Buiding upon Thursday's seminar, "The Noncommutative Geometry of Real Calculi," this presentation explores the future of real calculi. By considering what has already been accomplished we shall discuss several interesting prospects for future research and discuss the hurdles that may lie ahead, offering a roadmap for the next phase in our exploration of this algebraic framework for noncommutative geometry.

A soldering form on a principal G-bundle P over a manifold M is a horizontal and G-equivariant differential 1-form on P with values in a linear representation V of G that solders (or attaches) P to M by identifying the associated vector bundle (PxV)/G with the tangent bundle TM. This allows one, among other things, to introduce the notion of torsion (of a connection on P) which is a way of characterizing a twist or screw of a moving frame around a curve. In this talk we reason that derivation-based differential calculus d’après Dubois-Violette provides a natural framework for generalizing the theory of soldering forms to the noncommutative setting. Furthermore, we explain how our findings may be utilized to construct new examples of pseudo-Riemannian calculi.

P. Ševera's approach in Sel. Math. 22 (2016) to the quantization of Lie bialgebras is an important simplification and precision of the classical results by Etingof-Kazhdan's in 1996. The result is used as a tool in some approaches of deformation quantization. In my talk I also sketch a pedestrian way to the understanding of the Drinfel'd associator and its identities. Our work is in collaboration with Andrea Rivezzi and Thomas Weigel from Milano-Bicocca.

We will explain the following remarkable fact. For any commutative nonassociative algebra over a field satisfying a univariate identity and any its idempotent c, its Peirce spectrum necessarily contains the universal eigenvalue 1/2. Furthermore, the corresponding Peirce 1/2-subspace satisfies the Jordan algebra fusion rules.

The symmetric algebra S(g) of a lie algebra g=Lie G carries the standard Lie-Poisson structure. A subalgebra C of S(g) is Poisson-commutative, if the Poisson brackets vanishes on it. The interest in these objects appeared initially because of their role in complete integrability of Hamiltonian systems. Poisson-commutative subalgebras of S(g) are also important tools for the study of geometry of the coadjoint representation of G. We present a construction of a Poisson-commutative subalgebra associated with a finite order automorphism of g. Then discuss finite order automorphisms of reductive Lie algebras, in particular, classification in terms of Kac’s diagrams.

Some constructions, examples and open problems on Hom-algebras and twisted derivations will be presented.

Differential calculus in noncommutative geometry come in several different flavors, and one of the more concrete versions goes by the name of derivation based differential calculus. This calculus is built from a disinguished Lie algebra of derivations, and lead to the formulation of differential forms, cohomology and connections. A fundamental question in noncommutative Riemannian geometry is the existence and uniqueness of a torsion free and metric compatible connection; i.e a Levi-Civita connection. For the moment, there are no general results addressing this question in this context, and I will present a case study based on a simple quiver path algebra, and show how the existence of a Levi-Civita connection depend on the choice of a Lie algebra of derivations.