Related papers: Lie Algebraical Aspects of the Quantum Statistics.…
An infinite-dimensional Lie Algebra is proposed which includes, in its subalgebras and limits, most Lie Algebras routinely utilized in physics. It relies on the finite oscillator Lie group, and appears applicable to twisted noncommutative…
We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring $k$ with identity. These notions are applied to the study of pre-Lie $k$-algebras and, more generally, Lie-admissible…
The Fubini-Study metric of quantum state manifold generated by the operators which satisfy the Heisenberg Lie algebra is calculated. The similar problem is studied for the manifold generated by the so(3) Lie algebra operators. Using these…
We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is…
This is an addition to a series of papers [FL1, FL2, FL3, FL4], where we develop quaternionic analysis from the point of view of representation theory of the conformal Lie group and its Lie algebra. In this paper we develop split…
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…
The Lie algebra $so(2n+1)$ and the Lie superalgebra $osp(1/2n)$ are quantized in terms of $3n$ generators, called preoscillator generators. Apart from $n$ "Cartan" elements the preoscillator generators are deformed para-Fermi operators in…
In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra $\textbf{${\mathfrak g}_{\mathsf u}$}$ that extends $\mathbf{e_9}$. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds…
Quantum field theory can be physically regularized by modularizing it on several levels of aggregation. Since computation is already thoroughly modularized, physical experiments are treated here as quantum relativistic cellular computations…
The paper contains further development of the idea of field quantization introduced in M. Czachor, J. Phys. A: Math. Gen. {\bf 33} (2000) 8081-8103. The formalism is extended to the relativistic domain. The link to the standard theory is…
In this work we study the generation of photons inside an ideal cavity with resonantly oscillating boundaries in the presence of a two-level atom. We make use of Lie algebraic techniques to obtain an approximate time-evolution operator and…
We develop a rigorous framework for constructing Fock representations of quantum fields obeying generalized statistics associated with certain solutions of the spectral quantum Yang-Baxter equation. The main features of these…
This survey is about the fundamentals of the theory of finite dimensional Lie groups over the field of real numbers. The notion of the tangent space of a manifold at a point is considered to be defined via the well known chart and vector…
We introduce the theory of local minimal models for Kan simplicial manifolds, which provide the appropriate generalization of minimal Kan simplicial sets to geometric contexts. We use this to obtain the first proof of Lie's third theorem…
Fr\"olicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as C^\infty(M,G) or \Diff(M), but also projective limits of Lie groups are in a…
Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group $SU_q(2)$ is such…
Quantum real numbers are proposed by performing a quantum deformation of the standard real numbers $\R$. We start with the q-deformed Heisenberg algebra $\cLLq$ which is obtained by the Moyal $\ast$-deformation of the Heisenberg algebra…
The microscopic and the macroscopic properties of A-superstatistics, related to the class A(0,n-1)\equiv sl(1|n) of simple Lie superalgebras are investigated. The algebra sl(1|n) is described in terms of generators f_1^\pm, >..., f_n^\pm,…
Inspired by the quantum McKay correspondence, we consider the classical $ADE$ Lie theory as a quantum theory over $\mathfrak{sl}_2$. We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier…
Given a vector space with an action of a semi-simple Lie algebra, we can try to "categorify" this representation, which means finding a category where the generators of the Lie algebra act by functors. Such categorical representations arise…