Related papers: Categorification of level two representations of q…
We give elementary geometric proofs of the main theorems about the (small) quantum cohomology of partial flag varieties SL(n)/P, including the quantum Pieri and quantum Giambelli formulas and the presentation.
We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank $2$ and type $\mathbb{B}_2$), more precisely in the (large) sub theory of skew translation generalized quadrangles ("STGQs"). Some of…
In this article, we introduce the first degrees of a cochain complex associated to a strict Lie 2-group whose cohomology is shown to extend the classical cohomology theory of Lie groups. In particular, we show that the second cohomology…
The theory of $N$-complexes is a generalization of both ordinary chain complexes and graded objects. Hence it yields deeper insight in the structure of these and offers a broader range of applications. This work generalizes the tensor…
We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The…
We compactify M(atrix) theory on Riemann surfaces Sigma with genus g>1. Following [1], we construct a projective unitary representation of pi_1(Sigma) realized on L^2(H), with H the upper half-plane. As a first step we introduce a suitably…
We study branching problem of the metaplectic representation of $Sp(2, \mathbb R)$ under its principle subgroup $SL(2, \mathbb R)$. We find the complete decomposition.
We calculate the structure of the finitely-generated groups H_2(SL_2(Z[1/m])) when m is a multiple of 6. We construct explicit homology classes which generate these groups and have prescribed orders. When n is at least 2 and m is the…
We define a bicomplex whose Euler characteristic is the idempotented version of the ribbon element of quantum sl(2). We show that properties of this bicomplex descend to the centrality, invertibility and symmetries of the ribbon element…
We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component…
The universal enveloping algebra U(g) of a Lie algebra g acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or "quantum group") is a deformation of a universal…
We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by…
It is shown, using level-rank duality that a universal topological quantum computer based on Chern-Simons theory for SU(2)$_3$ also implies an analogous universal quantum computer based on SU(3)$_2$. Suggestions are made for the possible…
Using a homological invariant together with an obstruction class in a certain Ext^2-group, we may classify objects in triangulated categories that have projective resolutions of length two. This invariant gives strong classification results…
After recalling briefly some basic properties of the quantum group $GL_q(2)$, we study the quantum sphere $S_q^2$, quantum projective space $CP_q(N)$ and quantum Grassmannians as examples of complex (K\"{a}hler) quantum manifolds. The…
This note presents a general theorem about the cohomology of finite dimensional Lie algebras of arbitrary characteristic. As an application we compute the cohomology of the Borel subalgebra of sl(N).
We define families of invariants for elements of the mapping class group of S, a compact orientable surface. Fix any characteristic subgroup H of pi_1(S) and restrict to J(H), any subgroup of mapping classes that induce the identity modulo…
To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph.
We develop the theory of special biserial and string coalgebras and other concepts from the representation theory of quivers. These tools are then used to describe the finite dimensional comodules and Auslander-Reiten quiver for the…
We introduce global model categories as a general framework to capture several phenomena in global equivariant homotopy theory. We then construct genuine stabilizations of these, generalizing the usual passage from unstable to stable global…