Related papers: Quantum de Rham complex on ${\mathbb A}_{h}^{1|2}$
We show that a suitable deformation of the algebra $h_k(1)$ of the creation and annihilation operators for a complex scalar field, initially quantized in Minkowski space--time, induces the canonical quantization of the same field in a…
We employ holography to calculate the quantum complexity of $T\bar{T}$-deformation, utilizing the complexity equals volume (CV) and the complexity equals action (CA) proposals within the bulk spacetime with a finite radius cutoff. We find…
Those elements of the second de Rham cohomology group of a connected, oriented Riemannian manifold which map its second homotopy group to zero or to a discrete subgroup of the reals induce deformations of the path algebra of the manifold.…
The space $\Gamma_X$ of all locally finite configurations in a Riemannian manifold $X$ of infinite volume is considered. The deRham complex of square-integrable differential forms over $\Gamma_X$, equipped with the Poisson measure, and the…
A bi-invariant differential 2-form on a Lie group G is a highly constrained object, being determined by purely linear data: an Ad-invariant alternating bilinear form on the Lie algebra of G. On a compact connected Lie group these have an…
We describe the obstruction to decomposing in degrees $\leq p$ the de Rham complex of a smooth variety over a perfect field $k$ of characteristic $p$ that lifts over $W_2(k)$, and show that there exist liftable smooth projective varieties…
For the case of algebraic curves - compact Riemann surfaces - it is shown that de Rham cohomology group $H^{1}_{\mathrm{dR}}(X,\mathbb{C})$ of a genus $g$ Riemann surface $X$ has a natural structure of a symplectic vector space. Every…
A general deformation of the Heisenberg algebra is introduced with two deformed operators instead of just one. This is generalised to many variables, and permits the simultaneous existence of coherent states, and the transposition of…
This paper suveys some recent algebraic developments in two parameter Quantum deformations and their Nonstandard (or Jordanian) counterparts. In particular, we discuss the contraction procedure and the quantum group homomorphisms associated…
Let $i: \mathrm{L} \hookrightarrow \mathrm{X}$ be a compact K\"{a}hler Lagrangian in a holomorphic symplectic variety $\mathrm{X}/\mathbf{C}$. We use deformation quantisation to show that the endomorphism differential graded algebra…
We review several procedures of quantization formulated in the framework of (classical) phase space M. These quantization methods consider Quantum Mechanics as a "deformation" of Classical Mechanics by means of the "transformation" of the…
We generalize the classical de Rham decomposition theorem for Riemannian manifolds to the setting of geodesic metric spaces of finite dimension.
Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the…
We consider deformed special relativity (DSR) theories on commutative space-time, perhaps as an first approximation to a noncommutative space-time formulation. The corresponding field theories in general possess derivatives of all orders.…
We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity. The relation of the structure of…
In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra A are controlled by the second Hochschild cohomology group of A. More generally, any n-parameter first order deformation of A gives, due to…
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…
Denote $\fm_2$ the infinite dimensional $\N$-graded Lie algebra defined by the basis $e_i$ for $i\geq 1$ and by relations $[e_1,e_i]=e_{i+1}$ for all $i\geq 2$, $[e_2,e_j]=e_{j+2}$ for all $j\geq 3$. We compute in this article the bracket…
The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical…
Starting from the classical r-matrix of the non-standard (or Jordanian) quantum deformation of the sl(2,R) algebra, new triangular quantum deformations for the real Lie algebras so(2,2), so(3,1) and iso(2,1) are simultaneously constructed…