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| {{Lie groups}}
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| {{redirect|Lie bracket|the operation on vector fields|Lie bracket of vector fields}}
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| In [[mathematics]], '''Lie algebras''' ({{IPAc-en|ˈ|l|iː}}, not {{IPAc-en|ˈ|l|aɪ}}) are [[algebraic structure]]s which were introduced to study the concept of [[infinitesimal transformation]]s. The term "Lie algebra" (after [[Sophus Lie]]) was introduced by [[Hermann Weyl]] in the 1930s. In older texts, the name "'''infinitesimal group'''" is used.
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| Related mathematical concepts include [[Lie group]]s and [[differentiable manifold]]s.
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| == Definitions ==
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| A '''Lie algebra''' is a [[vector space]] <math>\,\mathfrak{g}</math> over some [[field (mathematics)|field]] ''F'' together with a [[binary operation]] <math>[\cdot,\cdot]: \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}</math> called the '''Lie bracket''', which satisfies the following axioms:
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| * [[Bilinear operator|Bilinearity]]:
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| ::<math> [a x + b y, z] = a [x, z] + b [y, z], \quad [z, a x + b y] = a[z, x] + b [z, y] </math>
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| :for all scalars ''a'', ''b'' in ''F'' and all elements ''x'', ''y'', ''z'' in <math>\mathfrak{g}</math>.
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| * [[Exterior product#Duality|Alternating]] on <math>\,\mathfrak{g}</math>:
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| ::<math> [x,x]=0\ </math>
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| :for all ''x'' in <math>\mathfrak{g}</math>.
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| * The [[Jacobi identity]]:
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| :: <math> [x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0 \quad </math>
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| :for all ''x'', ''y'', ''z'' in <math>\mathfrak{g}</math>.
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| Note that the bilinearity and alternating properties imply [[anticommutativity]], i.e., {{math| [''x,y''] {{=}} −[''y,x'']}}, for all elements ''x'', ''y'' in <math>\mathfrak{g}</math>, while anticommutativity only implies the alternating property if the field's [[Characteristic (algebra)|characteristic]] is not 2.<ref>Humphreys p. 1</ref>
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| It is customary to express a Lie algebra in lower-case [[fraktur]], like <math>\mathfrak{g}</math>. If a Lie algebra is associated with a [[Lie group]], then the spelling of the Lie algebra is the same as that Lie group. For example, the Lie algebra of [[special unitary group|SU(''n'')]] is written as <math>\mathfrak{su}(n)</math>.
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| === Generators and dimension ===
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| A collection of elements of a Lie algebra <math>\mathfrak{g}</math> are said to be '''[[Generator (mathematics)|generators]]''' of the Lie algebra if the smallest subalgebra of <math>\mathfrak{g}</math> containing them is <math>\mathfrak{g}</math> itself. The '''dimension''' of a Lie algebra is its dimension as a vector space over ''F''. The cardinality of a minimal generating set is always less than or equal to its dimension.
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| === Homomorphisms, subalgebras, and ideals ===
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| The Lie bracket is not [[associative]] in general, meaning that <math>[[x,y],z]</math> need not equal <math>[x,[y,z]]</math>. Nonetheless, much of the terminology that was developed in the theory of associative [[ring (mathematics)|rings]] or [[associative algebra]]s is commonly applied to Lie algebras. A subspace <math>\mathfrak{h} \subseteq \mathfrak{g}</math> that is closed under the Lie bracket is called a '''Lie subalgebra'''. If a subspace <math>I\subseteq\mathfrak{g}</math> satisfies a stronger condition that
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| : <math>[\mathfrak{g},I]\subseteq I,</math>
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| then ''I'' is called an '''ideal''' in the Lie algebra <math>\mathfrak{g}</math>.<ref>Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.</ref> A '''homomorphism''' between two Lie algebras (over the same [[base field]]) is a linear map that is compatible with the respective commutators:
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| : <math> f: \mathfrak{g}\to\mathfrak{g'}, \quad f([x,y])=[f(x),f(y)], </math>
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| for all elements ''x'' and ''y'' in <math>\mathfrak{g}</math>. As in the theory of associative rings, ideals are precisely the kernels of homomorphisms, given a Lie algebra <math>\mathfrak{g}</math> and an ideal ''I'' in it, one constructs the '''factor algebra''' <math>\mathfrak{g}/I</math>, and the [[first isomorphism theorem]] holds for Lie algebras.
