Algebraic structure → Group theoryGroup theory |
---|

Lie groups and Lie algebras |
---|

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 the identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) 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.

- General linear group of a vector space
- In terms of determinants
- As a Lie group
- Real case
- Complex case
- Over finite fields
- History
- Special linear group
- Other subgroups
- Diagonal subgroups
- Classical groups
- Related groups and monoids
- Projective linear group
- Affine group
- General semilinear group
- Full linear monoid
- Infinite general linear group
- See also
- Notes
- References
- External links

To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over **R** (the set of real numbers) is the group of *n*×*n* invertible matrices of real numbers, and is denoted by GL_{n}(**R**) or GL(*n*, **R**).

More generally, the general linear group of degree *n* over any field *F* (such as the complex numbers), or a ring *R* (such as the ring of integers), is the set of *n*×*n* invertible matrices with entries from *F* (or *R*), again with matrix multiplication as the group operation.^{ [1] } Typical notation is GL_{n}(*F*) or GL(*n*, *F*), or simply GL(*n*) if the field is understood.

More generally still, the general linear group of a vector space GL(*V*) is the automorphism group, not necessarily written as matrices.

The ** special linear group **, written SL(*n*, *F*) or SL_{n}(*F*), is the subgroup of GL(*n*, *F*) consisting of matrices with a determinant of 1.

The group GL(*n*, *F*) and its subgroups are often called **linear groups** or **matrix groups** (the automorphism group GL(*V*) is a linear group but not a matrix group). These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear group SL(2, **Z**).

If *n* ≥ 2, then the group GL(*n*, *F*) is not abelian.

If *V* is a vector space over the field *F*, the general linear group of *V*, written GL(*V*) or Aut(*V*), is the group of all automorphisms of *V*, i.e. the set of all bijective linear transformations *V* → *V*, together with functional composition as group operation. If *V* has finite dimension *n*, then GL(*V*) and GL(*n*, *F*) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in *V*. Given a basis (*e*_{1}, ..., *e*_{n}) of *V* and an automorphism *T* in GL(*V*), we have then for every basis vector *e*_{i} that

for some constants *a*_{ij} in *F*; the matrix corresponding to *T* is then just the matrix with entries given by the *a*_{ji}.

In a similar way, for a commutative ring *R* the group GL(*n*, *R*) may be interpreted as the group of automorphisms of a * free **R*-module *M* of rank *n*. One can also define GL(*M*) for any *R*-module, but in general this is not isomorphic to GL(*n*, *R*) (for any *n*).

Over a field *F*, a matrix is invertible if and only if its determinant is nonzero. Therefore, an alternative definition of GL(*n*, *F*) is as the group of matrices with nonzero determinant.

Over a commutative ring *R*, more care is needed: a matrix over *R* is invertible if and only if its determinant is a unit in *R*, that is, if its determinant is invertible in *R*. Therefore, GL(*n*, *R*) may be defined as the group of matrices whose determinants are units.

Over a non-commutative ring *R*, determinants are not at all well behaved. In this case, GL(*n*, *R*) may be defined as the unit group of the matrix ring M(*n*, *R*).

The general linear group GL(*n*, **R**) over the field of real numbers is a real Lie group of dimension *n*^{2}. To see this, note that the set of all *n*×*n* real matrices, M_{n}(**R**), forms a real vector space of dimension *n*^{2}. The subset GL(*n*, **R**) consists of those matrices whose determinant is non-zero. The determinant is a polynomial map, and hence GL(*n*, **R**) is an open affine subvariety of M_{n}(**R**) (a non-empty open subset of M_{n}(**R**) in the Zariski topology), and therefore^{ [2] } a smooth manifold of the same dimension.

The Lie algebra of GL(*n*, **R**), denoted consists of all *n*×*n* real matrices with the commutator serving as the Lie bracket.

As a manifold, GL(*n*, **R**) is not connected but rather has two connected components: the matrices with positive determinant and the ones with negative determinant. The identity component, denoted by GL^{+}(*n*, **R**), consists of the real *n*×*n* matrices with positive determinant. This is also a Lie group of dimension *n*^{2}; it has the same Lie algebra as GL(*n*, **R**).

The polar decomposition, which is unique for invertible matrices, shows that there is a homeomorphism between GL(*n*, **R**) and the Cartesian product of O(*n*) with the set of positive-definite symmetric matrices. Similarly, it shows that there is a homeomorphism between GL^{+}(*n*, **R**) and the Cartesian product of SO(*n*) with the set of positive-definite symmetric matrices. Because the latter is contractible, the fundamental group of GL^{+}(*n*, **R**) is isomorphic to that of SO(*n*).

