Related papers: Universal Quantum Dimensions: I. $\gamma$-Independ…
We present the universal, in Vogel's sense, expression for the quantum dimension of Cartan product of an arbitrary number of adjoint and $X_2$ representations of simple Lie algebras. The same formula mysteriously gives quantum dimensions of…
In his study of finite (Vassiliev's) knot invariants,Vogel introduced the so-called universal parameters, belonging to the projective plane, which particularly parameterize the simple Lie algebras by the Vogel's table. Subsequently a number…
The problem of uniqueness of universal formulae for (quantum) dimensions of simple Lie algebras is investigated. We present generic functions, which multiplied by a universal (quantum) dimension formula, preserve both its structure and its…
We conjecture the connection between $su$ and $so$ members of universal, in Vogel's sense, multiplets. The key element is the notion of the {\it vertical componentwise sum} $\oplus_v$ of Young diagrams. Representations in the decomposition…
Global dimensions for fusion categories defined by a pair (G,k), where G is a Lie group and k a positive integer, are expressed in terms of Lie quantum superfactorial functions. The global dimension is defined as the square sum of quantum…
We represent in the universal form restricted one-instanton partition function of supersymmetric Yang-Mills theory. It is based on the derivation of universal expressions for quantum dimensions (universal characters) of Cartan powers of…
The method of four-dimensional Causal Dynamical Triangulations provides a background-independent definition of the sum over geometries in quantum gravity, in the presence of a positive cosmological constant. We present the evidence…
The concept of universality has shaped our understanding of many-body physics, but is mostly limited to homogenous systems. Here, we present a study of universality on a non-homogeneous graph, the long-range diluted graph (LRDG). Its…
The purpose of this work is to present a method based on the factorizations used in one dimensional quantum mechanics in order to find the symmetries of quantum and classical superintegrable systems in higher dimensions. We apply this…
Loop Quantum Gravity heavily relies on a connection formulation of General Relativity such that 1. the connection Poisson commutes with itself and 2. the corresponding gauge group is compact. This can be achieved starting from the Palatini…
We discuss algebraic universality in the sense of P. Vogel for the simplest refined quantity, the Macdonald dimensions. The main known source of universal quantities is given by Chern-Simons theory. Refinement of Chern-Simons theory means…
A variety of three-dimensional left-covariant differential calculi on the quantum group $SU_q(2)$ is considered using an approach based on global $ U(1) $ -covariance. Explicit representations of possible $q $-Lie algebras are constructed…
We study the quantum cosmology of a flat Friedmann-Lema\^{i}tre-Robertson-Walker universe filled with a (free) massless scalar field and a perfect fluid that represents radiation or a cosmological constant whose value is not fixed by the…
We present a universal formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra (i.e., a universal formula for the Hilbert functions of homogeneous complex contact manifolds), as well as…
The antisymmetric square of the adjoint representation of any simple Lie algebra is equal to the sum of adjoint and $X_2$ representations. We present universal formulae for quantum dimensions of an arbitrary Cartan power of $X_2$. They are…
Quantum relativity as a generalized, or rather deformed, version of Einstein relativity with a linear realization on a classical six-geometry beyond the familiar setting of space-time offer a new framework to think about the quantum…
Considering a n-dimensional general spacetime, we deduce its 4-dimensional Einstein equation and Friedman equations, and discover a general dual relation between the scale factor $a(t)$ of our universe and the scale factor $B(t)$ of extra…
In accordance with P. Vogel, a set of algebra structures in Chern-Simons theory can be made universal, independent of a particular family of simple Lie algebras. In particular, this means that various quantities in the adjoint…
A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is…
Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics…