Operator algebra

About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.

Bicovariant quantum algebras and quantum Lie algebras

Abstract: A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun $$(\mathfrak{G}_q )$$ toU q g, given by elements of the pure braid group. These operators—the “reflection matrix”Y≡L + SL − being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO q (N).

Rigidity results for Bernoulli actions and their von Neumann algebras [after Sorin Popa]

Abstract: We survey Sorin Popa's recent work on Bernoulli actions. The paper was written on the occasion of the Bourbaki seminar. Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra, yielding the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II_1 factors with prescribed countable fundamental group.

Spectral theory of families of self-adjoint operators

Abstract: Comments to the introduction.- I Families of Commuting Normal Operators.- 1. Spectral Analysis of Countable Families of Commuting Self-Adjoint Operators (CSO).- 2. Unitary Representations of Inductive Limits of Commutative Locally Compact Groups.- 3. Differential Operators With Constant Coefficients In Spaces of Functions of Infinitely Many Variables.- Inductive Limits of Finite-Dimensional Lie Algebras and Their Representations.- 4. Canonical Commutation Relations (CCR) of Systems with Countable Degrees of Freedom.- 5. Unitary Representations of The Group of Finite SU(2)-Currents on A Countable Set.- 6. Representations of The Group of Upper Triangular Matrices.- 7. A Class of Inductive Limits of Groups and Their Representations.- Collections of Unbounded Self-Adjoint operators Satisfying General Relations.- 8. Anticommuting Self-Adjoint Operators.- 9. Finite and Countable Collections of Gradedcommuting Self-Adjoint Operators (GCSO).- 10. Collections Of Unbounded CSO (Ak) And CSO (Bk) Satisfying General Commutation Relations.- Representations of Operator Algebras And Non-Commutative Random Sequences.- 11. C* -ALGEBRASU0? And Their Representations.- 12. Non-Commutative Random Sequences and Methods for Their Construction.

Virasoro Algebra, Vertex Operators, Quantum {Sine-Gordon} and Solvable Quantum Field Theories

Abstract: The relationship between the conformal field theories and the soliton equations (KdV, MKdV and Sine–Gordon, etc.) at both quantum and classical levels is discussed. The quantum Sine–Gordon theory is formulated canonically. Its Hamiltonian is the vertex operator with respect to the Feigin–Fuchs–Miura form of the Virasoro algebra with central charge $c\le1$. It is found that the quantum conserved quantities of the Sine–Gordon-MKdV hierarchy are expressed as polynomial functions of the Virasoro generators. In other words, an infinite set of mutually commutative polynomial functions of the Virasoro generators is obtained. A very simple recursion formula for the quantum conserved quantities is found for the special case of $\beta^2_c=8\pi$ ($\beta_c$ is the coupling constant in Coleman’s theory of quantum Sine–Gordon).