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| Let ''S'' be a subset of <math>\mathfrak{g}</math>. The set of elements ''x'' such that <math>[x, s] = 0</math> for all ''s'' in ''S'' forms a subalgebra called the [[centralizer]] of ''S''. The centralizer of <math>\mathfrak{g}</math> itself is called the [[center (algebra)|center]] of <math>\mathfrak{g}</math>. Similar to centralizers, if ''S'' is a subspace,<ref>{{harvnb|Jacobson|1962|loc=pg. 28}}</ref> then the set of ''x'' such that <math>[x, s]</math> is in ''S'' for all ''s'' in ''S'' forms a subalgebra called the [[normalizer]] of ''S''.
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| === Direct sum ===
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| Given two Lie algebras <math>\mathfrak{g}</math> and <math>\mathfrak{g'}</math>, their [[Direct sum of modules|direct sum]] is the Lie algebra consisting of the vector space
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| <math>\mathfrak{g}\oplus\mathfrak{g'}</math>, of the pairs <math>\mathfrak{}(x,x'), \,x\in\mathfrak{g}, x'\in\mathfrak{g'}</math>, with the operation
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| : <math> [(x,x'),(y,y')]=([x,y],[x',y']), \quad x,y\in\mathfrak{g},\, x',y'\in\mathfrak{g'}.</math>
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| == Properties ==
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| === Admits an enveloping algebra ===
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| {{See also|Enveloping algebra}}
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| For any [[associative algebra]] ''A'' with multiplication <math>*</math>, one can construct a Lie algebra ''L''(''A''). As a vector space, ''L''(''A'') is the same as ''A''. The Lie bracket of two elements of ''L''(''A'') is defined to be their [[commutator#Ring theory|commutator]] in ''A'':
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| : <math> [a,b]=a * b-b * a.\ </math>
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| The associativity of the multiplication * in ''A'' implies the Jacobi identity of the commutator in ''L''(''A''). For example, the associative algebra of ''n'' × ''n'' matrices over a field ''F'' gives rise to the [[general linear group|general linear Lie algebra]] <math>\mathfrak{gl}_n(F).</math> The associative algebra ''A'' is called an '''enveloping algebra''' of the Lie algebra ''L''(''A''). Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see [[universal enveloping algebra]].
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| === Representation ===
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| Given a vector space ''V'', let <math>\mathfrak{gl}(V)</math> denote the Lie algebra enveloped by the associative algebra of all linear [[endomorphism]]s of ''V''. A [[Lie algebra representation|representation]] of a Lie algebra <math>\mathfrak{g}</math> on ''V'' is a Lie algebra homomorphism
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| :<math>\pi: \mathfrak g \to \mathfrak{gl}(V).</math>
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| A representation is said to be faithful if its kernel is trivial. Every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space ([[Ado's theorem]]).<ref>{{harvnb|Jacobson|1962|loc=Ch. VI}}</ref>
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| For example, <math>\operatorname{ad}</math> given by <math>\operatorname{ad}(x)(y) = [x, y]</math> is a representation of <math>\mathfrak{g}</math> on the vector space <math>\mathfrak{g}</math> called the [[adjoint representation of a Lie algebra|adjoint representation]]. A [[derivation (abstract algebra)|derivation]] on the Lie algebra <math>\mathfrak{g}</math> (in fact on any [[non-associative algebra]]) is a [[linear map]] <math>\delta:\mathfrak{g}\rightarrow \mathfrak{g}</math> that obeys the [[General Leibniz rule|Leibniz' law]], that is, | |
| :<math>\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]</math>
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| for all ''x'' and ''y'' in the algebra. For any ''x'', <math>\operatorname{ad}(x)</math> is a derivation; a consequence of the Jacobi identity. Thus, the image of <math>\operatorname{ad}</math> lies in the subalgebra of <math>\mathfrak{gl}(\mathfrak{g})</math> consisting of derivations. A derivation that happens to be in the image of <math>\operatorname{ad}</math> is called an inner derivation. If <math>\mathfrak{g}</math> is semisimple, every derivation is inner.
