Lie algebras

In mathematics, a Lie algebra (pronounced ) is a vector space $\mathfrak g$ together with an operation called the Lie bracket, an alternating bilinear map $\mathfrak g \times \mathfrak g \rightarrow \mathfrak g$, that satisfies the Jacobi identity. The Lie bracket of two vectors $x$ and $y$ is denoted ,y/math>. The vector space $\mathfrak g$ together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras.

In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.

An elementary example is the space of three dimensional vectors $\mathfrak=\mathbb^3$ with the bracket operation defined by the cross product ,yx\times y. This is skew-symmetric since $x\times y = -y\times x$, and instead of associativity it satisfies the Jacobi identity:
:$x\times(y\times z) \ =\ (x\times y)\times z \ +\ y\times(x\times z).$
This is the Lie algebra of the Lie group of rotations of space, and each vector $v\in\R^3$ may be pictured as an infinitesimal rotation around the axis $v$, with velocity equal to the magnitude of $v$. The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property ,xx\times x = 0.

History

Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name ''Lie algebra'' was given by Hermann Weyl in the 1930s; in older texts, the term ''infinitesimal group'' is used.

Definitions

Definition of a Lie algebra

A Lie algebra is a vector space $\,\mathfrak$ over some field $F$ together with a binary operation ,\cdot\,,\cdot\, \mathfrak\times\mathfrak\to\mathfrak called the Lie bracket satisfying the following axioms:

* Bilinearity,
:: x + b y, z= a , z+ b , z
:: , a x + b y= a, x+ b , y
:for all scalars $a$, $b$ in $F$ and all elements $x$, ''$y$'', ''$z$'' in $\mathfrak$.

* Alternativity,
:: ,x0\
:for all $x$ in $\mathfrak$.

* The Jacobi identity,
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:for all $x$, ''$y$'', ''$z$'' in $\mathfrak$.

Using bilinearity to expand the Lie bracket $[x+y,x+y]$ and using alternativity shows that ,y+ [y,x]=0\ for all elements $x$, ''$y$'' in $\mathfrak$, showing that bilinearity and alternativity together imply
* Anticommutativity,
:: ,y= - ,x\
:for all elements $x$, ''$y$'' in $\mathfrak$. If the field's characteristic is not 2 then anticommutativity implies alternativity, since it implies ,x- ,x

It is customary to denote a Lie algebra by a lower-case fraktur letter such as $\mathfrak$. 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 $\mathfrak(n)$.

Generators and dimension

Elements of a Lie algebra $\mathfrak$ are said to generate it if the smallest subalgebra containing these elements is $\mathfrak$ itself. The ''dimension'' of a Lie algebra is its dimension as a vector space over ''$F$''. 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.

Subalgebras, ideals and homomorphisms

The Lie bracket is not required to be associative, meaning that x,yz] need not equal ,[y,z._However,_it_is_flexible_algebra, flexible.html" ;"title=",z.html" ;"title=",[y,z">,[y,z. However, it is flexible algebra, flexible">,z.html" ;"title=",[y,z">,[y,z. However, it is flexible algebra, flexible. Nonetheless, much of the terminology of associative ring (mathematics), rings and associative algebra, algebras is commonly applied to Lie algebras. A ''Lie subalgebra'' is a subspace $\mathfrak \subseteq \mathfrak$ which is closed under the Lie bracket. An ''ideal'' $\mathfrak i\subseteq\mathfrak$ is a subalgebra satisfying the stronger condition:

: mathfrak,\mathfrak isubseteq \mathfrak i.

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

: \phi: \mathfrak\to\mathfrak, \quad \phi( ,y= phi(x),\phi(y)\ \text\
x,y \in \mathfrak g.

As for associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebra $\mathfrak$ and an ideal $\mathfrak i$ in it, one constructs the ''factor algebra'' or ''quotient algebra'' $\mathfrak/\mathfrak i$, 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 $x,y\in\mathfrak g$ ''commute'' if their bracket vanishes: ,y0.

The centralizer subalgebra of a subset $S\subset \mathfrak$ is the set of elements commuting with ''$S$'': that is, $\mathfrak_(S) = \$. The centralizer of $\mathfrak$ itself is the ''center'' $\mathfrak(\mathfrak)$. Similarly, for a subspace ''S'', the normalizer subalgebra of ''$S$'' is $\mathfrak_(S) = \$. Equivalently, if $S$ is a Lie subalgebra, $\mathfrak_(S)$ is the largest subalgebra such that $S$ is an ideal of $\mathfrak_(S)$.

Examples

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

\begin
\left \begin
a & b \\
c & d
\end,
\begin
x & 0 \\
0 & y
\end
\right&= \begin
ax & by\\
cx & dy \\
\end - \begin
ax & bx\\
cy & dy \\
\end \\
&= \begin
0 & b(y-x) \\
c(x-y) & 0
\end

\end

shows $\mathfrak(2)$ 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.

Direct sum and semidirect product

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

: x,x'),(y,y')( ,y ',y',

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

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

Derivations

A ''derivation'' on the Lie algebra $\mathfrak$ (or on any non-associative algebra) is a linear map $\delta\colon\mathfrak\rightarrow \mathfrak$ that obeys the Leibniz law, that is,
:\delta ( ,y = delta(x),y+ , \delta(y)/math>

for all $x,y\in\mathfrak g$. The ''inner derivation'' associated to any $x\in\mathfrak g$ is the adjoint mapping $\mathrm_x$ defined by \mathrm_x(y):= ,y/math>. (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 $\mathfrak$ is semisimple, every derivation is inner.

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

Examples

For example, given a Lie algebra ideal $\mathfrak \subset \mathfrak$ the adjoint representation $\mathfrak_\mathfrak$ of $\mathfrak$ acts as outer derivations on $\mathfrak$ since ,i\subset \mathfrak for any $x \in \mathfrak$ and $i \in \mathfrak$. For the Lie algebra $\mathfrak_n$ of upper triangular matrices in $\mathfrak(n)$, it has an ideal $\mathfrak_n$ 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 $\mathfrak_3$ and $\mathfrak_3$ gives

\begin
\left \begin
a & b & c \\
0 & d & e \\
0 & 0 & f
\end,
\begin
0 & x & y \\
0 & 0 & z \\
0 & 0 & 0
\end
\right&= \begin
0 & ax & ay+bz \\
0 & 0 & dz \\
0 & 0 & 0
\end - \begin
0 & dx & ex+yf \\
0 & 0 & fz \\
0 & 0 & 0
\end \\
&= \begin
0 & (a-d)x & (a-f)y-ex+bz \\
0 & 0 & (d-f)z \\
0 & 0 & 0
\end
\end

shows there exist outer derivations from $\mathfrak_3$ in $\text(\mathfrak_3)$.

Split Lie algebra

Let ''V'' be a finite-dimensional vector space over a field ''F'', $\mathfrak(V)$ the Lie algebra of linear transformations and $\mathfrak \subseteq \mathfrak(V)$ a Lie subalgebra. Then $\mathfrak$ is said to be split if the roots of the characteristic polynomials of all linear transformations in $\mathfrak$ are in the base field ''F''. More generally, a finite-dimensional Lie algebra $\mathfrak$ is said to be split if it has a Cartan subalgebra whose image under the adjoint representation $\operatorname: \mathfrak \to \mathfrak(\mathfrak g)$ 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

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