Lie algebra, linear

over a field $K$

A Lie algebra $\mathfrak{g}$ whose elements are linear transformations of a vector space $V$ over $K$; addition of elements and multiplication of them by elements of $K$ are defined in the usual way, and the commutator $[x,y]$ of two elements $x,y \in \mathfrak{g}$ is given by the formula $$ [x,y] = xy - yx $$ ($xy$ and $yx$ are the usual products of linear transformations). The linear Lie algebra consisting of all linear transformations of $V$ is denoted by $\mathfrak{gl}(V)$. If $V = K^n$, then $\mathfrak{gl}(V)$ is naturally identified with the set of all square matrices of order $n$ over $K$ and is denoted by $\mathfrak{gl}(n,K)$. Any linear Lie algebra is a subalgebra of some Lie algebra $\mathfrak{gl}(V)$.

Examples. 1) Let $V$ be endowed with an associative algebra structure. Then all derivations (cf. Derivation in a ring) of $V$ form a linear Lie algebra. If $V$ is a Lie algebra, then for a fixed element $x \in V$ the linear transformation of $V$ adjoint to $x$, defined by the formula $y \mapsto [x,y]$, $y \in V$, is a derivation of $V$; it is denoted by $\mathrm{ad} x$. The set $$ \mathrm{ad} V = \{ \mathrm{ad} x : x \in V \} $$ is a linear algebra, called the adjoint linear Lie algebra or the Lie algebra of inner derivations of $V$.

2) Let $K$ be a field that is complete with respect to some non-trivial absolute value, let $V$ be a normed complete space over $K$ and let $G$ be a linear Lie group of transformations of $V$, that is, a Lie subgroup of the Lie group of all automorphisms of $V$. Then the Lie algebra of the analytic group $G$ (cf. Lie algebra of an analytic group) is naturally identified with a Lie subalgebra of $\mathfrak{gl}(V)$, that is, it is a linear Lie algebra.

The problem of the existence of an isomorphism of an arbitrary finite-dimensional Lie algebra to some linear Lie algebra had already arisen in the first papers on group theory and Lie algebras, but it was affirmatively solved only in 1935 by Ado's theorem (see [4]): Every finite-dimensional Lie algebra $\mathfrak{g}$ over a field of characteristic zero has a faithful finite-dimensional representation $\rho$ (moreover, if $\mathfrak{n}$ is the largest nilpotent ideal of $\mathfrak{g}$, then $\rho$ can be chosen so that all elements of $\rho(\mathfrak{n})$ are nilpotent, cf. also Representation of a Lie algebra). The analogue of this theorem for Lie groups does not hold, in general; for example, the universal covering of the group of real unimodular matrices of order 2 does not have a faithful linear representation.

See also Lie algebra, algebraic.

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