Lie algebra

A Lie algebra is a vector space ${\displaystyle \,{\mathfrak {g}}}$ over some field ${\displaystyle F}$ together with a binary operation ${\displaystyle [\,\cdot \,,\cdot \,]:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}}$ called the Lie bracket satisfying the following axioms:[lower-alpha 1]

• Bilinearity,

${\displaystyle [ax+by,z]=a[x,z]+b[y,z],}$

${\displaystyle [z,ax+by]=a[z,x]+b[z,y]}$

for all scalars ${\displaystyle a}$, ${\displaystyle b}$ in ${\displaystyle F}$ and all elements ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$ in ${\displaystyle {\mathfrak {g}}}$.

• Alternativity,

${\displaystyle [x,x]=0\ }$

for all ${\displaystyle x}$ in ${\displaystyle {\mathfrak {g}}}$.

• The Jacobi identity,

${\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\ }$

for all ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$ in ${\displaystyle {\mathfrak {g}}}$.

Using bilinearity to expand the Lie bracket ${\displaystyle [x+y,x+y]}$ and using alternativity shows that ${\displaystyle [x,y]+[y,x]=0\ }$ for all elements ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle {\mathfrak {g}}}$, showing that bilinearity and alternativity together imply

• Anticommutativity,

${\displaystyle [x,y]=-[y,x],\ }$

for all elements ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle {\mathfrak {g}}}$. If the field's characteristic is not 2 then anticommutativity implies alternativity, since it implies ${\displaystyle [x,x]=-[x,x].}$[3]

It is customary to denote a Lie algebra by a lower-case fraktur letter such as ${\displaystyle {\mathfrak {g,h,b,n}}}$. If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group: for example the Lie algebra of SU(n) is ${\displaystyle {\mathfrak {su}}(n)}$.

Elements of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ are said to generate it if the smallest subalgebra containing these elements is ${\displaystyle {\mathfrak {g}}}$ itself. The dimension of a Lie algebra is its dimension as a vector space over ${\displaystyle F}$. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension.

See the classification of low-dimensional real Lie algebras for other small examples.

The Lie bracket is not required to be associative, meaning that ${\displaystyle [[x,y],z]}$ need not equal ${\displaystyle [x,[y,z]]}$. However, it is flexible. Nonetheless, much of the terminology of associative rings and algebras is commonly applied to Lie algebras. A Lie subalgebra is a subspace ${\displaystyle {\mathfrak {h}}\subseteq {\mathfrak {g}}}$ which is closed under the Lie bracket. An ideal ${\displaystyle {\mathfrak {i}}\subseteq {\mathfrak {g}}}$ is a subalgebra satisfying the stronger condition:[4]

${\displaystyle [{\mathfrak {g}},{\mathfrak {i}}]\subseteq {\mathfrak {i}}.}$

A Lie algebra homomorphism is a linear map compatible with the respective Lie brackets:

${\displaystyle \phi$ :{\mathfrak {g}}\to {\mathfrak {g'}},\quad \phi ([x,y])=[\phi (x),\phi (y)]\ {\text{for all}}\ x,y\in {\mathfrak {g}}.}

As for associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebra ${\displaystyle {\mathfrak {g}}}$ and an ideal ${\displaystyle {\mathfrak {i}}}$ in it, one constructs the factor algebra or quotient algebra ${\displaystyle {\mathfrak {g}}/{\mathfrak {i}}}$, and the first isomorphism theorem holds for Lie algebras.

Since the Lie bracket is a kind of infinitesimal commutator of the corresponding Lie group, we say that two elements ${\displaystyle x,y\in {\mathfrak {g}}}$ commute if their bracket vanishes: ${\displaystyle [x,y]=0}$.

The centralizer subalgebra of a subset ${\displaystyle S\subset {\mathfrak {g}}}$ is the set of elements commuting with ${\displaystyle S}$: that is, ${\displaystyle {\mathfrak {z}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}\ \mid \ [x,s]=0\ {\text{ for all }}s\in S\}}$. The centralizer of ${\displaystyle {\mathfrak {g}}}$ itself is the center ${\displaystyle {\mathfrak {z}}({\mathfrak {g}})}$. Similarly, for a subspace S, the normalizer subalgebra of ${\displaystyle S}$ is ${\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}\ \mid \ [x,s]\in S\ {\text{ for all}}\ s\in S\}}$.[5] Equivalently, if ${\displaystyle S}$ is a Lie subalgebra, ${\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)}$ is the largest subalgebra such that ${\displaystyle S}$ is an ideal of ${\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)}$.

