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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of the general linear group given by the kernel of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
:\det\colon \operatorname(n, F) \to F^\times. where ''F''× is the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of ''F'' (that is, ''F'' excluding 0). These elements are "special" in that they form an
algebraic subvariety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When ''F'' is a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of order ''q'', the notation is sometimes used.


Geometric interpretation

The special linear group can be characterized as the group of ''
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
and orientation preserving'' linear transformations of R''n''; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.


Lie subgroup

When ''F'' is R or C, is a
Lie subgroup In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
of of dimension . The
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
\mathfrak(n, F) of SL(''n'', ''F'') consists of all matrices over ''F'' with vanishing trace. The Lie bracket is given by the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
.


Topology

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
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s. The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the unitary matrix is on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
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 In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...
hermitian matrix (or symmetric matrix in the real case) having determinant 1. Thus the topology of the group 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 -dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. Since SU(''n'') is
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
, we conclude that is also simply connected, for all ''n''. The topology of 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 -dimensional Euclidean space. Thus, the group has the same
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
as SO(''n''), that is, Z for and Z2 for . Sections 13.2 and 13.3 In particular this means that , unlike , is not simply connected, for ''n'' greater than 1.


Relations to other subgroups of GL(''n'',''A'')

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 L, GL≤ SL), but in general do not coincide with it. The group generated by transvections is denoted (for elementary matrices) or . By the second Steinberg relation, for , transvections are commutators, so for , . For , transvections need not be commutators (of matrices), as seen for example when ''A'' is F2, the field of two elements, then :\operatorname(3) \cong operatorname(2, \mathbf_2),\operatorname(2, \mathbf_2)< \operatorname(2, \mathbf_2) = \operatorname(2, \mathbf_2) = \operatorname(2, \mathbf_2) \cong \operatorname(3), where Alt(3) and Sym(3) denote the alternating resp.
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
on 3 letters. However, if ''A'' is a field with more than 2 elements, then , and if ''A'' is a field with more than 3 elements, . In some circumstances these coincide: the special linear group over a field or a
Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers ...
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 , where SL(''A'') and E(''A'') are the stable groups of the special linear group and elementary matrices.


Generators and relations

If working over a ring where SL is generated by transvections (such as a field or
Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers ...
), one can give a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
of SL using transvections with some relations. Transvections satisfy the
Steinberg relations In algebraic K-theory, a field of mathematics, the Steinberg group \operatorname(A) of a ring A is the universal central extension of the commutator subgroup of the stable general linear group of A . It is named after Robert Steinberg, and ...
, 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 for is given by two of the Steinberg relations, plus a third relation . Let (1) be the elementary matrix with 1's on the diagonal and in the ''ij'' position, and 0's elsewhere (and ''i'' ≠ ''j''). Then :\begin \left T_,T_ \right&= T_ && \text i \neq k \\ pt \left T_,T_ \right&= \mathbf && \text i \neq \ell, j \neq k \\ pt \left(T_T_^T_\right)^4 &= \mathbf \end are a complete set of relations for SL(''n'', Z), ''n'' ≥ 3.


SL±(''n'',''F'')

In characteristic other than 2, the set of matrices with determinant 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: :\mathrm(n, F) \to \mathrm^(n, F) \to \. This sequence splits by taking any matrix with determinant , for example the diagonal matrix (-1, 1, \dots, 1). If n = 2k + 1 is odd, the negative identity matrix -I is in but not in and thus the group splits as an
internal direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
SL^\pm(2k + 1, F) \cong SL(2k + 1, F) \times \. However, if n = 2k is even, -I is already in , does not split, and in general is a non-trivial group extension. Over the real numbers, has two connected components, corresponding to and another component, which are isomorphic with identification depending on a choice of point (matrix with determinant ). In odd dimension these are naturally identified by -I, but in even dimension there is no one natural identification.


Structure of GL(''n'',''F'')

The group splits over its determinant (we use as the monomorphism from ''F''× to , see
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
), and therefore can be written as a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
of by ''F''×: :GL(''n'', ''F'') = SL(''n'', ''F'') ⋊ ''F''×.


See also

* SL(2, R) * SL(2, C) *
Modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
* Projective linear group *
Conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
* Representations of classical Lie groups


References

* *{{Citation, last=Hall, first=Brian C., title=Lie groups, Lie algebras, and representations: An elementary introduction, edition=2nd, series=Graduate Texts in Mathematics, volume=222, publisher=Springer, year=2015 Linear algebra Lie groups Linear algebraic groups