ec 2 00 3 A utom orphism s ofassociative algebras and noncom m utative geom etry (original) (raw)

Coordinate calculi on associative algebras

1995

A new notion of an optimal algebra for a first order free differential was introduced in [1]. Some relevant examples are indicated. Quadratic identities in the optimal algebras and calculi on quadratic algebras are studied. Canonical construction of a quantum de Rham complex for the coordinate differential is proposed. The relations between calculi and various generalizations of the Yang-Baxter equation are established. * Invited talk presented by V.K. Kharchenko at the XXXq Winter School in Karpacz.

Classification of differentials on quantum doubles and finite noncommutative geometry

2002

We discuss the construction of finite noncommutative geometries on Hopf algebras and finite groups in the 'quantum groups approach'. We apply the author's previous classification theorem, implying that calculi in the factorisable case correspond to blocks in the dual, to classify differential calculi on the quantum codouble D * (G) = kG>◭k(G) of a finite group G. We give D * (S 3) as an example including its exterior algebra and lower cohomology. We also study the calculus on D * (A) induced from one on a general Hopf algebra A in general and specialise to D * (G) = U (g)>◭k[G] as a noncommutative isometry group of an enveloping algebra U (g) as a noncommutative space.

Differential calculi on some quantum prehomogeneous vector spaces

Journal of Mathematical Physics, 2007

This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices.

A class of bicovariant differential calculi on hopf algebras

Letters in Mathematical Physics, 1992

We introduce a large class of bicovariant differential calculi on any quantum group A, associated to Ad-invariant elements. For example, the deformed trace element on SL q (2) recovers Woronowicz' 4D ± calculus. More generally, we obtain a sequence of differential calculi on each quantum group A(R), based on the theory of the corresponding braided groups B(R).

Nonassociative algebras: a framework for differential geometry

International Journal of Mathematics and Mathematical Sciences, 2003

A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the “torsionless” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie algeb...

Quantum maps for deformed algebras

1990

We find explicit functionals that map SU(2) algebra generators to those of several quantum deformations of that algebra. We indicate how any such quantized algebra can be mapped to any other, and how representations of any such algebra can be expressed as simple functions of SU(2) representations. The representation theory and its comultiplication rules are thus systematized and streamlined by direct reference to their SU(2) correspondents. We speculate on quantum deformations of the Virasoro algebra.

Canonical differential calculus on quantum general linear groups and supergroups

Physics Letters B, 1992

We specify a set of relations between non-commuting matrix elements and their differentials, defined in terms of an R-matrix satlS~,lng the braid relation, which are unlquel) determined by the requirements of conststenc) with the relations between noncommuting coordinates and their differentials We also give a necessaD' condlllOn for the existence of a matrix inverse (antipode) in the form of an additional equation to be satisfied by the R-matrix Recently the theory of quantum groups has given rise to considerable interest among both physicists and mathematicians in spaces described b~ noncommuting coordinates, and In the differential calculus on such spaces. There have been two main lines of development. One [ 1-5 ], initiated by Wess and Zumlno, follows Mamn's emphasis [6,7] on quantum spaces as the pnma~ objects, differential forms arc defined in terms of non-commuting coordinates, and the algebraic and differential properties of quantum groups acting on these spaces are deduced from the properties of the spaces. In the other approach

Quantum Calculi: differential forms and vector fields in noncommutative geometry

arXiv (Cornell University), 2022

In this paper, we revise the concept of noncommutative vector fields introduced previously in [1, 2], extending the framework, adding new results and clarifying the old ones. Using appropriate algebraic tools certain shortcomings in the previous considerations are filled and made more precise. We focus on the correspondence between so-called Cartan pairs and first-order differentials. The case of free bimodules admitting more friendly "coordinate description" and their braiding is considered in more detail. Bimodules of right/left universal vector fields are explicitly constructed.

Remarks on bicovariant differential calculi and exterior Hopf algebras

1993

We show that every bicovariant differential calculus over the quantum group d defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra of ~ defines bicovariant bimodule over d. We also study a quasitriangular structure on exterior Hopf algebras in some detail. (1991). 81R50, 16W30, 58B30, 46L87, 05A30.