These notes are based on the Number Theory seminar I gave at Purdue University on September 19, 2013 titled “Hilbert’s theorem 90 and generalization”. The proof of Hilbert’s 90 is taken from an answer I found on MathOverflow at http://mathoverflow.net/a/21117. In the seminar, Professor Goins asked an interesting question about the isomorphism classes of rank one tori which I have blogged here. ...

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact triangulated categories. In this paper, we show that the idempotent completion an extriangulated category admits natural structure. As application, prove recollement is still recollement.

x⊗ f(t), y ⊗ g(t) 7→ −κ(x, y) · Rest=0 fdg. Denote by ĝκ–mod the category of ĝκ-modules which are discrete, i.e., any vector is annihilated by the Lie subalgebra g ⊗ tC[[t]] for sufficiently large N ≥ 0, and on which 1 ∈ C ⊂ ĝκ acts as the identity. We will refer to objects of these category as modules at level κ. Let GrG = G((t))/G[[t]] be the affine Grassmannian of G. For each κ there is a ca...

‎This paper develops a basic theory of $H$-groups‎. ‎We‎ ‎introduce a special quotient of $H$-groups and‎ ‎extend some algebraic constructions of topological groups to the category‎ ‎of H-groups and H-maps and then present a functor from this category to the category of quasitopological groups‎.

in addition to exploring constructions and properties of limits and colimits in categories of topologicalalgebras, we study special subcategories of topological algebras and their properties. in particular, undercertain conditions, reflective subcategories when paired with topological structures give rise to reflectivesubcategories and epireflective subcategories give rise to epireflective subc...

these are notes from introductory survey lectures given at the institute for studies in theoretical physics and mathematics (ipm), teheran, in 2008 and 2010. we present the definition and the fundamental properties of fomin-zelevinsky’s cluster algebras. then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of ...

Bivariant (equivariant) K-theory is the standard setting for noncommutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, projective resolutions, and derived functors. We introduce these notions and apply them to examples from bivar...

of the Dissertation Self-dual Hall Modules by Matthew Bruce Young Doctor of Philosophy in Mathematics Stony Brook University 2013 In the past twenty years Hall algebras have played an important role in many areas of mathematics and physics, including the theory of quantum groups and string theory. In its original setting the Hall algebra is constructed from a finitary exact category, the multip...

The notion of quantum group equivariant homogeneous vector bundles on quantum homogeneous spaces is introduced. The category of such quantum vector bundles is shown to be exact, and its Grothendieck group is determined. It is also shown that the algebras of functions on quantum homogeneous spaces are noetherian.

2 The category of small abelian categories and exact functors 4 2.1 Categorical properties of ABEX . . . . . . . . . . . . . . . . . . . 5 2.2 Pullbacks in ABEX . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 ABEX is finitely accessible . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Abelian categories as schemes . . . . . . . . . . . . . . . . . . . . 16 2.4.1 The functor of point...