Related papers: Non-commutative Geometry of Homogenized Quantum $\…
The paper is devoted to the investigation of finite dimensional commutative nilpotent (associative) algebras N over an arbitrary base field of characteristic zero. Due to the lack of a general structure theory for algebras of this type (as…
We study a three dimensional non-commutative space emerging in the context of three dimensional Euclidean quantum gravity. Our starting point is the assumption that the isometry group is deformed to the Drinfeld double D(SU(2)). We…
An algebra homomorphism $\psi$ from the q-deformed algebra $U_q({\rm iso}_2)$ with generating elements $I$, $T_1$, $T_2$ and defining relations $[I,T_2]_q=T_1$, $[T_1,I]_q=T_2$, $[T_2,T_1]_q=0$ (where $[A,B]_q=q^{1/2}AB-q^{-1/2}BA$) to the…
The ``local'' structure of a quantum group G_q is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra U_q(g), which is a Hopf algebra deformation of the universal enveloping…
We explicitly construct an L$_\infty$ algebra that defines U$_{\star}(1)$ gauge transformations on a space with an arbitrary non-commutative and even non-associative star product. Matter fields are naturally incorporated in this scheme as…
By "parallelogram geometry" we mean the elementary, "commutative", geometry corresponding to vector addition, and by "trapezoid geometry" a certain "non-commutative deformation" of the former. This text presents an elementary approach via…
Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper…
We study noncommutative differential structures on the group of permutations $S_N$, defined by conjugacy classes. The 2-cycles class defines an exterior algebra $\Lambda_N$ which is a super analogue of the Fomin-Kirillov algebra $\CE_N$ for…
Trying to connect a fundamentally non-commutative spacetime with the conservative perturbative approach to quantum gravity, we are led to the natural question: are non-commutative geometrical effects already present in the regime where…
In our previous publications we have introduced a differential calculus on the algebra $U(gl(m))$ based on a new form of the Leibniz rule which differs from that usually employed in Noncommutative Geometry. This differential calculus…
In the present paper we construct all typical finite-dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ at generic deformation parameter $q$. As in the non-deformed case the finite-dimensional…
We study real nonsingular projective cubic fourfolds up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in the…
Recently B. Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra via generators and relations. To obtain this presentation they defined an algebra $\boxtimes$ by generators and relations, and displayed…
The nonstandard q-deformation $U'_q({\rm so}_n)$ of the universal enveloping algebra $U({\rm so}_n)$ has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional…
The FRT quantum Euclidean spaces $O_q^N$ are formulated in terms of Cartesian generators. The quantum analogs of N-dimensional Cayley-Klein spaces are obtained by contractions and analytical continuations. Noncommutative constant curvature…
In our previous publications we have introduced analogs of partial derivatives on the algebras U(gl(N)). In the present paper we compare two methods of introducing these analogs: via the so-called quantum doubles and by means of a…
We survey some aspects of the theory of noncommutative manifolds focusing on the noncommutative analogs of two-dimensional tori and low-dimensional spheres. We are particularly interested in those aspects of the theory that link the…
The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a…
Having in mind the significance of parity (reflection) in various areas of physics, the single-mode and two-mode Wigner algebras are considered adding to them a reflection operator. The associated deformed $sl(2, R)$ algebra,…
Hilbert space representations of the cross product *-algebras of the Hopf *-algebra U_q(su_2) and its module *-algebras O(S^2_{qr}) of Podles spheres are investigated and classified by describing the action of generators. The…