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In mathematics, the **special linear group**SL(*n*, *F*) of degree *n* over a field *F* is the set of *n* × *n* matrices with determinant 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

- Geometric interpretation
- Lie subgroup
- Topology
- Relations to other subgroups of GL(n,A)
- Generators and relations
- SL±(n,F)
- Structure of GL(n,F)
- See also
- References

where we write *F*^{×} for the multiplicative group of *F* (that is, *F* excluding 0).

These elements are "special" in that they form a algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).

The special linear group SL(*n*, **R**) can be characterized as the group of * volume and orientation preserving* linear transformations of **R**^{n}; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.

When *F* is **R** or **C**, SL(*n*, *F*) is a Lie subgroup of GL(*n*, *F*) of dimension *n*^{2} − 1. The Lie algebra of SL(*n*, *F*) consists of all *n* × *n* matrices over *F* with vanishing trace. The Lie bracket is given by the commutator.

Any invertible matrix can be uniquely represented according to the polar decomposition as the product of a unitary matrix and a hermitian matrix with positive eigenvalues. The determinant of the unitary matrix is on the unit circle while that of the hermitian matrix is real and positive and since in the case of a matrix from the special linear group the product of these two determinants must be 1, then each of them must be 1. Therefore, a special linear matrix can be written as the product of a special unitary matrix (or special orthogonal matrix in the real case) and a positive definite hermitian matrix (or symmetric matrix in the real case) having determinant 1.

Thus the topology of the group SL(*n*, **C**) is the product of the topology of SU(*n*) and the topology of the group of hermitian matrices of unit determinant with positive eigenvalues. A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of (*n*^{2} − 1)-dimensional Euclidean space.^{ [1] } Since SU(*n*) is simply connected,^{ [2] } we conclude that SL(*n*, **C**) is also simply connected, for all *n*.

The topology of SL(*n*, **R**) is the product of the topology of SO(*n*) and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of (*n* + 2)(*n* − 1)/2-dimensional Euclidean space. Thus, the group SL(*n*, **R**) has the same fundamental group as SO(*n*), that is, **Z** for *n* = 2 and **Z**_{2} for *n*> 2.^{ [3] } In particular this means that SL(*n*, **R**), unlike SL(*n*, **C**), is not simply connected, for *n* greater than 1.

Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroup of GL, and the group generated by transvections. These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so [GL, GL] ≤ SL), but in general do not coincide with it.

The group generated by transvections is denoted E(*n*, *A*) (for elementary matrices) or TV(*n*, *A*). By the second Steinberg relation, for *n* ≥ 3, transvections are commutators, so for *n* ≥ 3, E(*n*, *A*) ≤ [GL(*n*, *A*), GL(*n*, *A*)].

For *n* = 2, transvections need not be commutators (of 2 × 2 matrices), as seen for example when *A* is **F**_{2}, the field of two elements, then

where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.

However, if *A* is a field with more than 2 elements, then E(2, *A*) = [GL(2, *A*), GL(2, *A*)], and if *A* is a field with more than 3 elements, E(2, *A*) = [SL(2, *A*), SL(2, *A*)]. ^{[ dubious – discuss ]}

In some circumstances these coincide: the special linear group over a field or a Euclidean domain is generated by transvections, and the *stable* special linear group over a Dedekind domain is generated by transvections. For more general rings the stable difference is measured by the special Whitehead group SK_{1}(*A*) := SL(*A*)/E(*A*), where SL(*A*) and E(*A*) are the stable groups of the special linear group and elementary matrices.

If working over a ring where SL is generated by transvections (such as a field or Euclidean domain), one can give a presentation of SL using transvections with some relations. Transvections satisfy the Steinberg relations, but these are not sufficient: the resulting group is the Steinberg group, which is not the special linear group, but rather the universal central extension of the commutator subgroup of GL.

A sufficient set of relations for SL(*n*, **Z**) for *n* ≥ 3 is given by two of the Steinberg relations, plus a third relation ( Conder, Robertson & Williams 1992 , p. 19). Let *T _{ij}* :=

are a complete set of relations for SL(*n*, **Z**), *n* ≥ 3.

In characteristic other than 2, the set of matrices with determinant ±1 form another subgroup of GL, with SL as an index 2 subgroup (necessarily normal); in characteristic 2 this is the same as SL. This forms a short exact sequence of groups:

This sequence splits by taking any matrix with determinant −1, for example the diagonal matrix If is odd, the negative identity matrix is in SL^{±}(*n*,*F*) but not in SL(*n*,*F*) and thus the group splits as an internal direct product . However, if is even, is already in SL(*n*,*F*) , SL^{±} does not split, and in general is a non-trivial group extension.

