Related papers: Homomorphisms of quantum groups
Quantum families of maps between quantum spaces are defined and studied. We prove that quantum semigroup (and sometimes quantum group) structures arise naturally on such objects out of more fundamental properties. As particular cases we…
We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a gamma-equivariant G-module A, when a separate group "gamma" acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology of…
We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in…
In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott-Samelson bimodules…
We present two examples of actions of non-regular locally compact quantum groups on their homogeneous spaces. The homogeneous spaces are defined in a way specific to these examples, but the definitions we use have the advantage of being…
It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled `generalized global symmetries.' In this paper, we describe how those higher-form symmetries can be understood mathematically as…
The characterization of the quantum ensemble is a fundamental issue in quantum information theory and foundations. The ensemble is also useful for various quantum information processing. To characterize the quantum ensemble, in this…
A brief exposition of the general theory of characteristic classes of quantum principal bundles is given. The theory of quantum characteristic classes incorporates ideas of classical Weil theory into the conceptual framework of…
The "quantum duality principle" states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie…
Wave-particle duality relations express the fact that knowledge about the path a particle took suppresses information about its wave-like properties, in particular, its ability to generate an interference pattern. Recently, duality…
We show how classical and quantum dualities, as well as duality relations that appear only in a sector of certain theories ("emergent dualities"), can be unveiled, and systematically established. Our method relies on the use of morphisms of…
Classification of different forms of quantum entanglement is an active area of research, central to development of effective quantum computers, and similar to classification of error-correction codes, where code duality is broadened to…
A simpler definition for a class of two-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form. Their positive parts turn out to be 2-cocycle deformations of each other under some conditions. An…
Induced representations for quantum groups are defined starting from coisotropic quantum subgroups and their main properties are proved. When the coisotropic quantum subgroup has a suitably defined section such representations can be…
Looking for a quantum-mechanical implementation of duality, we formulate a relation between coherent states and complex-differentiable structures on classical phase space ${\cal C}$. A necessary and sufficient condition for the existence of…
We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups…
We classify the compact quantum groups $A_u(Q)$ (resp. $B_u(Q)$) up to isomorphism when $Q>0$ (resp. when $Q \bar{Q} \in {\mathbb R} I_n$). We show that the general $A_u(Q)$'s and $B_u(Q)$'s for arbitrary $Q$ have explicit decompositions…
We prove a duality theorem for quantum groupoid (weak Hopf algebra) actions that extends the well-known result for usual Hopf algebras.
We study quantization of a class of inhomogeneous Lie bialgebras which are crossproducts in dual sectors with Abelian invariant parts. This class forms a category stable under dualization and the double operations. The quantization turns…
We define concepts of amenability and co-amenability for algebraic quantum groups in the sense of A. Van Daele. We show that co-amenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or…