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| == Examples ==
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| === Vector spaces ===
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| *Any vector space ''V'' endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called [[Abelian group|abelian]], cf. below. Any one-dimensional Lie algebra over a field is abelian, by the [[#Definitions|antisymmetry of the Lie bracket]].
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| *The real vector space of all ''n'' × ''n'' [[skew-hermitian]] matrices is closed under the commutator and forms a real Lie algebra denoted <math>\mathfrak{u}(n)</math>. This is the Lie algebra of the [[unitary group]] ''U''(''n'').
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| <!--- This is a bit of an overkill, and slightly wrong
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| The vector space of left-invariant vector fields on a [[Lie group]] is closed under this operation and is therefore a finite dimensional Lie algebra. One may alternatively think of the underlying vector space of the Lie algebra belonging to a Lie group as the tangent space at the group's identity element. The multiplication is the differential of the group [[commutator]], (''a'',''b'') → ''aba''<sup>−1</sup>''b''<sup>−1</sup>, at the identity element.
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| --->
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| === Subspaces ===
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| * The subspace of the general linear Lie algebra <math>\mathfrak{gl}_n(F)</math> consisting of matrices of [[Trace (linear algebra)|trace]] zero is a subalgebra,<ref>Humphreys p.2</ref> the [[special linear Lie algebra]], denoted <math>\mathfrak{sl}_n(F).</math>
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| === Real matrix groups ===
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| * Any [[Lie group]] {{mvar|G}} defines an associated real Lie algebra <math>\mathfrak{g}</math>=Lie({{mvar|G}}). The definition in general is somewhat technical, but in the case of real [[matrix group]]s, it can be formulated via the [[exponential map]], or the matrix exponent. The Lie algebra <math>\mathfrak{g}</math> consists of those matrices {{mvar|X}} for which {{math| exp(''tX'') ∈ ''G''}}, ∀ real numbers {{mvar|t}}.
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| :The Lie bracket of <math>\mathfrak{g}</math> is given by the commutator of matrices. As a concrete example, consider the [[special linear group]] SL(''n'','''R'''), consisting of all ''n'' × ''n'' matrices with real entries and determinant 1. This is a matrix Lie group, and its Lie algebra consists of all ''n'' × ''n'' matrices with real entries and trace 0,
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| :: <math> L_{[X,Y]}f=L_X(L_Y f)-L_Y(L_X f)~.</math>
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| :This Lie algebra is related to the [[pseudogroup]] of [[diffeomorphism]]s of {{mvar|M}}.
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| === Three dimensions ===
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| *The three-dimensional [[Euclidean space]] '''R'''<sup>3</sup> with the Lie bracket given by the [[cross product]] of [[Vector (geometric)|vectors]] becomes a three-dimensional Lie algebra.
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| *The [[Heisenberg algebra]] ''H''<sub>3</sub>(R) is a three-dimensional Lie algebra generated by elements {{mvar|x}}, {{mvar|y}} and {{mvar|z}} with Lie brackets
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| :: <math>[x,y]=z,\quad [x,z]=0, \quad [y,z]=0</math> .
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| :It is explicitly realized as the space of 3×3 strictly upper-triangular matrices, with the Lie bracket given by the matrix commutator,
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| ::<math>
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| x = \left( \begin{array}{ccc}
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| 0&1&0\\
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| 0&0&0\\
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| 0&0&0
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| \end{array}\right),\quad
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| y = \left( \begin{array}{ccc}
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| 0&0&0\\
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| 0&0&1\\
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| 0&0&0
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| \end{array}\right),\quad
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| z = \left( \begin{array}{ccc}
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| 0&0&1\\
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| 0&0&0\\
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| 0&0&0
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| \end{array}\right)~.\quad
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| </math>
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| :Any element of the [[Heisenberg group]] is thus representable as a product of group generators, i.e., [[matrix exponential]]s of these Lie algebra generators,
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| ::<math>\left( \begin{array}{ccc}
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| 1&a&c\\
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| 0&1&b\\
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| 0&0&1
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| \end{array}\right)= e^{by} e^{cz} e^{ax}~.
