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…

Using some elementary methods from noncommutative geometry a structure is given to a point of space-time which is different from and simpler than that which would come from extra dimensions. The structure is described by a supplementary…

We compare the integral category O of shifted affine quantum groups of symmetric and non symmetric types. To do so we compute the K-theoretic analog of the Coulomb branches with symmetrizers introduced by Nakajima and Weekes. This yields an…

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra, called big algebra, attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are…

Noncommutative domain algebras are noncommutative analogues of the algebras of holomorphic functions on domains of $\C^n$ defined by holomorphic polynomials, and they generalize the noncommutative Hardy algebras. We present here a complete…

In a recent article, D. Kazhdan and A. Yom Din conjectured the validity of an asymptotic form of Schur's orthogonality for tempered irreducible unitary representations of semisimple groups defined over local fields. In the non-Archimedean…

Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, $\mathscr{P}$-affine schemes can be defined as objects of the opposite of the category of algebras over an operad…

We use compactifications of C*-algebras to introduce noncommutative coarse geometry. We transfer a noncommutative coarse structure on a C*-algebra with an action of a locally compact Abelian group by translations to Rieffel deformations and…

A self-dual algebras is one isomorphic as a module to the opposite of its dual; a quasi self-dual algebra is one whose cohomology with coefficients in itself is isomorphic to that with coefficients in the opposite of its dual. For these…

Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we…

A non-associative algebra over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space $A$ equipped with a bilinear operation \[ {A\times A\to A\colon\; (x,y)\mapsto x\cdot y=xy}. \] The collection of all non-associative algebras over…

A functor of sets $\mathbb X$ over the category of $K$-commutative algebras is said to be an affine functor if its functor of functions, $\mathbb A_{\mathbb X}$, is reflexive and $\mathbb X=\Spec \mathbb A_{\mathbb X}$. We prove that affine…

One particular approach to quantum groups (matrix pseudo groups) provides the Manin quantum plane. Assuming an appropriate set of non-commuting variables spanning linearly a representation space one is able to show that the endomorphisms on…