The homeomorphism also shows that the group GL(*n*, **R**) is noncompact. “The” ^{ [3] } maximal compact subgroup of GL(*n*, **R**) is the orthogonal group O(*n*), while "the" maximal compact subgroup of GL^{+}(*n*, **R**) is the special orthogonal group SO(*n*). As for SO(*n*), the group GL^{+}(*n*, **R**) is not simply connected (except when *n* = 1), but rather has a fundamental group isomorphic to **Z** for *n* = 2 or **Z**_{2} for *n* > 2.

The general linear group over the field of complex numbers, GL(*n*, **C**), is a *complex* Lie group of complex dimension *n*^{2}. As a real Lie group (through realification) it has dimension 2*n*^{2}. The set of all real matrices forms a real Lie subgroup. These correspond to the inclusions

- GL(
*n*,**R**) < GL(*n*,**C**) < GL(*2n*,**R**),

which have real dimensions *n*^{2}, 2*n*^{2}, and 4*n*^{2} = (2*n*)^{2}. Complex *n*-dimensional matrices can be characterized as real 2*n*-dimensional matrices that preserve a linear complex structure — concretely, that commute with a matrix *J* such that *J*^{2} = −*I*, where *J* corresponds to multiplying by the imaginary unit *i*.

The Lie algebra corresponding to GL(*n*, **C**) consists of all *n*×*n* complex matrices with the commutator serving as the Lie bracket.

Unlike the real case, GL(*n*, **C**) is connected. This follows, in part, since the multiplicative group of complex numbers **C**^{∗} is connected. The group manifold GL(*n*, **C**) is not compact; rather its maximal compact subgroup is the unitary group U(*n*). As for U(*n*), the group manifold GL(*n*, **C**) is not simply connected but has a fundamental group isomorphic to **Z**.

If *F* is a finite field with *q* elements, then we sometimes write GL(*n*, *q*) instead of GL(*n*, *F*). When *p* is prime, GL(*n*, *p*) is the outer automorphism group of the group **Z**_{p}^{n}, and also the automorphism group, because **Z**_{p}^{n} is abelian, so the inner automorphism group is trivial.

The order of GL(*n*, *q*) is:

This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general, the *k*th column can be any vector not in the linear span of the first *k* − 1 columns. In *q*-analog notation, this is .

For example, GL(3, 2) has order (8 − 1)(8 − 2)(8 − 4) = 168. It is the automorphism group of the Fano plane and of the group **Z**_{2}^{3}, and is also known as PSL(2, 7) .

More generally, one can count points of Grassmannian over *F*: in other words the number of subspaces of a given dimension *k*. This requires only finding the order of the stabilizer subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem.

These formulas are connected to the Schubert decomposition of the Grassmannian, and are *q*-analogs of the Betti numbers of complex Grassmannians. This was one of the clues leading to the Weil conjectures.

Note that in the limit *q* ↦ 1 the order of GL(*n*, *q*) goes to 0! – but under the correct procedure (dividing by (*q*− 1)^{n}) we see that it is the order of the symmetric group (See Lorscheid's article) – in the philosophy of the field with one element, one thus interprets the symmetric group as the general linear group over the field with one element: *S*_{n} ≅ GL(*n*, 1).

The general linear group over a prime field, GL(*ν*, *p*), was constructed and its order computed by Évariste Galois in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in the context of studying the Galois group of the general equation of order *p*^{ν}.^{ [4] }

The special linear group, SL(*n*, *F*), is the group of all matrices with determinant 1. They are special in that they lie on a subvariety – they satisfy a polynomial equation (as the determinant is a polynomial in the entries). Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix. SL(*n*, *F*) is a normal subgroup of GL(*n*, *F*).

If we write *F*^{×} for the multiplicative group of *F* (excluding 0), then the determinant is a group homomorphism

- det: GL(
*n*,*F*) →*F*^{×}.

that is surjective and its kernel is the special linear group. Therefore, by the first isomorphism theorem, GL(*n*, *F*)/SL(*n*, *F*) is isomorphic to *F*^{×}. In fact, GL(*n*, *F*) can be written as a semidirect product:

- GL(
*n*,*F*) = SL(*n*,*F*) ⋊*F*^{×}

The special linear group is also the derived group (also known as commutator subgroup) of the GL(*n*, *F*) (for a field or a division ring *F*) provided that or *k* is not the field with two elements.^{ [5] }

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.

