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A derivation on the Lie algebra g {\displaystyle {\mathfrak {g}}} (in fact on any non-associative algebra) is a linear map : g g {\displaystyle \delta :{\mathfrak {g}}\rightarrow {\mathfrak {g}}} that obeys the Leibniz law, that is, ( [ x , y ] ) = [ ( x ) , y ] + [ x , ( y ) ] {\displaystyle \delta ([x,y])=[\delta (x),y]+[x,\delta (y)]} for all x and y in the algebra. For any x, ad ( x ) {\displaystyle \operatorname {ad} (x)} is a derivation; a consequence of the Jacobi identity. Thus, the image of ad {\displaystyle \operatorname {ad} } lies in the subalgebra of g l ( g ) {\displaystyle {\mathfrak {gl}}({\mathfrak {g}})} consisting of derivations on g {\displaystyle {\mathfrak {g}}} . A derivation that happens to be in the image of ad {\displaystyle \operatorname {ad} } is called an inner derivation. If g {\displaystyle {\mathfrak {g}}} is semisimple, every derivation on g {\displaystyle {\mathfrak {g}}} is inner.

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 g {\displaystyle {\mathfrak {g}}} . Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand all possible representation of g {\displaystyle {\mathfrak {g}}} , up to the natural notion of equivalence. In the semisimple case, a basic theorem 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.

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 [ x , y ] = x y y x {\displaystyle [x,y]=xy-yx} . For an example of a Lie ring arising from the study of groups, let G {\displaystyle G} be a group with ( x , y ) = x 1 y 1 x y {\displaystyle (x,y)=x^{-1}y^{-1}xy} the commutator operation, and let G = G 0 G 1 G 2 G n {\displaystyle G=G_{0}\supseteq G_{1}\supseteq G_{2}\supseteq \cdots \supseteq G_{n}\supseteq \cdots } be a central series in G {\displaystyle G} that is the commutator subgroup ( G i , G j ) {\displaystyle (G_{i},G_{j})} is contained in G i + j {\displaystyle G_{i+j}} for any i , j {\displaystyle i,j} . Then L = G i / G i + 1 {\displaystyle L=\bigoplus G_{i}/G_{i+1}} is a Lie ring with addition supplied by the group operation (which will be commutative in each homogeneous part), and the bracket operation given by [ x G i , y G j ] = ( x , y ) G i + j {\displaystyle [xG_{i},yG_{j}]=(x,y)G_{i+j}\ } extended linearly. Note that the centrality of the series ensures the commutator ( x , y ) {\displaystyle (x,y)} gives the bracket operation the appropriate Lie theoretic properties.

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