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special linear group (Definition)

Given a vector space $ V$, the special linear group $ {\operatorname{SL}}(V)$ is defined to be the subgroup of the general linear group $ \operatorname{GL}(V)$ consisting of all invertible linear transformations $ T: V \longrightarrow V$ in $ \operatorname{GL}(V)$ that have determinant 1.

If $ V = \mathbb{F}^n$ for some field $ \mathbb{F}$, then the group $ {\operatorname{SL}}(V)$ is often denoted $ {\operatorname{SL}}(n,\mathbb{F})$ or $ {\operatorname{SL}}_n(\mathbb{F})$, and if one identifies each linear transformation with its matrix with respect to the standard basis, then $ {\operatorname{SL}}(n,\mathbb{F})$ consists of all $ n \times n$ matrices with entries in $ \mathbb{F}$ that have determinant 1.



"special linear group" is owned by djao.
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See Also: general linear group, group, unimodular matrix

Other names:  SL

Attachments:
theorems of special linear group over a finite field (Theorem) by Daume
projective special linear group (Definition) by alozano
irreducible representations of the special linear group over $\mathbb{F}_p$ (Theorem) by alozano
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Cross-references: standard basis, matrix, linear transformation, group, field, determinant, invertible linear transformations, general linear group, subgroup, vector space
There are 12 references to this entry.

This is version 4 of special linear group, born on 2002-02-22, modified 2005-05-04.
Object id is 2463, canonical name is SpecialLinearGroup.
Accessed 7883 times total.

Classification:
AMS MSC: 20G15 (Group theory and generalizations :: Linear algebraic groups :: Linear algebraic groups over arbitrary fields)
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