The special linear group SL(*n*, **R**) can be characterized as the group of * volume and orientation-preserving * linear transformations of **R**^{n}.

The group SL(*n*, **C**) is simply connected, while SL(*n*, **R**) is not. SL(*n*, **R**) has the same fundamental group as GL^{+}(*n*, **R**), that is, **Z** for *n* = 2 and **Z**_{2} for *n* > 2.

The set of all invertible diagonal matrices forms a subgroup of GL(*n*, *F*) isomorphic to (*F*^{×})^{n}. In fields like **R** and **C**, these correspond to rescaling the space; the so-called dilations and contractions.

A **scalar matrix** is a diagonal matrix which is a constant times the identity matrix. The set of all nonzero scalar matrices forms a subgroup of GL(*n*, *F*) isomorphic to *F*^{×}. This group is the center of GL(*n*, *F*). In particular, it is a normal, abelian subgroup.

The center of SL(*n*, *F*) is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of *n*th roots of unity in the field *F*.

The so-called classical groups are subgroups of GL(*V*) which preserve some sort of bilinear form on a vector space *V*. These include the

**orthogonal group**, O(*V*), which preserves a non-degenerate quadratic form on*V*,**symplectic group**, Sp(*V*), which preserves a symplectic form on*V*(a non-degenerate alternating form),**unitary group**, U(*V*), which, when*F*=**C**, preserves a non-degenerate hermitian form on*V*.

These groups provide important examples of Lie groups.

The projective linear group PGL(*n*, *F*) and the projective special linear group PSL(*n*, *F*) are the quotients of GL(*n*, *F*) and SL(*n*, *F*) by their centers (which consist of the multiples of the identity matrix therein); they are the induced action on the associated projective space.

The affine group Aff(*n*, *F*) is an extension of GL(*n*, *F*) by the group of translations in *F*^{n}. It can be written as a semidirect product:

- Aff(
*n*,*F*) = GL(*n*,*F*) ⋉*F*^{n}

where GL(*n*, *F*) acts on *F*^{n} in the natural manner. The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space *F*^{n}.

One has analogous constructions for other subgroups of the general linear group: for instance, the special affine group is the subgroup defined by the semidirect product, SL(*n*, *F*) ⋉ *F*^{n}, and the Poincaré group is the affine group associated to the Lorentz group, O(1, 3, *F*) ⋉ *F*^{n}.

The general semilinear group ΓL(*n*, *F*) is the group of all invertible semilinear transformations, and contains GL. A semilinear transformation is a transformation which is linear “up to a twist”, meaning “up to a field automorphism under scalar multiplication”. It can be written as a semidirect product:

- ΓL(
*n*,*F*) = Gal(*F*) ⋉ GL(*n*,*F*)

where Gal(*F*) is the Galois group of *F* (over its prime field), which acts on GL(*n*, *F*) by the Galois action on the entries.

The main interest of ΓL(*n*, *F*) is that the associated projective semilinear group PΓL(*n*, *F*) (which contains PGL(*n*, *F*)) is the collineation group of projective space, for *n* > 2, and thus semilinear maps are of interest in projective geometry.

The Full Linear Monoid, derived upon removal of the determinant's non-zero restriction, forms an algebraic structure akin to a monoid, often referred to as the full linear monoid or occasionally as the full linear semigroup or general linear monoid. Notably, it constitutes a regular semigroup.

If one removes the restriction of the determinant being non-zero, the resulting algebraic structure is a monoid, usually called the **full linear monoid**,^{ [6] }^{ [7] }^{ [8] } but occasionally also *full linear semigroup*,^{ [9] }*general linear monoid*^{ [10] }^{ [11] } etc. It is actually a regular semigroup.^{ [7] }

The **infinite general linear group** or ** stable general linear group** is the direct limit of the inclusions GL(*n*, *F*) → GL(*n* + 1, *F*) as the upper left block matrix. It is denoted by either GL(*F*) or GL(∞, *F*), and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.^{ [12] }

It is used in algebraic K-theory to define K_{1}, and over the reals has a well-understood topology, thanks to Bott periodicity.

It should not be confused with the space of (bounded) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible – see Kuiper's theorem.

