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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the special linear group \operatorname(n,R) of degree n over a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
R is the set of n\times n
matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...
with
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
1, with the group operations of ordinary
matrix multiplication In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
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 ...
of the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
given by the kernel of the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
:\det\colon \operatorname(n, R) \to R^\times. where R^\times 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 R (that is, R excluding 0 when 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 is the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
of order q, the notation \operatorname(n,q) is sometimes used.


Geometric interpretation

The special linear group \operatorname(n,\R) can be characterized as the group of ''
volume Volume is a measure of regions in 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 Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
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, \operatorname(n,F) is a
Lie subgroup In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
of \operatorname(n,F) of dimension n^2-1. 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 ident ...
\mathfrak(n, F) of \operatorname(n,F) consists of all n\times n matrices over F with vanishing trace. The
Lie bracket 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 identit ...
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 In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is a unitary matrix, and P is a positive semi-definite Hermitian matrix (U is an orthogonal matrix, and P is a posit ...
as the product of a
unitary matrix In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if U^* U = UU^* = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate ...
and a
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
with positive
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s. The
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
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. 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 linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...
in the real case) having determinant 1. It follows that the topology of the group \operatorname(n,\C) is the product of the topology of \operatorname(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 Exponential may refer to any of several mathematical topics related to exponentiation, including: * Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value * Ex ...
of a traceless Hermitian matrix, and therefore the topology of this is that of (n^2-1)-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. Since \operatorname(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 (topology), path between two points can be continuously transformed into any other such path while preserving ...
, then \operatorname(n,\C) is also simply connected, for all n\geq 2. The topology of \operatorname(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 -dimensional Euclidean space. Thus, the group \operatorname(n,\R) has the same
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
as \operatorname(n); that is, \Z for n=2 and \Z_2 for n>2. Sections 13.2 and 13.3 In particular this means that \operatorname(n,\R), unlike \operatorname(n,\C), is not simply connected, for n>1.


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

Two related subgroups, which in some cases coincide with \operatorname, and in other cases are accidentally conflated with \operatorname, are the
commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
of \operatorname, and the group generated by transvections. These are both subgroups of \operatorname (transvections have determinant 1, and det is a map to an abelian group, so operatorname,\operatorname\operatorname), but in general do not coincide with it. The group generated by transvections is denoted \operatorname(n,A) (for elementary matrices) or \operatorname(n,A). By the second Steinberg relation, for n\geq 3, transvections are commutators, so for n\geq 3, \operatorname(n,A)< operatorname(n,A),\operatorname(n,A)/math>. For n=2, transvections need not be commutators (of 2\times 2 matrices), as seen for example when A is \mathbb_2, the field of two elements. In that case :A_3 \cong operatorname(2, \mathbb_2),\operatorname(2, \mathbb_2)< \operatorname(2, \mathbb_2) = \operatorname(2, \mathbb_2) = \operatorname(2, \mathbb_2) \cong S_3, where A_3 and S_3 respectively denote the alternating and
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 grou ...
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 Euclidean division of integers. Th ...
is generated by transvections, and the ''stable'' special linear group over a
Dedekind domain In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...
is generated by transvections. For more general rings the stable difference is measured by the special Whitehead group SK_1(A)=\operatorname(A)/\operatorname(A), where \operatorname(A) and \operatorname(A) are the stable groups of the special linear group and elementary matrices.


Generators and relations

If working over a ring where \operatorname 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 Euclidean division of integers. Th ...
), 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 \operatorname using transvections with some relations. Transvections satisfy the
Steinberg relations Steinberg Media Technologies GmbH (trading as Steinberg; ) is a German musical software and hardware company based in Hamburg. It develops software for writing, recording, arranging and editing music, most notably Steinberg Cubase, Cubase, Stein ...
, 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 In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \op ...
of the commutator subgroup of \operatorname. 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 In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
of groups: :1\to\operatorname(n, F) \to \operatorname^(n, F) \to \\to1. 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 \operatorname^\pm(2k + 1, F) \cong \operatorname(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 In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ove ...
. 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 \operatorname(n,F) splits over its determinant (we use F^\times = \operatorname(1,F)\to \operatorname(n,F) as the
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphis ...
from F^\times to \operatorname(n,F), see
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...
), and therefore \operatorname(n,F) can be written as a semidirect product of \operatorname(n,F) by F^\times: :\operatorname(n,F)=\operatorname(n,F)\rtimes F^\times.


See also

* SL(2, R) * SL(2, C) *
Modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
(PSL(2, Z)) *
Projective linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
*
Conformal map In mathematics, a conformal map is a function (mathematics), 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-prese ...
* 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