For ${\displaystyle {\mathfrak {d}}(2)\subset {\mathfrak {gl}}(2)}$, the commutator of two elements ${\displaystyle g\in {\mathfrak {gl}}(2)}$ and ${\displaystyle d\in {\mathfrak {d}}(2)}$:

${\displaystyle {\begin{aligned}\left[{\begin{bmatrix}a&b\\c&d\end{bmatrix}},{\begin{bmatrix}x&0\\0&y\end{bmatrix}}\right]&={\begin{bmatrix}ax&by\\cx&dy\\\end{bmatrix}}-{\begin{bmatrix}ax&bx\\cy&dy\\\end{bmatrix}}\\&={\begin{bmatrix}0&b(y-x)\\c(x-y)&0\end{bmatrix}}\end{aligned}}}$

shows ${\displaystyle {\mathfrak {d}}(2)}$ is a subalgebra, but not an ideal. In fact, every one-dimensional linear subspace of a Lie algebra has an induced abelian Lie algebra structure, which is generally not an ideal. For any simple Lie algebra, all abelian Lie algebras can never be ideals.

For two Lie algebras ${\displaystyle {\mathfrak {g^{}}}}$ and ${\displaystyle {\mathfrak {g'}}}$, their direct sum Lie algebra is the vector space ${\displaystyle {\mathfrak {g}}\oplus {\mathfrak {g'}}}$consisting of all pairs ${\displaystyle {\mathfrak {}}(x,x'),\,x\in {\mathfrak {g}},\ x'\in {\mathfrak {g'}}}$, with the operation

${\displaystyle [(x,x'),(y,y')]=([x,y],[x',y']),}$

so that the copies of ${\displaystyle {\mathfrak {g}},{\mathfrak {g}}'}$ commute with each other: ${\displaystyle [(x,0),(0,x')]=0.}$

Let ${\displaystyle {\mathfrak {g}}}$ be a Lie algebra and ${\displaystyle {\mathfrak {i}}}$ an ideal of ${\displaystyle {\mathfrak {g}}}$. If the canonical map ${\displaystyle {\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}}$ splits (i.e., admits a section), then ${\displaystyle {\mathfrak {g}}}$ is said to be a semidirect product of ${\displaystyle {\mathfrak {i}}}$ and ${\displaystyle {\mathfrak {g}}/{\mathfrak {i}}}$, ${\displaystyle {\mathfrak {g}}={\mathfrak {g}}/{\mathfrak {i}}\ltimes {\mathfrak {i}}}$. See also semidirect sum of Lie algebras.

Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra (Levi subalgebra).

A derivation on the Lie algebra ${\displaystyle {\mathfrak {g}}}$ (or on any non-associative algebra) is a linear map ${\displaystyle \delta \colon {\mathfrak {g}}\rightarrow {\mathfrak {g}}}$ that obeys the Leibniz law, that is,

${\displaystyle \delta ([x,y])=[\delta (x),y]+[x,\delta (y)]}$

for all ${\displaystyle x,y\in {\mathfrak {g}}}$. The inner derivation associated to any ${\displaystyle x\in {\mathfrak {g}}}$ is the adjoint mapping ${\displaystyle \mathrm {ad} _{x}}$ defined by ${\displaystyle \mathrm {ad} _{x}(y):=[x,y]}$. (This is a derivation as a consequence of the Jacobi identity.) The outer derivations are derivations which do not come from the adjoint representation of the Lie algebra. If ${\displaystyle {\mathfrak {g}}}$ is semisimple, every derivation is inner.

The derivations form a vector space ${\displaystyle \mathrm {Der} ({\mathfrak {g}})}$, which is a Lie subalgebra of ${\displaystyle {\mathfrak {gl}}({\mathfrak {g}})}$; the bracket is commutator. The inner derivations form a Lie subalgebra of ${\displaystyle \mathrm {Der} ({\mathfrak {g}})}$.

