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

Year 2021, , 145 - 156, 15.04.2021

Mircea Crasmareanu

https://doi.org/10.36890/iejg.754557

Cited By: 2

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 $Sym(2)$ 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 $S^3\rightarrow S^2(\frac{1}{2})$. Also, the product $<\cdot , \cdot >_{det}$ is treated with complex numbers by using the Hopf invariant map of $Sym(2)$ and this complex approach yields a Heisenberg product on $Sym(2)$. Moreover, the quadratic equation of critical points for a rational Morse function of height type generates a cosymplectic structure on $Sym(2)$ 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

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