Abstract: We develop a generalization of the $Q$-construction of the first author, Diemer, and the third author for Grassmann flops over an arbitrary field of characteristic zero. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. This idempotent kernel, after restriction, induces a derived equivalence over any twisted form of a Grassmann flop. Furthermore its image, after restriction to the geometric invariant theory semistable locus, ``opens'' a canonical ``window'' in the derived category of the quotient stack. We check this window coincides with the set of representations used by Kapranov to form a full exceptional collection on Grassmannians. Even in the well-studied special case of standard Atiyah flops, the arguments yield a new proof of the derived equivalence.

Abstract: In this article we will build a universal imbedding of a regular Hom- Lie triple system into a Lie algebra and show that the category of regular Hom-Lie triple systems is equivalent to a full subcategory of pairs of $\mathbb{Z}_2$- graded Lie algebras and Lie algebra automorphism, then finally give some characterizations of this subcategory.

Abstract: We Show that every finite $\mathbb{Z}$-grading on the algebra of compact operators is induced by a decomposition of the underlying Complex Hilbert space. We do this by first showing that every $\mathbb{Z}$-grading of the compoact operators defines a Peirce decomposition of a certain ideal. Then show this decomposition induces the $\mathbb{Z}$-grading on the compact operators. While finally as a Corollary we will show that every finite $\mathbb{Z}$-grading on the bounded operators is also induced by a decomposition of the underlying Hilbert Space