For example, given a Lie algebra ideal ${\displaystyle {\mathfrak {i}}\subset {\mathfrak {g}}}$ the adjoint representation ${\displaystyle {\mathfrak {ad}}_{\mathfrak {g}}}$ of ${\displaystyle {\mathfrak {g}}}$ acts as outer derivations on ${\displaystyle {\mathfrak {i}}}$ since ${\displaystyle [x,i]\subset {\mathfrak {i}}}$ for any ${\displaystyle x\in {\mathfrak {g}}}$ and ${\displaystyle i\in {\mathfrak {i}}}$. For the Lie algebra ${\displaystyle {\mathfrak {b}}_{n}}$ of upper triangular matrices in ${\displaystyle {\mathfrak {gl}}(n)}$, it has an ideal ${\displaystyle {\mathfrak {n}}_{n}}$ of strictly upper triangular matrices (where the only non-zero elements are above the diagonal of the matrix). For instance, the commutator of elements in ${\displaystyle {\mathfrak {b}}_{3}}$ and ${\displaystyle {\mathfrak {n}}_{3}}$ gives

${\displaystyle {\begin{aligned}\left[{\begin{bmatrix}a&b&c\\0&d&e\\0&0&f\end{bmatrix}},{\begin{bmatrix}0&x&y\\0&0&z\\0&0&0\end{bmatrix}}\right]&={\begin{bmatrix}0&ax&ay+bz\\0&0&dz\\0&0&0\end{bmatrix}}-{\begin{bmatrix}0&dx&ex+yf\\0&0&fz\\0&0&0\end{bmatrix}}\\&={\begin{bmatrix}0&(a-d)x&(a-f)y-ex+bz\\0&0&(d-f)z\\0&0&0\end{bmatrix}}\end{aligned}}}$

shows there exist outer derivations from ${\displaystyle {\mathfrak {b}}_{3}}$ in ${\displaystyle {\text{Der}}({\mathfrak {n}}_{3})}$.

Let V be a finite-dimensional vector space over a field F, ${\displaystyle {\mathfrak {gl}}(V)}$ the Lie algebra of linear transformations and ${\displaystyle {\mathfrak {g}}\subseteq {\mathfrak {gl}}(V)}$ a Lie subalgebra. Then ${\displaystyle {\mathfrak {g}}}$ is said to be split if the roots of the characteristic polynomials of all linear transformations in ${\displaystyle {\mathfrak {g}}}$ are in the base field F.[6] More generally, a finite-dimensional Lie algebra ${\displaystyle {\mathfrak {g}}}$ is said to be split if it has a Cartan subalgebra whose image under the adjoint representation ${\displaystyle \operatorname {ad}$ :{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})} is a split Lie algebra. A split real form of a complex semisimple Lie algebra (cf. #Real form and complexification) is an example of a split real Lie algebra. See also split Lie algebra for further information.

For practical calculations, it is often convenient to choose an explicit vector space basis for the algebra. A common construction for this basis is sketched in the article structure constants.

Although the definitions above are sufficient for a conventional understanding of Lie algebras, once this is understood, additional insight can be gained by using notation common to category theory, that is, by defining a Lie algebra in terms of linear maps—that is, morphisms of the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is supposed to be of characteristic different from two.)

For the category-theoretic definition of Lie algebras, two braiding isomorphisms are needed. If A is a vector space, the interchange isomorphism ${\displaystyle \tau :A\otimes A\to A\otimes A}$ is defined by

${\displaystyle \tau (x\otimes y)=y\otimes x.}$

The cyclic-permutation braiding ${\displaystyle \sigma :A\otimes A\otimes A\to A\otimes A\otimes A}$ is defined as

${\displaystyle \sigma =(\mathrm {id} \otimes \tau )\circ (\tau \otimes \mathrm {id} ),}$

where ${\displaystyle \mathrm {id} }$ is the identity morphism. Equivalently, ${\displaystyle \sigma }$ is defined by

${\displaystyle \sigma (x\otimes y\otimes z)=y\otimes z\otimes x.}$

With this notation, a Lie algebra can be defined as an object ${\displaystyle A}$ in the category of vector spaces together with a morphism

${\displaystyle [\cdot ,\cdot ]:A\otimes A\rightarrow A}$

that satisfies the two morphism equalities

${\displaystyle [\cdot ,\cdot ]\circ (\mathrm {id} +\tau )=0,}$

and

${\displaystyle [\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes \mathrm {id} )\circ (\mathrm {id} +\sigma +\sigma ^{2})=0.}$

Any vector space ${\displaystyle V}$ endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field of characteristic different from 2 is abelian, by the alternating property of the Lie bracket.

