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We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If $K$ is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra $K^n$:…
We formulate semi-classical field theory as an approximate decoherence-free-subspace of a finite-dimensional quantum-gravity hilbert space. A complementarity construction can be realized as a unitary transformation which changes the…
For applications of group theory in quantum mechanics, one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some Hamiltonian of interest. Here we use…
We introduce a systematic approach for treating the large N limit of matrix field theories.
Mathematical tools related to coherence theory and classical-quantum equivalence, due to Wigner and Glauber, are essential to modern, practical and empirical understanding of electromagnetics in areas like quantum optics and…
A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in Z_{n_{i}}, i=1,...,k, is considered. Its Hilbert space is given by a k-fold tensor product of Hilbert spaces…
We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum…
We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the…
This paper is concerned with the structures introduced recently by the authors of the current paper concerning the multiplier Hopf $*$-graph algebras and also the Cuntz-Krieger algebras and their relations with the $C^*$-graph algebras, and…
Quantum circuits have been widely used as a platform to simulate generic quantum many-body systems. In particular, random quantum circuits provide a means to probe universal features of many-body quantum chaos and ergodicity. Some such…
I describe, in the simplified context of finite groups and their representations, a mathematical model for a physical system that contains both its quantum and classical aspects. The physically observable system is associated with the space…
A modular quantum architecture is given for the space-time, particles, and fields of the Standard Model and General Relativity. It assumes a right-handed neutrino, so that based on their multiplet structure all fundamental fermions have…
Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems…
We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules…
In the standard model matter fields form complete representations of a grand unified group whereas Higgs fields belong to incomplete `split' multiplets. This remarkable fact is naturally explained by `local grand unification' in…
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we…
We use the mathematical structure of group algebras and $H^{+}$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are ${\rm SU}(2)$…
The 1/N expansion in quantum field theory is formulated within an algebraic framework. For a scalar field taking values in the $N$ by $N$ hermitian matrices, we rigorously construct the gauge invariant interacting quantum field operators in…
In this work we analyze complex scalar fields using a new framework where the object of noncommutativity $\theta^{\mu\nu}$ represents independent degrees of freedom. In a first quantized formalism, $\theta^{\mu\nu}$ and its canonical…
Inhomogeneous quantum groups are shown to be an effective algebraic tool in the study of integrable systems and to provide solutions equivalent to the Bethe ansatz. The method is illustrated on the 1D Heisenberg ferromagnet whose symmetry…