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| </math>
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| * The commutation relations between the ''x'', ''y'', and ''z'' components of the [[angular momentum]] operator in [[quantum mechanics]] form a representation of a complex three-dimensional Lie algebra, which is the [[complexification]] of the Lie algebra ''so''(3) of the three-dimensional [[Rotation group SO(3)|rotation group]]:
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| :: <math>[L_x, L_y] = i \hbar L_z</math>
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| :: <math>[L_y, L_z] = i \hbar L_x</math>
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| :: <math>[L_z, L_x] = i \hbar L_y</math> .
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| === Infinite dimensions ===
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| *An important class of infinite-dimensional real Lie algebras arises in [[differential topology]]. The space of smooth [[vector field]]s on a [[differentiable manifold]] ''M'' forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of [[Lie derivative]]s, which identifies a vector field ''X'' with a first order partial differential operator ''L''<sub>''X''</sub> acting on smooth functions by letting ''L''<sub>''X''</sub>(''f'') be the directional derivative of the function ''f'' in the direction of ''X''. The Lie bracket [''X'',''Y''] of two vector fields is the vector field defined through its action on functions by the formula:
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|
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| :: <math> L_{[X,Y]}f=L_X(L_Y f)-L_Y(L_X f).\,</math>
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| *A [[Kac–Moody algebra]] is an example of an infinite-dimensional Lie algebra.
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| *The [[Moyal bracket|Moyal algebra]] is an infinite-dimensional Lie algebra which contains all [[Classical_Lie_groups#Relationship_with_bilinear_forms|classical Lie algebra]]s as subalgebras.
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| == Structure theory and classification ==
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| Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.
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| === Abelian, nilpotent, and solvable ===
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| Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.
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| A Lie algebra <math>\mathfrak{g}</math> is '''abelian{{anchor|abelian}}''' if the Lie bracket vanishes, i.e. [''x'',''y''] = 0, for all ''x'' and ''y'' in <math>\mathfrak{g}</math>. Abelian Lie algebras correspond to commutative (or [[abelian group|abelian]]) connected Lie groups such as vector spaces <math>K^n</math> or [[torus|tori]] <math>T^n,</math> and are all of the form <math>\mathfrak{k}^n,</math> meaning an ''n''-dimensional vector space with the trivial Lie bracket.
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| A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra <math>\mathfrak{g}</math> is '''[[nilpotent Lie algebra|nilpotent]]''' if the [[lower central series]]
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| :<math> \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots</math>
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| becomes zero eventually. By [[Engel's theorem]], a Lie algebra is nilpotent if and only if for every ''u'' in <math>\mathfrak{g}</math> the [[adjoint endomorphism]]
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| :<math>\operatorname{ad}(u):\mathfrak{g} \to \mathfrak{g}, \quad \operatorname{ad}(u)v=[u,v]</math>
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| is nilpotent.
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| More generally still, a Lie algebra <math>\mathfrak{g}</math> is said to be '''[[solvable Lie algebra|solvable]]''' if the [[derived series]]:
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| :<math> \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]] > [[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]] > \cdots</math>
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| becomes zero eventually.
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| Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its [[radical of a Lie algebra|radical]]. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.
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| === Simple and semisimple ===
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| A Lie algebra is "[[Simple Lie algebra|simple]]" if it has no non-trivial ideals and is not abelian.
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| A Lie algebra <math>\mathfrak{g}</math> is called '''[[semisimple Lie algebra|semisimple]]''' if its radical is zero. Equivalently, <math>\mathfrak{g}</math> is semisimple if it does not contain any non-zero abelian ideals. In particular, a simple Lie algebra is semisimple. Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras.
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| The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field ''F'' has [[characteristic (field)|characteristic]] zero, any finite-dimensional representation of a semisimple Lie algebra is semisimple (i.e., direct sum of irreducible representations.) In general, a Lie algebra is called [[reductive Lie algebra|reductive]] if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.<!--
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| of a Lie algebra <math>\mathfrak{g}</math> over ''F'' is equivalent to the complete reducibility of all finite-dimensional [[Lie algebra representation|representations]] of <math>\mathfrak{g}.</math> An early proof of this statement proceeded via connection with compact groups ([[Weyl's unitary trick]]), but later entirely algebraic proofs were found.-->
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| === Cartan's criterion ===
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| [[Cartan's criterion]] gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the [[Killing form]], a [[symmetric bilinear form]] on <math>\mathfrak{g}</math> defined by the formula
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| : <math>K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),</math>
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| where tr denotes the [[Trace (linear algebra)|trace of a linear operator]]. A Lie algebra <math>\mathfrak{g}</math> is semisimple if and only if the Killing form is [[nondegenerate form|nondegenerate]]. A Lie algebra <math>\mathfrak{g}</math> is solvable if and only if <math>K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0.</math>
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| === Classification ===
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| The [[Levi decomposition]] expresses an arbitrary Lie algebra as a [[semidirect sum]] of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their [[root system]]s. However, the classification of solvable Lie algebras is a 'wild' problem, and cannot{{Clarify|date=April 2009}} be accomplished in general.