• On an associative algebra ${\displaystyle A}$ over a field ${\displaystyle F}$ with multiplication ${\displaystyle (x,y)\mapsto xy}$, a Lie bracket may be defined by the commutator ${\displaystyle [x,y]=xy-yx}$. With this bracket, ${\displaystyle A}$ is a Lie algebra.[7] The associative algebra A is called an enveloping algebra of the Lie algebra ${\displaystyle (A,[\,\cdot \,,\cdot \,])}$. Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see universal enveloping algebra.
• The associative algebra of the endomorphisms of an F-vector space ${\displaystyle V}$ with the above Lie bracket is denoted ${\displaystyle {\mathfrak {gl}}(V)}$.
• For a finite dimensional vector space ${\displaystyle V=F^{n}}$, the previous example is exactly the Lie algebra of n × n matrices, denoted ${\displaystyle {\mathfrak {gl}}(n,F)}$ or ${\displaystyle {\mathfrak {gl}}_{n}(F)}$,[8] and with bracket ${\displaystyle [X,Y]=XY-YX}$ where adjacency indicates matrix multiplication. This is the Lie algebra of the general linear group, consisting of invertible matrices.

Two important subalgebras of ${\displaystyle {\mathfrak {gl}}_{n}(F)}$ are:

• The matrices of trace zero form the special linear Lie algebra ${\displaystyle {\mathfrak {sl}}_{n}(F)}$, the Lie algebra of the special linear group ${\displaystyle \mathrm {SL} _{n}(F)}$.[9]
• The skew-hermitian matrices form the unitary Lie algebra ${\displaystyle {\mathfrak {u}}(n)}$, the Lie algebra of the unitary group U(n).

A complex matrix group is a Lie group consisting of matrices, ${\displaystyle G\subset M_{n}(\mathbb {C} )}$, where the multiplication of G is matrix multiplication. The corresponding Lie algebra ${\displaystyle {\mathfrak {g}}}$ is the space of matrices which are tangent vectors to G inside the linear space ${\displaystyle M_{n}(\mathbb {C} )}$: this consists of derivatives of smooth curves in G at the identity:

${\displaystyle {\mathfrak {g}}=\{X=c'(0)\in M_{n}(\mathbb {C} )\ \mid \ {\text{ smooth }}c:\mathbb {R} \to G,\ c(0)=I\}.}$

The Lie bracket of ${\displaystyle {\mathfrak {g}}}$ is given by the commutator of matrices, ${\displaystyle [X,Y]=XY-YX}$. Given the Lie algebra, one can recover the Lie group as the image of the matrix exponential mapping ${\displaystyle \exp :M_{n}(\mathbb {C} )\to M_{n}(\mathbb {C} )}$ defined by ${\displaystyle \exp(X)=I+X+{\tfrac {1}{2!}}X^{2}+\cdots }$, which converges for every matrix ${\displaystyle X}$: that is, ${\displaystyle G=\exp({\mathfrak {g}})}$.

The following are examples of Lie algebras of matrix Lie groups:[10]

• The special linear group ${\displaystyle {\rm {SL}}_{n}(\mathbb {C} )}$, consisting of all n × n matrices with determinant 1. Its Lie algebra ${\displaystyle {\mathfrak {sl}}_{n}(\mathbb {C} )}$consists of all n × n matrices with complex entries and trace 0. Similarly, one can define the corresponding real Lie group ${\displaystyle {\rm {SL}}_{n}(\mathbb {R} )}$ and its Lie algebra ${\displaystyle {\mathfrak {sl}}_{n}(\mathbb {R} )}$.
• The unitary group ${\displaystyle U(n)}$ consists of n × n unitary matrices (satisfying ${\displaystyle U^{*}=U^{-1}}$). Its Lie algebra ${\displaystyle {\mathfrak {u}}(n)}$ consists of skew-self-adjoint matrices (${\displaystyle X^{*}=-X}$).
• The special orthogonal group ${\displaystyle \mathrm {SO} (n)}$, consisting of real determinant-one orthogonal matrices (${\displaystyle A^{\mathrm {T} }=A^{-1}}$). Its Lie algebra ${\displaystyle {\mathfrak {so}}(n)}$ consists of real skew-symmetric matrices (${\displaystyle X^{\rm {T}}=-X}$). The full orthogonal group ${\displaystyle \mathrm {O} (n)}$, without the determinant-one condition, consists of ${\displaystyle \mathrm {SO} (n)}$ and a separate connected component, so it has the same Lie algebra as ${\displaystyle \mathrm {SO} (n)}$. See also infinitesimal rotations with skew-symmetric matrices. Similarly, one can define a complex version of this group and algebra, simply by allowing complex matrix entries.