Over the real numbers, SL^{±}(*n*, *R*) has two connected components, corresponding to SL(*n*, *R*) and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant −1). In odd dimension these are naturally identified by , but in even dimension there is no one natural identification.

The group GL(*n*, *F*) splits over its determinant (we use *F*^{×} ≅ GL(1, *F*) → GL(*n*, *F*) as the monomorphism from *F*^{×} to GL(*n*, *F*), see semidirect product), and therefore GL(*n*, *F*) can be written as a semidirect product of SL(*n*, *F*) by *F*^{×}:

- GL(
*n*,*F*) = SL(*n*,*F*) ⋊*F*^{×}.

In mathematics, the **determinant** is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible, and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants . The determinant of a matrix *A* is denoted det(*A*), det *A*, or |*A*|.

In mathematics, a **Lie group** is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.

In linear algebra, the **trace** of a square matrix **A**, denoted tr(**A**), is defined to be the sum of elements on the main diagonal of **A**.

In mathematics, the **general linear group** of degree *n* is the set of *n*×*n* invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

In mathematics, a **square matrix** is a matrix with the same number of rows and columns. An *n*-by-*n* matrix is known as a square matrix of order . Any two square matrices of the same order can be added and multiplied.

In linear algebra, the **transpose** of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix **A** by producing another matrix, often denoted by **A**^{T}.

In mathematics, the **orthogonal group** in dimension *n*, denoted O(*n*), is the group of distance-preserving transformations of a Euclidean space of dimension *n* that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the **general orthogonal group**, by analogy with the general linear group. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

In mathematics, the name **symplectic group** can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2*n*, **F**) and Sp(*n*) for positive integer *n* and field **F**. The latter is called the **compact symplectic group**. Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2*n*, **C**) is denoted *C _{n}*, and Sp(

In mathematics, the **unitary group** of degree *n*, denoted U(*n*), is the group of *n* × *n* unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(*n*, **C**). **Hyperorthogonal group** is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

In mathematics, the **special unitary group** of degree *n*, denoted SU(*n*), is the Lie group of *n* × *n* unitary matrices with determinant 1.

In mathematics, particularly in linear algebra, a **skew-symmetric****matrix** is a square matrix whose transpose equals its negative. That is, it satisfies the condition

In mathematics, an **Hermitian matrix** is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:

In mathematics, the **affine group** or **general affine group** of any affine space over a field K is the group of all invertible affine transformations from the space into itself.

In the mathematical discipline of linear algebra, a **triangular matrix** is a special kind of square matrix. A square matrix is called **lower triangular** if all the entries *above* the main diagonal are zero. Similarly, a square matrix is called **upper triangular** if all the entries *below* the main diagonal are zero.

In mathematics, especially in the group theoretic area of algebra, the **projective linear group** is the induced action of the general linear group of a vector space *V* on the associated projective space P(*V*). Explicitly, the projective linear group is the quotient group

In linear algebra, a square matrix with complex entries is said to be **skew-Hermitian** or **anti-Hermitian** if its conjugate transpose is the negative of the original matrix. That is, the matrix is skew-Hermitian if it satisfies the relation

In mathematics, more specifically in group theory, a group is said to be **perfect** if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients. In symbols, a perfect group is one such that *G*^{(1)} = *G*, or equivalently one such that *G*^{ab} = {1}.

In projective geometry and linear algebra, the **projective orthogonal group** PO is the induced action of the orthogonal group of a quadratic space *V* = (*V*,*Q*) on the associated projective space P(*V*). Explicitly, the projective orthogonal group is the quotient group

In mathematics, the **classical groups** are defined as the special linear groups over the reals **R**, the complex numbers **C** and the quaternions **H** together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the **complex classical Lie groups** are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The **compact classical groups** are compact real forms of the complex classical groups. The finite analogues of the classical groups are the **classical groups of Lie type**. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph *The Classical Groups*.

In mathematics, the special linear group **SL(2,R)** or **SL _{2}(R)** is the group of 2×2 real matrices with determinant one:

This article needs additional citations for verification .(January 2008) |

- Conder, Marston; Robertson, Edmund; Williams, Peter (1992), "Presentations for 3-dimensional special linear groups over integer rings",
*Proceedings of the American Mathematical Society*, American Mathematical Society,**115**(1): 19–26, doi:10.2307/2159559, JSTOR 2159559, MR 1079696 - Hall, Brian C. (2015),
*Lie groups, Lie algebras, and representations: An elementary introduction*, Graduate Texts in Mathematics,**222**(2nd ed.), Springer

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