One other parameterization of SU(4) group

• URL: http://arxiv.org/abs/2408.14888v1
• Date: Tue, 27 Aug 2024 09:03:14 GMT
• Title: One other parameterization of SU(4) group
• Authors: Arsen Khvedelidze, Dimitar Mladenov, Astghik Torosyan,
• Abstract summary: decomposition of Lie $mathfraksu(4)$ algebra into the direct sum of subspaces. $ with $mathfrakk=mathfraksu(2)oplusmathfraksu(2)$ and a triplet of 3-dimensional Abelian subalgebras.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a special decomposition of the Lie $\mathfrak{su}(4)$ algebra into the direct sum of orthogonal subspaces, $\mathfrak{su}(4)=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{a}^\prime\oplus\mathfrak{t}\,,$ with $\mathfrak{k}=\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ and a triplet of 3-dimensional Abelian subalgebras $(\mathfrak{a}, \mathfrak{a}^{\prime}, \mathfrak{t})\,,$ such that the exponential mapping of a neighbourhood of the $0\in \mathfrak{su}(4)$ into a neighbourhood of the identity of the Lie group provides the following factorization of an element of $SU(4)$ \[ g = k\,a\,t\,, \] where $k \in \exp{(\mathfrak{k})} = SU(2)\times SU(2) \subset SU(4)\,,$ the diagonal matrix $t$ stands for an element from the maximal torus $T^3=\exp{(\mathfrak{t})},$ and the factor $a=\exp{(\mathfrak{a})}\exp{(\mathfrak{a}^\prime)}$ corresponds to a point in the double coset $SU(2)\times SU(2)\backslash SU(4)/T^3.$ Analyzing the uniqueness of the inverse of the above exponential mappings, we establish a logarithmic coordinate chart of the $SU(4)$ group manifold comprising 6 coordinates on the embedded manifold $ SU(2)\times SU(2) \subset SU(4)$ and 9 coordinates on three copies of the regular octahedron with the edge length $2\pi\sqrt{2}\,$.

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