Definition
SL(4,C)
SL(4,C) as a separately identified complex Lie group, with its defining operations and dimension.
The complex special linear group consists of the determinant-one matrices in , with matrix multiplication.
Dimensions and structure
This group has real dimension , complex dimension . It is connected and simply connected. Its tangent algebra is the trace-zero 4 by 4 matrices; the central quotient by {±I} gives SO(6,C).
Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.
References
- Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §31.2, Example 31.3 and Proposition 31.4; §31.3, Example 31.10, pp. 166–168.