Related papers: A classification of generalized quantum statistics…
Let g be a complex, semisimple Lie algebra. Drinfeld showed that the quantum group associated to g is isomorphic as an algebra to the trivial deformation of the universal enveloping algebra of g. In this paper we construct explicitly such…
In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra $\mathfrak{g}$. This problem reduces to the classification of all Lie bialgebra structures on…
We show explicitly a generalised Lie algebra embedded in the positive and negative parts of the Drinfeld-Jimbo quantum groups of type A_n. Such a generalised Lie algebra satisfy axioms closely related to the ones found by S.L. Woronowicz.…
We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n \geq 2$, and on the simple associative superalgebras $M(m,n)$, $m, n \geq 1$, over an algebraically closed field: fine gradings up to…
We consider the compatibility conditions for a N-particle D-dimensional Wigner quantum oscillator. These conditions can be rewritten as certain triple relations involving anticommutators, so it is natural to look for solutions in terms of…
We introduce the notion of a generalized metric n-Leibniz algebra and show that there is a one-to-one correspondence between generalized metric n-Leibniz algebras and faithful generalized orthogonal representations of metric Lie algebras…
Multi-parameter versions U_p(g) and C_p[G] of the standard quantum groups U_q(g) and C_q[G] are considered where G is a semi-simple connected complex algebraic group and g is the Lie algebra of G. The primitive spectrum of C_p[G] is…
A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra…
Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic…
A geometric framework for quantum statistical estimation is used to establish a series of higher order corrections to the Heisenberg uncertainty relations associated with pairs of canonically conjugate variables. These corrections can be…
Interacting systems of particles with generalized statistics are considered on both classical and quantum level. It is shown that all possible quantum states and corresponding processes can be represented in terms of certain specific…
The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at $q=1$. We explain the relationship between the structure constants of…
Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for…
The coefficient algebra of a finite-dimensional Lie algebra on a finite-dimensional representation is defined as the subalgebra generated by all coefficients of the corresponding characteristic polynomial. We explore connections between…
We describe the definition of Jacobi (generalized)-Lie bialgebras $(({\bf{g}},\phi_{0}),({\bf{g}}^{*},X_{0}))$ in terms of structure constants of the Lie algebras ${\bf{g}}$ and ${\bf{g}}^{*}$ and components of their 1-cocycles $X_{0}\in…
We classify good Z-gradings of basic Lie superalgebras over an algebraically closed field of characteristic zero. Good Z-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the…
We overcome one of Bell's objections to `quantum measurement' by generalizing the definition to include systems outside the laboratory. According to this definition a {\sl generalized quantum measurement} takes place when the value of a…
In this didactical note I review in depth the rationale for using generalised canonical distributions in quantum statistics. Particular attention is paid to the proper definitions of quantum entropy and quantum relative entropy, as well as…
We define a quantum analogue of a class of generalized cluster algebras which can be viewed as a generalization of quantum cluster algebras defined in \cite{berzel}. In the case of rank two, we extend some structural results from the…
We extend the notion of the quantization of the coefficients of the ordinary cluster algebras to the generalized cluster algebras by Chekhov and Shapiro. In parallel to the ordinary case, it is tightly integrated with certain…