In mathematics, the special linear group SL ( n , R ) {\displaystyle \operatorname {SL} (n,R)} of degree n {\displaystyle n} over a commutative ring R {\displaystyle R} is the set of n × n {\displaystyle n\times n} matrices with determinant 1 {\displaystyle 1} , with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant
det : GL ( n , R ) → R × . {\displaystyle \det \colon \operatorname {GL} (n,R)\to R^{\times }.}
where R × {\displaystyle R^{\times }} is the multiplicative group of R {\displaystyle R} (that is, R {\displaystyle R} excluding 0 {\displaystyle 0} when R {\displaystyle R} is a field). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When R {\displaystyle R} is the finite field of order q {\displaystyle q} , the notation SL ( n , q ) {\displaystyle \operatorname {SL} (n,q)} is sometimes used.
Geometric interpretation The special linear group SL ( n , R ) {\displaystyle \operatorname {SL} (n,\mathbb {R} )} can be characterized as the group of volume- and orientation-preserving linear transformations of R n {\displaystyle \mathbb {R} ^{n}} . This corresponds to the interpretation of the determinant as measuring change in volume and orientation.
Lie subgroup
When F {\displaystyle F} is R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } , SL ( n , F ) {\displaystyle \operatorname {SL} (n,F)} is a Lie subgroup of GL ( n , F ) {\displaystyle \operatorname {GL} (n,F)} of dimension n 2 − 1 {\displaystyle n^{2}-1} . The Lie algebra s l ( n , F ) {\displaystyle {\mathfrak {sl}}(n,F)} of SL ( n , F ) {\displaystyle \operatorname {SL} (n,F)} consists of all n × n {\displaystyle n\times n} matrices over F {\displaystyle F} with vanishing trace. The Lie bracket is given by the commutator.
… excerpt ends here. Continue reading the full article.
