Suppose G is a reductive algebraic group, T is a Cartan subgroup of G, N= Norm (T), and W= N/T is the Weyl group. If w W has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.

For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, the decomposition of $\mathcal{C}$ is determined by the decomposition of $\mathcal{T}$ satisfying "vanishing condition" of negative extension groups, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $\mbox{Hom} _{\mathcal{C}}(\mathcal{T} _i[t],\mathcal{T} _j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that for any two cluster tilting objects $T, T'$ in a $2-$Calabi-Yau triangulated category $\mathcal{C}$, the Gabriel quivers of endomorphism algebra End$_{\mathcal{C}}T$ of $T$ is connected if and only if so is End$_{\mathcal{C}}T'$. As an application, we prove that indecomposable $2-$Calabi-Yau triangulated categories with cluster tilting objects have no non-trivial t-structures and no non-trivial co-t-structures. This allows us to give a classification of torsion pairs in those triangulated categories, and to determine further the hearts of torsion pairs in the sense of Nakaoka, which are equivalent to the module categories over the endomorphism algebras of the cores of the torsion pairs.

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In this paper, we generalize the categorifical construction of a quantum group and its canonical basis by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.

We show that Hecke algebra of type B and a coideal subalgebra of the type A quantum group satisfy a double centralizer property, generalizing the Schur-Jimbo duality in type A. The quantum group of type A and its coideal subalgebra form a quantum symmetric pair. A new theory of canonical bases arising from quantum symmetric pairs is initiated. It is then applied to formulate and establish for the first time a Kazhdan-Lusztig theory for the BGG category O of the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$. In particular, our approach provides a new formulation of the Kazhdan-Lusztig theory for Lie algebras of type B/C.