- ↑ Here rings are assumed to be associative and unital.
- ↑ Since the Zariski topology is coarser than the metric topology; equivalently, polynomial maps are continuous.
- ↑ A maximal compact subgroup is not unique, but is essentially unique, hence one often refers to “the” maximal compact subgroup.
- ↑ Galois, Évariste (1846). "Lettre de Galois à M. Auguste Chevalier".
*Journal de Mathématiques Pures et Appliquées*.**XI**: 408–415. Retrieved 2009-02-04, GL(*ν*,*p*) discussed on p. 410.`{{cite journal}}`

: CS1 maint: postscript (link) - ↑ Suprunenko, D.A. (1976),
*Matrix groups*, Translations of Mathematical Monographs, American Mathematical Society, Theorem II.9.4 - ↑ Jan Okniński (1998).
*Semigroups of Matrices*. World Scientific. Chapter 2: Full linear monoid. ISBN 978-981-02-3445-4. - 1 2 Meakin (2007). "Groups and Semigroups: Connections and contrast". In C. M. Campbell (ed.).
*Groups St Andrews 2005*. Cambridge University Press. p. 471. ISBN 978-0-521-69470-4. - ↑ John Rhodes; Benjamin Steinberg (2009).
*The q-theory of Finite Semigroups*. Springer Science & Business Media. p. 306. ISBN 978-0-387-09781-7. - ↑ Eric Jespers; Jan Okniski (2007).
*Noetherian Semigroup Algebras*. Springer Science & Business Media. 2.3: Full linear semigroup. ISBN 978-1-4020-5810-3. - ↑ Meinolf Geck (2013).
*An Introduction to Algebraic Geometry and Algebraic Groups*. Oxford University Press. p. 132. ISBN 978-0-19-967616-3. - ↑ Mahir Bilen Can; Zhenheng Li; Benjamin Steinberg; Qiang Wang (2014).
*Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics*. Springer. p. 142. ISBN 978-1-4939-0938-4. - ↑ Milnor, John Willard (1971).
*Introduction to algebraic K-theory*. Annals of Mathematics Studies. Vol. 72. Princeton, NJ: Princeton University Press. p. 25. MR 0349811. Zbl 0237.18005.

In mathematics, a **Lie algebra** is a vector space together with an operation called the **Lie bracket**, an alternating bilinear map , that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, .

In mathematics, a **Lie group** is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

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

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 (an orthogonal matrix is a real matrix whose inverse equals its transpose). 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** (usually **C** or **R**). The latter is called the **compact symplectic group** and is also denoted by . 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, an **algebraic group** is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.

In mathematics, the **affine group** or **general affine group** of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space, the affine group consists of those functions from the space to itself such that the image of every line is a line.

In mathematics, the **adjoint representation** of a Lie group *G* is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if *G* is , the Lie group of real *n*-by-*n* invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible *n*-by-*n* matrix to an endomorphism of the vector space of all linear transformations of defined by: .

In mathematics, the **modular group** is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices *A* and −*A* are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.

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 associated projective space P(*V*). Explicitly, the projective linear group is the quotient group

In mathematics, a **generalized flag variety** is a homogeneous space whose points are flags in a finite-dimensional vector space *V* over a field **F**. When **F** is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a **real** or **complex****flag manifold**. Flag varieties are naturally projective varieties.

In mathematics, a **linear algebraic group** is a subgroup of the group of invertible matrices that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of .

In mathematics, a **reductive group** is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group *G* over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group *GL*(*n*) of invertible matrices, the special orthogonal group *SO*(*n*), and the symplectic group *Sp*(2*n*). **Simple algebraic groups** and (more generally) **semisimple algebraic groups** are reductive.

In mathematics, the **Bruhat decomposition***G* = *BWB* of certain algebraic groups *G* into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this.

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*.

**Representation theory** is a branch of mathematics that studies abstract algebraic structures by *representing* their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.

In linear algebra, particularly projective geometry, a **semilinear map** between vector spaces *V* and *W* over a field *K* is a function that is a linear map "up to a twist", hence *semi*-linear, where "twist" means "field automorphism of *K*". Explicitly, it is a function *T* : *V* → *W* that is:

In mathematics, the **automorphism group** of an object *X* is the group consisting of automorphisms of *X* under composition of morphisms. For example, if *X* is a finite-dimensional vector space, then the automorphism group of *X* is the group of invertible linear transformations from *X* to itself. If instead *X* is a group, then its automorphism group is the group consisting of all group automorphisms of *X*.

- Springer, Tonny Albert (1998).
*Linear Algebraic Groups*(2nd ed.). Birkhäuser. ISBN 978-0-8176-4839-8.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.