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Starting from a Unified Field Theory (UFT) proposed previously by the author, the possible fermionic representations arising from the same spacetime are considered from the algebraic and geometrical viewpoint. We specifically demonstrate in…

Under suitable hypotheses on the ground field and on the matrix $M$, we discuss existence, uniqueness and properties of some additive decompositions of $M$ and of its image through a convergent series.

We endow a non-semisimple category of modules of unrolled quantum sl(2) with a Hermitian structure. We also prove that the TQFT constructed in arXiv:1202.3553 using this category is Hermitian. This gives rise to projective representations…

These are the notes from my courses on the arithmetic of quadratic forms.

This is a philosophical paper. It claims that there is a gap to be filled in the relationship between complexity theory (CT) and quantum theory (QT). This gap concerns two very distinct understandings of time. The paper provides the ground…

We construct two-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent…

An algebraic quantum field theory (AQFT) may be expressed as a functor from a category of spacetimes to a category of algebras of observables. However, a generic category $\mathsf{C}$ whose objects admit interpretation as spacetimes is not…

Withdrawn paper because the results are published in math.AT/0308054 and math.AT/0308063.

In previous papers (arxiv:math/0612370 and arxiv:0909.1342) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,F). Here we construct the analytic index of an elliptic operator as a…

It has been observed that, given an algebraic quantum field theory (AQFT) on a manifold $M$ and an open cover $\{M_\alpha\}$ of $M$, it is typically not possible to recover the global algebra of observables on $M$ by simply gluing the…

Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the…

The goal of this article is to present a survey of the recent theory of plurisubharmonic functions of quaternionic variables, and its applications to theory of valuations on convex sets and HKT-geometry (HyperK\"ahler with Torsion). The…

The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency…

This is an expository article for the Encyclopedia of Mathematical Physics on the subject in the title.

We give an overview of our effort to introduce (dual) semicanonical bases in the setting of symmetrizable Cartan matrices.

We have introduced q-analogues of bounded symmetric domains in our work q-alg/9703005. Given the simplest ones among those, the works q-alg/9603012 and math.QA/9803110 announce the relations describing the algebras of functions,…

We give an account of the current state of the approch to quantum field theory via Hopf algebras and Hochschild cohomology. We emphasize the versatility and mathematical foundation of this algebraic structure, and collect algebraic…

Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such…

This paper has been removed by arXiv administrators because it plagiarizes astro-ph/0611537, astro-ph/0506256, astro-ph/0203033, astro-ph/0311128, gr-qc/0505144, astro-ph/0611460, and astro-ph/0610840.

We investigate several Hopf algebras of diagrams related to Quantum Field Theory of Partitions and whose product comes from the Hopf algebras WSym or WQSym respectively built on integer set partitions and set compositions. Bases of these…