General linear Lie algebra

The general linear Lie algebra is examined in the mathematical theory of Lie algebras , it is in a sense the prototype of a Lie algebra. The general linear Lie algebra belongs to every vector space . ${\displaystyle V}$ ${\displaystyle {\mathfrak {gl}}(V)}$

Definitions

It is a vector space over a field and the K -algebra of K -linear figures , the so-called endomorphisms on . For two endomorphisms one defines the commutator by ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle \mathrm {End} (V)}$ ${\displaystyle V\rightarrow V}$ ${\displaystyle V}$ ${\displaystyle x,y}$

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

Then a bilinear mapping is on and it holds ${\displaystyle (x,y)\mapsto [x,y]}$ ${\displaystyle \mathrm {End} (V)}$

${\displaystyle [x,x]=0}$ for all endomorphisms ${\displaystyle x}$

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

The last equation is called Jacobi identity ; it is instructive to confirm them briefly.

${\displaystyle [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=x(yz-zy)-(yz-zy)x+y(zx-xz)-(zx-xz)y+z(xy-yx)-(xy-yx)z}$

${\displaystyle =x(yz)-x(zy)-(yz)x+(zy)x+y(zx)-y(xz)-(zx)y+(xz)y+z(xy)-z(yx)-(xy)z+(yx)z=0}$

The first equal sign only uses the definition of the commutator and the second uses the distributive law . Regarding the last equals sign, one can see that among the 12 products, each of the 6 possible permutations of occurs exactly twice, each with a different sign and different brackets. So it is essentially the associative law in algebra that leads to the validity of the Jacobi identity. ${\displaystyle x,y,z}$ ${\displaystyle \mathrm {End} (V)}$

A vector space with a bilinear mapping that has the two properties mentioned above is called a Lie algebra . We have therefore shown that together with the commutator there is such a Lie algebra. To distinguish them from associative algebra , they are called , the exact reason for this naming is given below. ${\displaystyle L}$ ${\displaystyle [\cdot ,\cdot ]:L\times L\rightarrow L}$ ${\displaystyle \mathrm {End} (V)}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle \mathrm {End} (V)}$ ${\displaystyle {\mathfrak {gl}}(V)}$

The case is particularly important , you then write . If the endomorphisms are represented as matrices with respect to the standard basis of , then Lie algebras of matrices are obtained. ${\displaystyle V\cong K^{n}}$ ${\displaystyle {\mathfrak {gl}}(n,K)}$ ${\displaystyle K^{n}}$

In general, any associative algebra can be made into a Lie algebra with the construction described here using commutator formation.

The Lie algebra of the general linear group

The name general linear Lie algebra comes from the theory of Lie groups . As is known, referred to the group of invertible linear operators on , the so-called general linear group (English: G eneral L inear Group), and this is a Lie group. Each Lie group is assigned a Lie algebra in a certain way, the so-called Lie algebra of the Lie group , and it is common practice to use the name of the group in lowercase letters for this. ${\displaystyle {\mathfrak {gl}}(V)}$ ${\displaystyle GL(n,\mathbb {R} )}$ ${\displaystyle \mathbb {R} ^{n}}$

With this assignment, the group is mapped to the tangential space on the neutral element with a certain Lie product. Is so be ${\displaystyle X\in \mathrm {End} (\mathbb {R} ^{n})}$

${\displaystyle \gamma _{X}:\mathbb {R} \rightarrow GL(n,\mathbb {R} ),\quad \gamma _{X}(t):=\exp(tX)}$

with the matrix exponential . Then the neutral element and therefore an element of the tangent space on the neutral element, i.e. an element of the associated Lie algebra. By identifying with , the set is obtained as a tangent space; together with the commutator as the Lie product, this gives the Lie algebra we are looking for. Hence the general linear Lie algebra is the general linear group Lie algebra , which explains its name. ${\displaystyle \exp }$ ${\displaystyle \gamma (0)=\mathrm {id} }$ ${\displaystyle \gamma _{X}'(0)}$ ${\displaystyle X}$ ${\displaystyle \gamma _{X}'(0)}$ ${\displaystyle \mathrm {End} (\mathbb {R} ^{n})}$ ${\displaystyle {\mathfrak {gl}}(n,\mathbb {R} )}$ ${\displaystyle GL(n,\mathbb {R} )}$

