Special linear group

Linking table from ${\displaystyle \operatorname {SL} (2,\mathbb {F} _{3})}$

The special linear group of degree over a body (or more generally a commutative, unitary ring ) is the group of all matrices with coefficients whose determinant is 1; these are also called unimodular matrices . The group linkage is the matrix multiplication . ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle n\times n}$ ${\displaystyle K}$

The special linear group of degrees above is denoted by. If it is clear from the context that the body is the set of real or complex numbers, one also writes or . ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle \operatorname {SL} (n,K)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \operatorname {SL} (n)}$ ${\displaystyle \operatorname {SL} _{n}}$

properties

The special linear group is a normal divisor of the general linear group . ${\displaystyle \operatorname {SL} (n,K)}$ ${\displaystyle \operatorname {GL} (n,K)}$

The factor group is isomorphic to the unit group of (for a body is the same ). The proof takes place via the homomorphism theorem with the determinant as homomorphism . ${\displaystyle \operatorname {GL} (n,K)/\operatorname {SL} (n,K)}$ ${\displaystyle K^{*}}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K^{*}}$ ${\displaystyle K\setminus \{0\}}$

Important subgroups of the are for the special orthogonal group and for the special unitary group . ${\displaystyle \operatorname {SL} (n,K)}$ ${\displaystyle K=\mathbb {R} }$ ${\displaystyle \operatorname {SO} (n)}$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle \operatorname {SU} (n)}$

The special linear group over the body or is a Lie group over the dimension . ${\displaystyle \operatorname {SL} (n,K)}$ ${\displaystyle K=\mathbb {R} }$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle K}$ ${\displaystyle n^{2}-1}$

The special linear groups are algebraic groups since the condition that the determinant must be equal to 1 can be expressed by a polynomial equation in the matrix coefficients.

The special linear group contains all orientation- true and volume- preserving linear maps . ${\displaystyle \operatorname {SL} (n,K)}$

See also

• SL (2, R)
• Modular form

Individual evidence

1. ↑ Eric W. Weisstein : Determinant . In: MathWorld (English).