Related papers: Lusztig's Canonical Quotient and Generalized Duali…
In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can…
Gelfand-Naimark duality (Commutative $C^*$-algebras $\equiv$ Locally compact Hausdorff spaces) is extended to $C^*$-algebras $\equiv$ Quotient maps on locally compact Hausdorff spaces. Using this duality, we give for an \emph{arbitrary}…
Like quantum groups, quantum groupoids frequently appear in pairs of mutually dual objects. We develop a general Pontrjagin duality theory for quantum groupoids in the algebraic setting that extends Van Daele's duality theory for multiplier…
The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a…
We describe the duality between different geometries which arises by considering the classical and quantum harmonic map problem. To appear in ``Essays on Mirror Manifolds II''.
The main result of this paper is an explicit formula for the Fourier transform of the canonical measure on a nilpotent coadjoint orbit for GL(n,R). This paper also includes some results on limit formulas for reductive Lie groups including…
A canonical basis in the sense of Lusztig is a basis of a free module over a ring of Laurent polynomials that is invariant under a certain semilinear involution and is obtained from a fixed "standard basis" through a triangular base change…
It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Gei{\ss}, Leclerc and Schr\"{o}er has…
The modified quantized enveloping algebra $\dot{\mathbf{U}}$ has a remarkable canonical basis, which was introduced by Lusztig. In this paper, we give an explicit description of all elements of the canonical basis of $\dot{\mathbf{U}}$ for…
We introduce a relation on real conjugacy classes of SL(2)-orbits in a Mumford-Tate domain D which is compatible with natural partial orders on the sets of nilpotent orbits in the corresponding Lie algebra and boundary orbits in the compact…
We introduce the notions of a commutative square ring $R$ and of a quadratic map between modules over $R$, called $R$-quadratic map. This notion generalizes various notions of quadratic maps between algebraic objects in the literature. We…
Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $\eta\colon Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the associated…
In studying the unusual properties of a special Witt basis of a Clifford geometric algebra with a Lorentz metric, a new concept of local duality makes it possible to define any real geometric algebra by complexifying this structure. Whereas…
We show that Lusztig's canonical basis for the degree two part of the Grassmannian coordinate ring is given by SL(k) web diagrams. Equivalently, we show that every SL(2) web immanant of a plabic graph for Gr(k,n) is an SL(k) web invariant.
Pairings are particular bilinear maps, and as any bilinear maps they factor through the tensor product as group homomorphisms. Besides, nothing seems to prevent us to construct pairings on other abelian groups than elliptic curves or more…
Building on the iHopf algebra realization of quasi-split universal iquantum groups developed in a prequel, we construct the dual canonical basis for a universal iquantum group of arbitrary finite type, which are further shown to be…
Let $(G,G')$ be a reductive dual pair of a symplectic group and an orthogonal group over a finite field of odd characteristic. The Howe correspondence establishes a correspondence between a subset of irreducible characters of $G$ and a…
Let $G$ be a classical linear algebraic group over an algebraically closed field, and let $\mathfrak{n}$ denote the subset of nilpotent elements in its Lie algebra. In this paper we study a partial order on the $G$-orbits in $\mathfrak{n}$…
We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov's first and second Poincare duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz…
We consider categories of rational maps to algebraic groups and study existence and construction of universal objects for such categories, using the duality theory of Laumon 1-motives. In particular, we obtain functorial descriptions of the…