• On any field ${\displaystyle F}$ there is, up to isomorphism, a single two-dimensional nonabelian Lie algebra. With generators x, y, its bracket is defined as ${\displaystyle \left[x,y\right]=y}$. It generates the affine group in one dimension.

This can be realized by the matrices:

${\displaystyle x=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad y=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right).}$

Since

${\displaystyle \left({\begin{array}{cc}1&c\\0&0\end{array}}\right)^{n+1}=\left({\begin{array}{cc}1&c\\0&0\end{array}}\right)}$

for any natural number ${\displaystyle n}$ and any ${\displaystyle c}$, one sees that the resulting Lie group elements are upper triangular 2×2 matrices with unit lower diagonal:

${\displaystyle \exp(a\cdot {}x+b\cdot {}y)=\left({\begin{array}{cc}e^{a}&{\tfrac {b}{a}}(e^{a}-1)\\0&1\end{array}}\right)=1+{\tfrac {e^{a}-1}{a}}\left(a\cdot {}x+b\cdot {}y\right).}$

• The Heisenberg algebra ${\displaystyle {\rm {H}}_{3}(\mathbb {R} )}$ is a three-dimensional Lie algebra generated by elements x, y, and z with Lie brackets

${\displaystyle [x,y]=z,\quad [x,z]=0,\quad [y,z]=0}$.

It is usually realized as the space of 3×3 strictly upper-triangular matrices, with the commutator Lie bracket and the basis

${\displaystyle x=\left({\begin{array}{ccc}0&1&0\\0&0&0\\0&0&0\end{array}}\right),\quad y=\left({\begin{array}{ccc}0&0&0\\0&0&1\\0&0&0\end{array}}\right),\quad z=\left({\begin{array}{ccc}0&0&1\\0&0&0\\0&0&0\end{array}}\right)~.\quad }$

Any element of the Heisenberg group has a representation as a product of group generators, i.e., matrix exponentials of these Lie algebra generators,

${\displaystyle \left({\begin{array}{ccc}1&a&c\\0&1&b\\0&0&1\end{array}}\right)=e^{by}e^{cz}e^{ax}~.}$

• The Lie algebra ${\displaystyle {\mathfrak {so}}(3)}$ of the group SO(3) is spanned by the three matrices[11]

${\displaystyle F_{1}=\left({\begin{array}{ccc}0&0&0\\0&0&-1\\0&1&0\end{array}}\right),\quad F_{2}=\left({\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}}\right),\quad F_{3}=\left({\begin{array}{ccc}0&-1&0\\1&0&0\\0&0&0\end{array}}\right)~.\quad }$

The commutation relations among these generators are

${\displaystyle [F_{1},F_{2}]=F_{3},}$

${\displaystyle [F_{2},F_{3}]=F_{1},}$

${\displaystyle [F_{3},F_{1}]=F_{2}.}$

The three-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ with the Lie bracket given by the cross product of vectors has the same commutation relations as above: thus, it is isomorphic to ${\displaystyle {\mathfrak {so}}(3)}$. This Lie algebra is unitarily equivalent to the usual Spin (physics) angular-momentum component operators for spin-1 particles in quantum mechanics.

• An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields 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 derivatives, which identifies a vector field X with a first order partial differential operator L_X acting on smooth functions by letting L_X(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:

${\displaystyle L_{[X,Y]}f=L_{X}(L_{Y}f)-L_{Y}(L_{X}f).\,}$

• Kac–Moody algebras are a large class of infinite-dimensional Lie algebras whose structure is very similar to the finite-dimensional cases above.
• The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.
• The Virasoro algebra is of paramount importance in string theory.

Given a vector space V, let ${\displaystyle {\mathfrak {gl}}(V)}$ denote the Lie algebra consisting of all linear endomorphisms of V, with bracket given by ${\displaystyle [X,Y]=XY-YX}$. A representation of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ on V is a Lie algebra homomorphism

${\displaystyle \pi$ :{\mathfrak {g}}\to {\mathfrak {gl}}(V).}

A representation is said to be faithful if its kernel is zero. Ado's theorem[12] states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.

For any Lie algebra ${\displaystyle {\mathfrak {g}}}$, we can define a representation

${\displaystyle \operatorname {ad} \colon {\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})}$

given by ${\displaystyle \operatorname {ad} (x)(y)=[x,y]}$; it is a representation on the vector space ${\displaystyle {\mathfrak {g}}}$ called the adjoint representation.

