Horizontally Affine Functions on Step-2 Carnot Algebras

Area of Science:

• Mathematics
• Geometric Analysis
• Algebraic Geometry

Background:

• Carnot algebras are fundamental in geometric analysis.
• Horizontally affine functions (h-affine) have emerging applications.
• Precisely monotone sets and minimal hypersurfaces are active research areas.

Purpose of the Study:

• Introduce and characterize horizontally affine (h-affine) functions on step-2 Carnot algebras.
• Establish an isomorphism between the vector space of h-affine functions and exterior algebras.
• Describe h-affine functions on arbitrary step-2 Carnot algebras by leveraging results from free Carnot algebras.

Main Methods:

• Definition and analysis of horizontally affine functions.
• Isomorphism theorems relating function spaces to exterior algebras.
• Deductive reasoning from free Carnot algebras to arbitrary ones.

Main Results:

• A complete description of h-affine functions on step-2 Carnot algebras.
• The vector space of h-affine functions on free step-2 rank-n Carnot algebras is isomorphic to the exterior algebra of $$.
• Characterizations of step-2 Carnot algebras where h-affine functions are affine in the usual sense.

Conclusions:

• H-affine functions provide a new perspective on Carnot algebra structures.
• The findings link h-affine functions to precisely monotone sets and minimal hypersurfaces.
• This work offers a foundation for further research into h-affine functions and their applications.