Editing Lie algebra (section)

=== Matrix Lie algebras ===
A [[Linear group|matrix group]] is a Lie group consisting of invertible matrices, <math>G\subset \mathrm{GL}(n,\mathbb{R})</math>, where the group operation of ''G'' is matrix multiplication. The corresponding Lie algebra <math>\mathfrak g</math> is the space of matrices which are tangent vectors to ''G'' inside the linear space <math>M_n(\mathbb{R})</math>: this consists of derivatives of smooth curves in ''G'' at the [[identity matrix]] <math>I</math>:
:<math>\mathfrak{g} = \{ X = c'(0) \in M_n(\mathbb{R}) : \text{ smooth } c: \mathbb{R}\to G, \ c(0) = I \}.</math>
The Lie bracket of <math>\mathfrak{g}</math> is given by the commutator of matrices, <math>[X,Y]=XY-YX</math>. Given a Lie algebra <math>\mathfrak{g}\subset \mathfrak{gl}(n,\mathbb{R})</math>, one can recover the Lie group as the subgroup generated by the [[matrix exponential]] of elements of <math>\mathfrak{g}</math>.<ref>{{harvnb|Varadarajan|1984|loc=section 2.10, Remark 2.}}</ref> (To be precise, this gives the [[identity component]] of ''G'', if ''G'' is not connected.) Here the exponential mapping <math>\exp: M_n(\mathbb{R})\to M_n(\mathbb{R})</math> is defined by <math>\exp(X) = I + X + \tfrac{1}{2!}X^2 + \tfrac{1}{3!}X^3 + \cdots</math>, which converges for every matrix <math>X</math>.

The same comments apply to complex Lie subgroups of <math>GL(n,\mathbb{C})</math> and the complex matrix exponential, <math>\exp: M_n(\mathbb{C})\to M_n(\mathbb{C})</math> (defined by the same formula).

Here are some matrix Lie groups and their Lie algebras.<ref>{{harvnb|Hall|2015|loc=§3.4.}}</ref>

* For a positive integer ''n'', the [[special linear group]] <math>\mathrm{SL}(n,\mathbb{R})</math> consists of all real {{math|''n''&nbsp;×&nbsp;''n''}} matrices with determinant 1. This is the group of linear maps from <math>\mathbb{R}^n</math> to itself that preserve volume and [[orientability|orientation]]. More abstractly, <math>\mathrm{SL}(n,\mathbb{R})</math> is the [[commutator subgroup]] of the general linear group <math>\mathrm{GL}(n,\R)</math>. Its Lie algebra <math>\mathfrak{sl}(n,\mathbb{R})</math> consists of all real {{math|''n''&nbsp;×&nbsp;''n''}} matrices with [[Trace (linear algebra)|trace]] 0. Similarly, one can define the analogous complex Lie group <math>{\rm SL}(n,\mathbb{C})</math> and its Lie algebra <math>\mathfrak{sl}(n,\mathbb{C})</math>.
* The [[orthogonal group]] <math>\mathrm{O}(n)</math> plays a basic role in geometry: it is the group of linear maps from <math>\mathbb{R}^n</math> to itself that preserve the length of vectors. For example, rotations and reflections belong to <math>\mathrm{O}(n)</math>. Equivalently, this is the group of ''n'' x ''n'' orthogonal matrices, meaning that <math>A^{\mathrm{T}}=A^{-1}</math>, where <math>A^{\mathrm{T}}</math> denotes the [[transpose]] of a matrix. The orthogonal group has two connected components; the identity component is called the ''special orthogonal group'' <math>\mathrm{SO}(n)</math>, consisting of the orthogonal matrices with determinant 1. Both groups have the same Lie algebra <math>\mathfrak{so}(n)</math>, the subspace of skew-symmetric matrices in <math>\mathfrak{gl}(n,\mathbb{R})</math> (<math>X^{\rm T}=-X</math>). See also [[Skew-symmetric matrix#Infinitesimal rotations|infinitesimal rotations with skew-symmetric matrices]].
:The complex orthogonal group <math>\mathrm{O}(n,\mathbb{C})</math>, its identity component <math>\mathrm{SO}(n,\mathbb{C})</math>, and the Lie algebra <math>\mathfrak{so}(n,\mathbb{C})</math> are given by the same formulas applied to ''n'' x ''n'' complex matrices. Equivalently, <math>\mathrm{O}(n,\mathbb{C})</math> is the subgroup of <math>\mathrm{GL}(n,\mathbb{C})</math> that preserves the standard [[symmetric bilinear form]] on <math>\mathbb{C}^n</math>.
* The [[unitary group]] <math>\mathrm{U}(n)</math> is the subgroup of <math>\mathrm{GL}(n,\mathbb{C})</math> that preserves the length of vectors in <math>\mathbb{C}^n</math> (with respect to the standard [[Hermitian inner product]]). Equivalently, this is the group of ''n''&nbsp;×&nbsp;''n'' unitary matrices (satisfying <math>A^*=A^{-1}</math>, where <math>A^*</math> denotes the [[conjugate transpose]] of a matrix). Its Lie algebra <math>\mathfrak{u}(n)</math> consists of the skew-hermitian matrices in <math>\mathfrak{gl}(n,\mathbb{C})</math> (<math>X^*=-X</math>). This is a Lie algebra over <math>\mathbb{R}</math>, not over <math>\mathbb{C}</math>. (Indeed, ''i'' times a skew-hermitian matrix is hermitian, rather than skew-hermitian.) Likewise, the unitary group <math>\mathrm{U}(n)</math> is a real Lie subgroup of the complex Lie group <math>\mathrm{GL}(n,\mathbb{C})</math>. For example, <math>\mathrm{U}(1)</math> is the [[circle group]], and its Lie algebra (from this point of view) is <math>i\mathbb{R}\subset \mathbb{C}=\mathfrak{gl}(1,\mathbb{C})</math>.
* The [[special unitary group]] <math>\mathrm{SU}(n)</math> is the subgroup of matrices with determinant 1 in <math>\mathrm{U}(n)</math>. Its Lie algebra <math>\mathfrak{su}(n)</math> consists of the skew-hermitian matrices with trace zero.
*The [[symplectic group]] <math>\mathrm{Sp}(2n,\R)</math> is the subgroup of <math>\mathrm{GL}(2n,\mathbb{R})</math> that preserves the standard [[symplectic vector space|alternating bilinear form]] on <math>\mathbb{R}^{2n}</math>. Its Lie algebra is the [[symplectic Lie algebra]] <math>\mathfrak{sp}(2n,\mathbb{R})</math>.
*The [[classical Lie algebra]]s are those listed above, along with variants over any field.