Related papers: Basic quantum algebra
Quantum field planes furnish a noncommutative differential algebra $\Omega$ which substitutes for the commutative algebra of functions and forms on a contractible manifold. The data required in their construction come from a quantum field…
Quantum groups and quantum homogeneous spaces - developed by several authors since the 80's - provide a large class of examples of algebras which for many reasons we interpret as `coordinate algebras' over noncommutative spaces. This…
Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras. On them the quantum Lie…
Some very elementary ideas about quantum groups and quantum algebras are introduced and a few examples of their physical applications are mentioned.
A very elementary introduction to quantum algebras is presented and a few examples of their physical applications are mentioned.
Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well defined mathematical structure from the quantum group symmetry. In turn such quantum spaces can be…
A construction of the noncommutative-geometric counterparts of classical classifying spaces is presented, for general compact matrix quantum structure groups. A quantum analogue of the classical concept of the classifying map is introduced…
The concept of a quantum algebra is made easy through the investigation of the prototype algebras $u_{qp}(2)$, $su_q(2)$ and $u_{qp}(1,1)$. The latter quantum algebras are introduced as deformations of the corresponding Lie algebras~; this…
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics.…
In the present paper we propose a new approach to quantum fields in terms of category algebras and states on categories. We define quantum fields and their states as category algebras and states on causal categories with partial involution…
A detailed account of the construction of a homogeneous space for the quantum "az+b" group is presented. The homogeneous space is described by a commutative C*-algebra which means that it is a classical space. Then a covariant differential…
The Lie and module (Rinehart) algebraic structure of vector fields of compact support over C infinity functions on a (connected) manifold M define a unique universal non-commutative Poisson * algebra. For a compact manifold, a…
Quantum Lie algebras are generalizations of Lie algebras whose structure constants are power series in $h$. They are derived from the quantized enveloping algebras $\uqg$. The quantum Lie bracket satisfies a generalization of antisymmetry.…
The purpose of this paper is to make the theory of vertex algebras trivial. We do this by setting up some categorical machinery so that vertex algebras are just ``singular commutative rings'' in a certain category. This makes it easy to…
We reconstruct a quantum group associated with any Lie algebra together with its representation theory from twisted homologies of generalized configuration spaces of disks. Along the way it brings new combinatorics to the theory, but our…
We consider a class of C*-algebras C(X) associated with quantum spaces such as spheres, projective spaces, and lens spaces. We introduce a non-self-adjoint operator algebra A together with an explicit functor from the category of…
We propose a general scheme for the "logic" of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non Commutative Geometry. It involves Baire*-algebras, the non-commutative…
In this paper we try to define the higher dimensional analogues of vertex algebras. In other words we define algebras which we hope have the same relation to higher dimensional quantum field theories that vertex algebras have to one…
We encapsulate the basic notions of the theory of vertex algebras into the construction of a comonad on an appropriate category of formal distributions. Vertex algebras are recovered as coalgebras over this comonad.
In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum…