The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field Fq and any sequence i of simple objects in C the element XV,i of the corresponding algebra PC,i of q-polynomials. We prove that if C was hereditary, then the assignments V 7→ XV,i define algebra homomorphisms from the (dual) Ha...

We investigate support schemes for infinitesimal unipotent supergroups and their representations. Our main results provide a non-cohomological description of these which generalizes the classical work Suslin, Friedlander, Bendel. As consequence, in this setting have desired features such theory, including naturality with respect to group homomorphisms, tensor product property, realizability. an...

The Auslander-Reiten quiver of a finite-dimensional associative algebra A encodes information about the indecomposable finite-dimensional representations of A and their homomorphisms. A component of the AuslanderReiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the AuslanderReiten translation. Preproj...

In this paper, we define trace-like operators on a subspace of the space derivations free Lie algebra generated by first homology group H surface Σ. This definition depends choice Lagrangian H, and call these traces. We suppose that Σ is boundary handlebody with H′, show that, in degree greater than 2, traces corresponding to Ker(H→H′) vanish image Johnson homomorphisms elements filtration exte...

Introduction 2 1. Thom Spaces 3 1.1. G-framed submanifolds. Classes of L-equivalent submanifolds 3 1.2. Thom spaces. The classifying properties of Thom spaces 4 1.3. The cohomologies of Thom spaces modulo p for p > 2 6 1.4. Cohomologies of Thom spaces modulo 2 8 1.5. Diagonal Homomorphisms 11 2. Inner Homology Rings 13 2.1. Modules with One Generator 13 2.2. Modules over the Steenrod Algebra. T...

We define for each group G the skein algebra of G. We show how it is related to the Kauffman bracket skein modules. We prove that skein algebras of abelian groups are isomorphic to symmetric subalgebras of corresponding group rings. Moreover, we show that, for any abelian group G, homomorphisms from the skein algebra of G to C correspond exactly to traces of SL(2, C)-representations of G. We al...

We consider the semigroup Ext(A, B) of extensions of a separable C *-algebra A by a stable C *-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semigroup contains the group Ext −1/2 (A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext 1/2 (A, B) is homotopy invariant and that it coincides with the functor of homoto...

Let X,Y be Banach modules over a C∗-algebra and let r1, . . . , rn ∈ R be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C∗-algebra: ∑n j 1 f −rjxj ∑ 1≤i≤n,i / j rixi 2 ∑n i 1 rif xi nf ∑n i 1 rixi . We show that if ∑n i 1 ri / 0, ri, rj / 0 for some 1 ≤ i < j ≤ n and a mapping f : X → Y satisfies the functional equation...

Abstract. Gelfand and Kolmogorov in 1939 proved that a compact Hausdorff topological space X can be canonically embedded into the infinite-dimensional vector space C(X)∗, the dual space of the algebra of continuous functions C(X), as an “algebraic variety", specified by an infinite system of quadratic equations. Buchstaber and Rees have recently extended this to all symmetric powers Symn(X) usi...

We categorify Lusztig’s U̇ – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2-category U̇ is constructed whose split Grothendieck ring is isomorphic to the algebra U̇. The indecomposable morphisms of this 2-category lift Lusztig’s canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on U̇. Graded lifts of various homomo...