The Special Linear Group of Complex Dimension Two

Three scenes are shown. The main scene represents the abstract vector space of complex dimension two. But, the side scenes show two copies of the special orthogonal group of real dimension three. First, we mark the boundary of one SO(3) copy with sixteen vertices of a hypercube. Second, make a tangent frame at origin O and parallel transport it to the marked points. After, mark the frames with identifier I. Next, parallel transport a tangent tetrad from O to a boundary point and then switch coordinate charts from N to S in order to parallel transport it to the center of the counterpart copy of SO(3). From the center of the other SO(3), parallel transport the frame to the boundary points. Mark the frames that are parallel transported from the center of chart S at origin O with identifier II, for comparing with frames originated from the center of chart N at origin O. The twist in frames I compared to frames II suggests the rotations of the two-dimensional sphere, or the reorientation of 3D objects in the Euclidean three-dimensional space. Even though both copies of SO(3) represent the same 3D orientation, they are distinguished from each other as points in the special linear group of complex dimension two. The symmetry group SL(2,C) is a set of two by two matrices with numbers of type complex, such that the determinant of the matrix is equal to unity. For example, alpha times delta minus beta times gamma equals zero, for the matrix [alpha beta; gamma delta], where alpha, beta, gamma and delta are complex numbers. The equivalence class of the twisted frames at the boundary points forms a group of rotations and determines the clutching function. The clutching construction makes an associated vector bundle on the three-dimensional sphere. The clutching function creates a vector bundle by resolving the conflicts of tangent frames as we parallel transport them and cross into the other coordinate chart.

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