The Determinant Inner Product and the Heisenberg Product of $Sym(2)$

Home Page
About
Submit A Journal
Submit A Conference
Submit Paper/Book
- Submit a Preprint
- Submit a Book
Contact

International Electronic Journal of Geometry
Volume:14 Issue:1
The Determinant Inner Product and the Heisenberg Product of $Sym(2)$

The Determinant Inner Product and the Heisenberg Product of $Sym(2)$

Pages : 145-156

View : 49 | Download : 12

Publication Date : 2021-04-15

Article Type : Research Paper

Abstract :The aim of this work is to introduce and study the nondegenerate inner product $<\cdot , \cdot >_{det}$ induced by the determinant map on the space $Syminsert ignore into journalissuearticles values(2);$ of symmetric $2\times 2$ real matrices. This symmetric bilinear form of index $2$ defines a rational symmetric function on the pairs of rays in the plane and an associated function on the $2$-torus can be expressed with the usual Hopf bundle projection $S^3\rightarrow S^2insert ignore into journalissuearticles values(\frac{1}{2});$. Also, the product $<\cdot , \cdot >_{det}$ is treated with complex numbers by using the Hopf invariant map of $Syminsert ignore into journalissuearticles values(2);$ and this complex approach yields a Heisenberg product on $Syminsert ignore into journalissuearticles values(2);$. Moreover, the quadratic equation of critical points for a rational Morse function of height type generates a cosymplectic structure on $Syminsert ignore into journalissuearticles values(2);$ with the unitary matrix as associated Reeb vector and with the Reeb $1$-form being half of the trace map.
Keywords : Symmetric matrix, determinant, Hopf bundle, Hopf invariant

ORIGINAL ARTICLE URL

* There may have been changes in the journal, article,conference, book, preprint etc. informations. Therefore, it would be appropriate to follow the information on the official page of the source. The information here is shared for informational purposes. IAD is not responsible for incorrect or missing information.