Appendix A Open Problems

A reader may already note numerous objects and results, which deserve a further consideration. It may also worth to state some open problems explicitly. In this section we indicate several directions for further work, which go through four main areas described in the paper.

A.1 Geometry

Geometry is most elaborated area so far, yet many directions are waiting for further exploration.

1. Möbius transformations (1) with three types of hypercomplex units appear from the action of the group SL₂(ℝ) on the homogeneous space SL₂(ℝ)/H [194], where H is any subgroup A, N, K from the Iwasawa decomposition (7). Which other actions and hypercomplex numbers can be obtained from other Lie groups and their subgroups?
2. Lobachevsky geometry of the upper half-plane is extremely beautiful and well-developed subject [26] [71]. However the traditional study is limited to one subtype out of nine possible: with the complex numbers for Möbius transformation and the complex imaginary unit used in FSCc (??). The remaining eight cases shall be explored in various directions, notably in the context of discrete subgroups [24].
3. The Fillmore-Springer-Cnops construction, see subsection ??, is closely related to the orbit method [161] applied to SL₂(ℝ). An extension of the orbit method from the Lie algebra dual to matrices representing cycles may be fruitful for semisimple Lie groups.
5. A development of a discrete version of the geometrical notions can be derived from suitable discrete groups. A natural first example is the group SL₂(F), where F is a finite field, e.g. ℤ_p the field of integers modulo a prime p.

A.2 Analytic Functions

It is known that in several dimensions there are different notions of analyticity, e.g. several complex variables and Clifford analysis. However, analytic functions of a complex variable are usually thought to be the only options in a plane domain. The following seems to be promising:

1. Development of the basic components of analytic function theory (the Cauchy integral, the Taylor expansion, the Cauchy-Riemann and Laplace equations, etc.) from the same construction and principles in the elliptic, parabolic and hyperbolic cases and respective subcases.
2. Identification of Hilbert spaces of analytic functions of Hardy and Bergman types, investigation of their properties. Consideration of the corresponding Toeplitz operators and algebras generated by them.
3. Application of analytic methods to elliptic, parabolic and hyperbolic equations and corresponding boundary and initial values problems.
4. Generalisation of the results obtained to higher dimensional spaces. Detailed investigation of physically significant cases of three and four dimensions.
5. There is a current interest in construction of analytic function theory on discrete sets. Our approach is ready for application to an analytic functions in discrete geometric set-up outlined in item 5 above.

A.3 Functional Calculus

The functional calculus of a finite dimensional operator considered in Section 1 is elementary but provides a coherent and comprehensive treatment. It shall be extended to further cases where other approaches seems to be rather limited.

1. Nilpotent and quasinilpotent operators have the most trivial spectrum possible (the single point {0}) while their structure can be highly non-trivial. Thus the standard spectrum is insufficient for this class of operators. In contract, the covariant calculus and the spectrum give complete description of nilpotent operators—the basic prototypes of quasinilpotent ones. For quasinilpotent operators the construction will be more complicated and shall use analytic functions mentioned in 1.
2. The version of covariant calculus described above is based on the discrete series representations of SL₂(ℝ) group and is particularly suitable for the description of the discrete spectrum (note the remarkable coincidence in the names).

   It is interesting to develop similar covariant calculi based on the two other representation series of SL₂(ℝ): principal and complementary [240]. The corresponding versions of analytic function theories for principal [170] and complementary series [191] were initiated within a unifying framework. The classification of analytic function theories into elliptic, parabolic, hyperbolic [191, 185] hints the following associative chains:

   Representations	—	Function Theory	—	Type of Spectrum
   discrete series	—	elliptic	—	discrete spectrum
   principal series	—	hyperbolic	—	continuous spectrum
   complementary series	—	parabolic	—	residual spectrum

3. Let a be an operator with a∈$ⅅ^_$ and ||a^k||< C k^p. It is typical to consider instead of a the power bounded operator ra, where 0<r< 1, and consequently develop its H_∞ calculus. However such a regularisation is very rough and hides the nature of extreme points of a. To restore full information a subsequent limit transition r→ 1 of the regularisation parameter r is required. This make the entire technique rather cumbersome and many results have an indirect nature.

   The regularisation a^k→ a^k/k^p is more natural and accurate for polynomially bounded operators. However it cannot be achieved within the homomorphic calculus Defn. 1 because it is not compatible with any algebra homomorphism. Albeit this may be achieved within the covariant calculus Defn. 4 and Bergman type space from 2.

4. Several non-commuting operators are especially difficult to treat with functional calculus Defn. 1 or a joint spectrum. For example, deep insights on joint spectrum of commuting tuples [319] refused to be generalised to non-commuting case so far. The covariant calculus was initiated [168] as a new approach to this hard problem and was later found useful elsewhere as well. Multidimensional covariant calculus [180] shall use analytic functions described in 4.
5. As we noted above there is a duality between the co- and contravariant calculi from Defins. 23 and 25. We also seen in Section 1 that functional calculus is an example of contravariant calculus and the functional model is a case of a covariant one. It is interesting to explore the duality between them further.

A.4 Quantum Mechanics

Due to the space restrictions we only touched quantum mechanics, further details can be found in [169] [181] [184] [183] [188] [199]. In general, Erlangen approach is much more popular among physicists rather than mathematicians. Nevertheless its potential is not exhausted even there.

1. There is a possibility to build representation of the Heisenberg group using characters of its centre with values in dual and double numbers rather than in complex ones. This will naturally unifies classical mechanics, traditional QM and hyperbolic QM [155]. In particular, a full construction of the corresponding Fock–Segal–Bargmann spaces would be of interest.
2. Representations of nilpotent Lie groups with multidimensional centres in Clifford algebras as a framework for consistent quantum field theories based on De Donder–Weyl formalism [183].