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| == Relation to Lie groups ==
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| Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups.
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| Lie's fundamental theorems describe a relation between [[Lie groups]] and Lie algebras. In particular, any Lie group gives rise to a canonically determined Lie algebra (concretely, ''the tangent space at the identity''); and, conversely, for any Lie algebra there is a corresponding connected Lie group ([[Lie's third theorem]]; see the [[Baker–Campbell–Hausdorff formula]]). This Lie group is not determined uniquely; however, any two connected Lie groups with the same Lie algebra are ''locally isomorphic'', and in particular, have the same [[universal cover]]. For instance, the [[special orthogonal group]] [[SO(3)]] and the [[special unitary group]] [[SU(2)]] give rise to the same Lie algebra, which is isomorphic to '''R'''<sup>3</sup> with the cross-product, while SU(2) is a simply-connected twofold cover of SO(3).
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| Given a Lie group, a Lie algebra can be associated to it either by endowing the [[tangent space]] to the [[identity element|identity]] with the [[pushforward (differential)|differential]] of the [[adjoint representation of a Lie group|adjoint map]], or by considering the left-invariant vector fields as mentioned in the examples. In the case of real [[matrix group]]s, the Lie algebra <math>\mathfrak{g}</math> consists of those matrices {{mvar|X}} for which {{math|exp(''tX'') ∈ ''G''}} for all real numbers {{mvar|t}}, where {{math|exp}} is the [[Matrix exponential|exponential map]].
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| Some examples of Lie algebras corresponding to Lie groups are the following:
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| * The Lie algebra <math>\mathfrak{gl}_n(\mathbb{C})</math> for the group <math>\mathrm{GL}_n(\mathbb{C})</math> is the algebra of complex {{math|''n×n''}} matrices
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| * The Lie algebra <math>\mathfrak{sl}_n(\mathbb{C})</math> for the group <math>\mathrm{SL}_n(\mathbb{C})</math> is the algebra of complex {{math|''n×n''}} matrices with trace 0
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| * The Lie algebras <math>\mathfrak{o}(n)</math> for the group <math>\mathrm{O}(n)</math> and <math>\mathfrak{so}(n)</math> for <math>\mathrm{SO}(n)</math> are both the algebra of real anti-symmetric {{math|''n×n''}} matrices (See [[Antisymmetric_matrix#Infinitesimal_rotations|Antisymmetric matrix: Infinitesimal rotations]] for a discussion)
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| * The Lie algebra <math>\mathfrak{u}(n)</math> for the group <math>\mathrm{U}(n)</math> is the algebra of skew-Hermitian complex {{math|''n×n''}} matrices while the Lie algebra <math>\mathfrak{su}(n)</math> for <math>\mathrm{SU}(n)</math> is the algebra of skew-Hermitian, traceless complex {{math|''n×n''}} matrices.
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| In the above examples, the Lie bracket <math>[X,Y]</math> (for <math>X</math> and <math>Y</math> matrices in the Lie algebra) is defined as <math>[X,Y] = XY - YX</math>.
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| Given a set of generators {{math|''T<sup>a</sup>''}}, the '''[[structure constants]]''' {{math|''f <sup>abc</sup>''}} express the Lie brackets of pairs of generators as linear combinations of generators from the set, i.e., {{math|[''T<sup>a</sup>, T<sup>b</sup>''] {{=}} ''f <sup>abc</sup> T<sup>c</sup>''}}. The structure constants determine the Lie brackets of elements of the Lie algebra, and consequently nearly completely determine the group structure of the Lie group. The structure of the Lie group near the identity element is displayed explicitly by the [[Baker–Campbell–Hausdorff formula]], an expansion in Lie algebra elements {{math|''X, Y''}} and their Lie brackets, all nested together within a single exponent, {{math|exp(''tX'') exp(''tY'') {{=}} exp(''tX''+''tY''+½ ''t<sup>2</sup>''[''X,Y''] + O(''t<sup>3</sup>'') )}}.