One important aspect of the study of Lie algebras (especially semisimple Lie algebras) is the study of their representations. (Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory.) Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra ${\displaystyle {\mathfrak {g}}}$. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand all possible representation of ${\displaystyle {\mathfrak {g}}}$, up to the natural notion of equivalence. In the semisimple case over a field of characteristic zero, Weyl's theorem[13] says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The irreducible representations, in turn, are classified by a theorem of the highest weight.

The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example would be the angular momentum operators, whose commutation relations are those of the Lie algebra ${\displaystyle {\mathfrak {so}}(3)}$ of the rotation group SO(3). Typically, the space of states is very far from being irreducible under the pertinent operators, but one can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the quantum hydrogen atom, for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra ${\displaystyle {\mathfrak {so}}(3)}$.

Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.

A Lie algebra ${\displaystyle {\mathfrak {g}}}$ is abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in ${\displaystyle {\mathfrak {g}}}$. Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces ${\displaystyle \mathbb {K} ^{n}}$ or tori ${\displaystyle \mathbb {T} ^{n}}$, and are all of the form ${\displaystyle {\mathfrak {k}}^{n},}$ meaning an n-dimensional vector space with the trivial Lie bracket.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra ${\displaystyle {\mathfrak {g}}}$ is nilpotent if the lower central series

${\displaystyle {\mathfrak {g}}>[{\mathfrak {g}},{\mathfrak {g}}]>[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}]>[[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}],{\mathfrak {g}}]>\cdots }$

becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in ${\displaystyle {\mathfrak {g}}}$ the adjoint endomorphism

${\displaystyle \operatorname {ad} (u):{\mathfrak {g}}\to {\mathfrak {g}},\quad \operatorname {ad} (u)v=[u,v]}$

is nilpotent.

More generally still, a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is said to be solvable if the derived series:

${\displaystyle {\mathfrak {g}}>[{\mathfrak {g}},{\mathfrak {g}}]>[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]>[[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]],[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]]>\cdots }$

becomes zero eventually.

Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.

A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. (This implies that a one-dimensional—necessarily abelian—Lie algebra is by definition not simple, even though it has no nontrivial ideals.) A Lie algebra ${\displaystyle {\mathfrak {g}}}$ is called semisimple if it is isomorphic to a direct sum of simple algebras. There are several equivalent characterizations of semisimple algebras, such as having no nonzero solvable ideals.

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 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 if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.

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 ${\displaystyle {\mathfrak {g}}}$ defined by the formula

${\displaystyle K(u,v)=\operatorname {tr} (\operatorname {ad} (u)\operatorname {ad} (v)),}$

where tr denotes the trace of a linear operator. A Lie algebra ${\displaystyle {\mathfrak {g}}}$ is semisimple if and only if the Killing form is nondegenerate. A Lie algebra ${\displaystyle {\mathfrak {g}}}$ is solvable if and only if ${\displaystyle K({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0.}$

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. (Such a decomposition exists for a finite-dimensional Lie algebra over a field of characteristic zero.[14]) Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems.

• Any Lie algebra over a general ring instead of a field is an example of a Lie ring. Lie rings are not Lie groups under addition, despite the name.
• Any associative ring can be made into a Lie ring by defining a bracket operator

${\displaystyle [x,y]=xy-yx.}$

• For an example of a Lie ring arising from the study of groups, let ${\displaystyle G}$ be a group with ${\displaystyle [x,y]=x^{-1}y^{-1}xy}$ the commutator operation, and let ${\displaystyle G=G_{0}\supseteq G_{1}\supseteq G_{2}\supseteq \cdots \supseteq G_{n}\supseteq \cdots }$ be a central series in ${\displaystyle G}$ — that is the commutator subgroup ${\displaystyle [G_{i},G_{j}]}$ is contained in ${\displaystyle G_{i+j}}$ for any ${\displaystyle i,j}$. Then

${\displaystyle L=\bigoplus G_{i}/G_{i+1}}$

is a Lie ring with addition supplied by the group operation (which is abelian in each homogeneous part), and the bracket operation given by

${\displaystyle [xG_{i},yG_{j}]=[x,y]G_{i+j}\ }$

extended linearly. The centrality of the series ensures that the commutator ${\displaystyle [x,y]}$ gives the bracket operation the appropriate Lie theoretic properties.