Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics

2014 ◽  
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Quantum Groups ◽  
Recent Advances

scholarly journals A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

2009 ◽  
Vol 52 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Jakob Cimprič
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Representation Theorem ◽  
Commutative Rings ◽  
Real Algebraic Geometry ◽  
New Approach ◽  
Noncommutative Version ◽  
Quadratic Modules ◽  
Rings With Involution ◽  
Associative Rings

AbstractWe present a new approach to noncommutative real algebraic geometry based on the representation theory of C*-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative C*-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

scholarly journals Relative Calabi–Yau structures

2019 ◽  
Vol 155 (2) ◽  
pp. 372-412 ◽  
Author(s):  
Christopher Brav ◽  
Tobias Dyckerhoff
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Riemann Surfaces ◽  
Fukaya Categories ◽  
Composition Law

We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

scholarly journals QUANTUM GROUPS AT ROOTS OF UNITY AND MODULARITY

2006 ◽  
Vol 15 (10) ◽  
pp. 1245-1277 ◽  
Author(s):  
STEPHEN F. SAWIN
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Roots Of Unity ◽  
Quantum Field ◽  
Root Of Unity ◽  
Basic Representation ◽  
Topological Quantum ◽  
Quantum Version ◽  
Simply Connected ◽  
Connected Lie Group

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations of representation theory of quantum groups at roots of unity which applied only to quantizations of the simplest groups, or to certain fractional levels, or only to the projective form of the group. The second half of this paper applies the representation theory to give a sequence of results crucial to applications in topology. In particular, for each compact, simple, simply-connected Lie group we show that at each integer level the quotient category is in fact modular (thus leading to a Topological Quantum Field Theory), we determine when at fractional levels the corresponding category is modular, and we give a quantum version of the Racah formula for the decomposition of the tensor product.

GEOMETRY OF QUANTUM HOMOGENEOUS VECTOR BUNDLES AND REPRESENTATION THEORY OF QUANTUM GROUPS I

1999 ◽  
Vol 11 (05) ◽  
pp. 533-552 ◽  
Author(s):  
A. R. GOVER ◽  
R. B. ZHANG
Keyword(s):  
Representation Theory ◽  
Finite Type ◽  
Quantum Groups ◽  
Vector Bundles ◽  
Homogeneous Spaces ◽  
Projective Modules ◽  
Geometrical Structures ◽  
Induced Representations ◽  
Compact Quantum Groups ◽  
Homogeneous Vector

Quantum homogeneous vector bundles are introduced in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies, and their sections furnish finite type projective modules over algebras of functions on quantum homogeneous spaces. Further properties of the quantum homogeneous vector bundles are investigated, and applied to the study of the geometrical structures of induced representations of quantum groups.

scholarly journals Two Maps on Affine Type $A$ Crystals and Hecke Algebras

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Nicolas Jacon
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Fock Space ◽  
Hecke Algebras ◽  
Type A ◽  
Crystal Basis ◽  
Affine Type

We use the crystal isomorphisms of the Fock space to describe two maps on partitions and multipartitions which naturally appear in the crystal basis theory for quantum groups in affine type $A$ and in the representation theory of Hecke algebras of type $G(l,l,n)$. 

scholarly journals THE MONOID OF FAMILIES OF QUIVER REPRESENTATIONS

2002 ◽  
Vol 84 (3) ◽  
pp. 663-685 ◽  
Author(s):  
MARCUS REINEKE
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Representation Theory ◽  
Quantum Group ◽  
Subject Classification ◽  
Quiver Representations ◽  
Enveloping Algebras ◽  
Geometric Methods ◽  
Quantized Enveloping Algebras ◽  
Monoid Structure

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, that is, subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analysed by representation-theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.

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