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| The mapping from Lie groups to Lie algebras is [[functorial]], which implies that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps Lie subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, quotients and cokernels of Lie algebras, respectively.
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| The [[functor]] '''L''' which takes each Lie group to its Lie algebra and each homomorphism to its differential is [[faithful functor|faithful]] and [[exact functor|exact]]. It is however not an [[equivalence of categories]]: different Lie groups may have isomorphic Lie algebras (for example [[special orthogonal group|SO(3)]] and [[special unitary group|SU(2)]] ), and there are (infinite dimensional) Lie algebras that are not associated to any Lie group.<ref>{{harvnb|Beltita|2005|loc=pg. 75}}</ref>
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| However, when the Lie algebra <math>\mathfrak{g}</math> is finite-dimensional, one can associate to it a [[simply connected space|simply connected]] Lie group having <math>\mathfrak{g}</math> as its Lie algebra. More precisely, the Lie algebra functor '''L''' has a [[adjoint functors|left adjoint functor]] '''Γ''' from finite-dimensional (real) Lie algebras to Lie groups, factoring through the full subcategory of simply connected Lie groups.<ref>Adjoint property is discussed in more general context in Hofman & Morris (2007) (e.g., page 130) but is a straightforward consequence of, e.g., Bourbaki (1989) Theorem 1 of page 305 and Theorem 3 of page 310.</ref> In other words, there is a natural isomorphism of bifunctors
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| ::<math> \mathrm{Hom}(\Gamma(\mathfrak{g}), H) \cong \mathrm{Hom}(\mathfrak{g},\mathrm{L}(H)).</math>
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| The adjunction <math>\mathfrak{g} \rightarrow \mathrm{L}(\Gamma(\mathfrak{g}))</math> (corresponding to the identity on <math>\Gamma(\mathfrak{g})</math>) is an isomorphism, and the other adjunction <math>\Gamma(\mathrm{L}(H)) \rightarrow H</math> is the projection homomorphism from the [[universal cover]] group of the identity component of {{mvar|H}} to {{mvar|H}}. It follows immediately that if {{mvar|G}} is simply connected, then the Lie algebra functor establishes a bijective correspondence between Lie group homomorphisms {{math|''G→H''}} and Lie algebra homomorphisms {{math|'''L'''(''G'')→'''L'''(''H'')}}.
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| The universal cover group above can be constructed as the image of the Lie algebra under the [[exponential map]]. More generally, we have that the Lie algebra is [[homeomorphic]] to a [[neighborhood (mathematics)|neighborhood]] of the identity. But globally, if the Lie group is compact, the exponential will not be [[injective]], and if the Lie group is not connected, [[simply connected]] or [[compact space|compact]], the exponential map need not be [[surjective]].
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| If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a [[homeomorphism]] (for example, in Diff('''S'''<sup>1</sup>), one may find diffeomorphisms arbitrarily close to the identity that are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.
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| The correspondence between Lie algebras and Lie groups is used in several ways, including in the [[list of simple Lie groups|classification of Lie groups]] and the related matter of the [[representation theory]] of Lie groups. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one to one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group.
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| As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the [[Center (algebra)|center]], once the classification of Lie algebras is known (solved by [[Élie Cartan|Cartan]] et al. in the [[Semisimple Lie algebra|semisimple]] case).
