On unipotent quotients and some A-contractible smooth schemes

Unipotent groups and some A1-contractible smooth schemes

Vector bundles on contractible smooth schemes

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are that of Spec k. One might hope that furthermore, and in analogy with the classification of topological vector bundles on m...

Algebraic Groups I. Unipotent radicals and reductivity

In class, we have proved the important fact that over any field k, a non-solvable connected reductive group containing a 1-dimensional split maximal k-torus is k-isomorphic to SL2 or PGL2. That proof relied on knowing that maximal tori remain maximal after a ground field extension to k, and so relies on Grothendieck’s theorem. But for algebraically closed fields there is no content to Grothendi...

A note on essentially left $phi$-contractible Banach algebras

In this note, we show that cite[Corollary 3.2]{sad} is not always true. In fact, we characterize essential left $phi$-contractibility of the group algebras in terms of compactness of its related locally compact group. Also, we show that for any compact commutative group $G$, $L^{2}(G)$ is always essentially left $phi$-contractible. We discuss the essential left $phi$-contractibility of some Fou...

Quotients by Reductive Group, Borel Subgroup, Unipotent Group and Maximal Torus

Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this paper, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup. Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal toru...