Linear Lie algebras

The sub-Lie algebras of the general linear Lie algebra are called linear Lie algebras , many important Lie algebras can easily be described as sub-Lie algebras . ${\displaystyle {\mathfrak {gl}}(n,K)}$

${\displaystyle {\mathfrak {t}}(n,K):=\{(a_{i,j})_{i,j}\in {\mathfrak {gl}}(n,K)|\,a_{i,j}=0{\text{ für alle }}i>j\}}$,

the Lie algebra of the upper triangular matrices .

${\displaystyle {\mathfrak {n}}(n,K):=\{(a_{i,j})_{i,j}\in {\mathfrak {gl}}(n,K)|\,a_{i,j}=0{\text{ für alle }}i\geq j\}}$,

the nilpotent Lie algebra of the strict, upper triangular matrices.

${\displaystyle {\mathfrak {sl}}(n,K):=\{(a_{i,j})_{i,j}\in {\mathfrak {gl}}(n,K)|\,a_{1,1}+\ldots +a_{n,n}=0\}}$,

the Lie algebra for the special linear group .

${\displaystyle {\mathfrak {sp}}(2n,K):=\{a\in {\mathfrak {gl}}(2n,K)|\,sa=-a^{t}s\}}$,

the symplectic Lie algebra , where with let be the identity matrix and denote the transpose of . ${\displaystyle s={\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}}\in {\mathfrak {gl}}(2n,K)}$ ${\displaystyle I_{n}}$ ${\displaystyle a^{t}}$ ${\displaystyle a}$

See the textbook below for more examples. Using universal enveloping algebra one can show that every Lie algebra is isomorphic to a linear algebra.

properties

center

The center of general linear Lie algebra is . This is a special case of Schur's lemma , but it can also easily be demonstrated directly by inspecting the commutators with the standard matrices. ${\displaystyle {\mathfrak {gl}}(n,K)}$ ${\displaystyle KI_{n}=\{\lambda I_{n}|\,\lambda \in K\}}$ ${\displaystyle E_{i,j}}$

Solvability

The general linear Lie algebra is not solvable . It applies ${\displaystyle {\mathfrak {gl}}(n,K)}$

${\displaystyle [{\mathfrak {gl}}(n,K),{\mathfrak {gl}}(n,K)]={\mathfrak {sl}}(n,K)}$ and

${\displaystyle [{\mathfrak {sl}}(n,K),{\mathfrak {sl}}(n,K)]={\mathfrak {sl}}(n,K)}$,

so that the descending chain of derived algebras is stuck at . ${\displaystyle {\mathfrak {sl}}(n,K)}$

radical

The general linear Lie algebra is not semi-simple , it breaks down into a direct sum , with the first summand simple and the second solvable. The radical that is . ${\displaystyle {\mathfrak {gl}}(n,K)}$ ${\displaystyle {\mathfrak {gl}}(n,K)={\mathfrak {sl}}(n,K)\oplus KI_{n}}$ ${\displaystyle {\mathfrak {gl}}(n,K)}$ ${\displaystyle KI_{n}}$

See also

• Representation (Lie algebra)

Individual evidence

1. ^ N. Jacobson: Lie Algebras. John Wiley & Sons, 1962, p. 6.
2. ↑ VV Gorbatsevich, AL Onishchik, EB Vinberg: Foundations of Lie Theory and Lie Transformation Groups. Springer-Verlag, 1997, ISBN 3-540-61222-X , p. 31.
3. ^ JE Humphreys: Introduction to Lie Algebras and Representation Theory. Springer-Verlag, 1972, ISBN 0-387-90052-7 , pp. 2 and 3
4. ↑ VV Gorbatsevich, AL Onishchik, EB Vinberg: Foundations of Lie Theory and Lie Transformation Groups. Springer-Verlag, 1997, ISBN 3-540-61222-X , p. 58, example 2