Symmetry often appears first as a group. A rotation group, a matrix group, or a group of transformations records finite motions and their composition. In analysis and geometry, however, many symmetries vary continuously. Near the identity transformation, such a group has first-order directions. Those infinitesimal directions form a [vector space](/page/Vector%20Space), and their commutator records the first non-commuting effect of moving in two directions. This infinitesimal algebraic object is a Lie algebra.

A basic example comes from square matrices. Let $M_n(F)$ be the vector space of $n\times n$ matrices over a field $F$. Matrix multiplication is not commutative in general. The expression

\begin{align*} [A,B]=AB-BA \end{align*}

measures the failure of $A$ and $B$ to commute. This commutator is bilinear, alternates in its two variables, and satisfies a special three-term relation. Those three features are the axioms of a Lie algebra.

The point of the theory is that the bracket is not a second multiplication of the same kind as a ring product. It is an operation designed to measure interaction between infinitesimal transformations. Many structural questions then become linear-algebraic while still remembering noncommutativity.

## Definition

The ambient scalar field matters throughout the subject. In many analytic applications the field is $\mathbb R$ or $\mathbb C$. In algebraic applications one often works over an arbitrary field, with extra care in small characteristic.

[definition: Lie Algebra] Let $F$ be a field. A Lie algebra over $F$ is an $F$-vector space $\mathfrak g$ equipped with a bilinear map \begin{align*} [\cdot,\cdot]\colon \mathfrak g\times \mathfrak g\to \mathfrak g \end{align*} such that, for all $x,y,z\in \mathfrak g$, \begin{align*} [x,x]=0. \end{align*} The Jacobi identity is \begin{align*} [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0. \end{align*} The map $[\cdot,\cdot]$ is called the Lie bracket. [/definition]

The identity $[x,x]=0$ is the alternating law. When the field has characteristic different from $2$, it is equivalent to skew-symmetry:

\begin{align*} [x,y]=-[y,x]. \end{align*}

The three-term identity is the Jacobi identity. Since a Lie bracket is not an associative operation, Jacobi plays the structural role that makes iterated brackets coherent.

Small characteristic changes some familiar shortcuts. The alternating law $[x,x]=0$ is part of the definition because skew-symmetry alone does not imply it in characteristic $2$. In characteristics $2$ and $3$, several classification theorems and representation-theoretic tests require extra hypotheses or modified statements, so results first learned over $\mathbb C$ should not be transferred to arbitrary fields without checking the characteristic.

The notation $[x,y]$ is part of the meaning of the structure. It suggests a commutator, but the definition does not require an associative multiplication behind the scenes. Some Lie algebras arise from associative algebras, while others are introduced directly by generators, relations, or geometric constructions.

## First Examples

### Matrix Lie Algebras

The matrix example gives a template for much of the subject.

[example: General Linear Lie Algebra] The vector space $M_n(F)$ becomes a Lie algebra when equipped with the commutator bracket \begin{align*} [A,B]=AB-BA. \end{align*} This Lie algebra is denoted $\mathfrak{gl}_n(F)$. The bracket is bilinear because matrix multiplication is bilinear: for $\lambda,\mu\in F$ and $A,A',B\in M_n(F)$, \begin{align*} [\lambda A+\mu A',B]=(\lambda A+\mu A')B-B(\lambda A+\mu A')=\lambda(AB-BA)+\mu(A'B-BA')=\lambda[A,B]+\mu[A',B]. \end{align*} The same calculation in the second variable gives \begin{align*} [A,\lambda B+\mu B']=A(\lambda B+\mu B')-(\lambda B+\mu B')A=\lambda(AB-BA)+\mu(AB'-B'A)=\lambda[A,B]+\mu[A,B']. \end{align*} It is alternating, since for every $A\in M_n(F)$, \begin{align*} [A,A]=AA-AA=0. \end{align*} It remains to verify the Jacobi identity. For $A,B,C\in M_n(F)$, using the definition of the commutator and associativity of matrix multiplication, \begin{align*} [A,[B,C]]=A(BC-CB)-(BC-CB)A=ABC-ACB-BCA+CBA. \end{align*} Similarly, \begin{align*} [B,[C,A]]=B(CA-AC)-(CA-AC)B=BCA-BAC-CAB+ACB. \end{align*} And \begin{align*} [C,[A,B]]=C(AB-BA)-(AB-BA)C=CAB-CBA-ABC+BAC. \end{align*} Adding the three displayed expressions, the terms cancel in pairs: \begin{align*} (ABC-ABC)+(-ACB+ACB)+(-BCA+BCA)+(CBA-CBA)+(-BAC+BAC)+(-CAB+CAB)=0. \end{align*} Thus \begin{align*} [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0. \end{align*} So $M_n(F)$ with the commutator bracket is a Lie algebra, and the example shows that ordinary associative matrix multiplication produces a Lie bracket by measuring noncommutativity. [/example]

The trace-zero matrices form a smaller Lie algebra that is central in representation theory.

[example: Special Linear Lie Algebra] The set \begin{align*} \mathfrak{sl}_n(F)=\{A\in M_n(F):\operatorname{tr}(A)=0\} \end{align*} is a vector subspace of $M_n(F)$. Indeed, if $A,B\in \mathfrak{sl}_n(F)$ and $\lambda,\mu\in F$, then linearity of trace gives \begin{align*} \operatorname{tr}(\lambda A+\mu B)=\lambda\operatorname{tr}(A)+\mu\operatorname{tr}(B)=\lambda\cdot 0+\mu\cdot 0=0. \end{align*} So $\lambda A+\mu B\in \mathfrak{sl}_n(F)$. It remains to check that this subspace is closed under the commutator bracket inherited from $\mathfrak{gl}_n(F)$. Let $A=(a_{ij})$ and $B=(b_{ij})$ be elements of $\mathfrak{sl}_n(F)$. The diagonal entries of $AB$ are \begin{align*} (AB)_{ii}=\sum_{j=1}^n a_{ij}b_{ji}. \end{align*} Hence \begin{align*} \operatorname{tr}(AB)=\sum_{i=1}^n(AB)_{ii}=\sum_{i=1}^n\sum_{j=1}^n a_{ij}b_{ji}. \end{align*} Similarly, \begin{align*} \operatorname{tr}(BA)=\sum_{j=1}^n\sum_{i=1}^n b_{ji}a_{ij}. \end{align*} Since entries lie in the field $F$, $a_{ij}b_{ji}=b_{ji}a_{ij}$, and the two finite double sums have the same terms. Therefore \begin{align*} \operatorname{tr}(AB)=\operatorname{tr}(BA). \end{align*} Thus \begin{align*} \operatorname{tr}([A,B])=\operatorname{tr}(AB-BA)=\operatorname{tr}(AB)-\operatorname{tr}(BA)=0. \end{align*} So $[A,B]\in\mathfrak{sl}_n(F)$. The commutator bracket on $\mathfrak{sl}_n(F)$ is the restriction of the commutator bracket on $M_n(F)$, so bilinearity, alternation, and the Jacobi identity are inherited from the same matrix identities. Hence $\mathfrak{sl}_n(F)$ is a Lie algebra, called the special linear Lie algebra. [/example]

There are also Lie algebras with no nonzero brackets.