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| == Category theoretic definition ==
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| Using the language of [[category theory]], a '''Lie algebra''' can be defined as an object ''A'' in '''Vec'''<sub>''k''</sub>, the [[category of vector spaces]] over a field ''k'' of characteristic not 2, together with a [[morphism]] [.,.]: ''A'' ⊗ ''A'' → ''A'', where ⊗ refers to the [[Monoidal category|monoidal product]] of '''Vec'''<sub>''k''</sub>, such that
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| *<math>[\cdot, \cdot] \circ (\mathrm{id} + \tau_{A,A}) = 0</math>
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| *<math>[\cdot, \cdot] \circ ([\cdot, \cdot] \otimes \mathrm{id}) \circ (\mathrm{id} + \sigma + \sigma^2) = 0</math>
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| where τ (''a'' ⊗ ''b'') := ''b'' ⊗ ''a'' and σ is the [[cyclic permutation]] braiding (id ⊗ τ<sub>''A'',''A''</sub>) ° (τ<sub>''A'',''A''</sub> ⊗ id). In [[Diagrammatic Notation|diagrammatic form]]:
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| :<center>[[File:Liealgebra.png]]</center>
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| ==See also==
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| {{Col-begin}}
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| {{Col-1-of-2}}
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| * [[Adjoint representation of a Lie algebra]]
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| * [[Anyonic Lie algebra]]
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| * [[Differential graded Lie algebra]]
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| * [[Index of a Lie algebra]]
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| * [[Killing form]]
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| * [[Lie algebra cohomology]]
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| * [[Lie algebra representation]]
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| * [[Lie bialgebra]]
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| * [[Lie coalgebra]]
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| * [[Particle physics and representation theory]]
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| * [[Glossary of group theory]]
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| {{Col-2-of-2}}
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| * [[Lie superalgebra]]
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| * [[Matrix (mathematics)]]
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| * [[Poisson algebra]]
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| * [[Quantum groups]]
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| * [[Moyal bracket|Moyal algebra]]
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| * [[Quasi-Frobenius Lie algebra]]
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| * [[Quasi-Lie algebra]]
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| * [[Restricted Lie algebra]]
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| * [[Symmetric Lie algebra]] <!-- missing article -->
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| {{col-end}}
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| == Notes ==
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| <references />
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| ==References==
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| * Beltita, Daniel. ''Smooth Homogeneous Structures in Operator Theory'', CRC Press, 2005. ISBN 978-1-4200-3480-6
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| * Boza, Luis; Fedriani, Eugenio M. & Núñez, Juan. ''A new method for classifying complex filiform Lie algebras'', Applied Mathematics and Computation, 121 (2-3): 169–175, 2001
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| * Bourbaki, Nicolas. "Lie Groups and Lie Algebras - Chapters 1-3", Springer, 1989, ISBN 3-540-64242-0
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| * [[Karin Erdmann|Erdmann, Karin]] & Wildon, Mark. ''Introduction to Lie Algebras'', 1st edition, Springer, 2006. ISBN 1-84628-040-0
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| * Hall, Brian C. ''Lie Groups, Lie Algebras, and Representations: An Elementary Introduction'', Springer, 2003. ISBN 0-387-40122-9
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| * Hofman, Karl & Morris, Sidney. "The Lie Theory of Connected Pro-Lie Groups", European Mathematical Society, 2007, ISBN 978-3-03719-032-6
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| * Humphreys, James E. ''Introduction to Lie Algebras and Representation Theory'', Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5
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| * Jacobson, Nathan, ''Lie algebras'', Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
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| * Kac, Victor G. et al. ''Course notes for MIT 18.745: Introduction to Lie Algebras'', [http://www.math.mit.edu/~lesha/745lec/ math.mit.edu]
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| * O'Connor, J.J. & Robertson, E.F. Biography of Sophus Lie, MacTutor History of Mathematics Archive, [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Lie.html www-history.mcs.st-andrews.ac.uk]
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| * O'Connor, J.J. & Robertson, E.F. Biography of Wilhelm Killing, MacTutor History of Mathematics Archive, [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Killing.html www-history.mcs.st-andrews.ac.uk]
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| * Serre, Jean-Pierre. "Lie Algebras and Lie Groups", 2nd edition, Springer, 2006. ISBN 3-540-55008-9
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| * Steeb, W.-H. ''Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra'', second edition, World Scientific, 2007, ISBN 978-981-270-809-0
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| * Varadarajan, V.S. ''Lie Groups, Lie Algebras, and Their Representations'', 1st edition, Springer, 2004. ISBN 0-387-90969-9
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| {{DEFAULTSORT:Lie Algebra}}
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| [[Category:Lie groups]]
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| [[Category:Lie algebras| ]]
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