Semi-simple Lie algebra

Semi-simple Lie algebras are studied in the mathematical theory of Lie algebras . The finite-dimensional , semi-simple, complex Lie algebras can be fully classified. They are made up of simple Lie algebras , which is where their name comes from. This theory goes back essentially to the work of Wilhelm Killing and Élie Cartan at the end of the 19th century. The Dynkin diagrams used today for classification were introduced by Eugene Dynkin in 1947 . Essential parts of the theory can be found in the standard work by James E. Humphreys on representations of Lie algebras from 1972, where the description of the so-called exceptional Lie algebras is missing. This can be found in an older textbook by Richard D. Schafer on non-associative algebras from 1966. The Roger Carter textbook given below has a more modern, easily accessible presentation.

Definitions and characterizations

Here we consider finite-dimensional Lie algebras over an algebraically closed field of the characteristic , the field of complex numbers is the most prominent example. Some of the following explanations make do with weaker requirements for the basic field, but at some points in the theory one needs the existence of eigenvalues and therefore the algebraic closure, and the division by integers and therefore the characteristic . ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle 0}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle 0}$

There are a number of equivalent ways of defining semisimple Lie algebras. A Lie algebra is called simple if it is not Abelian and contains no further ideals apart from the null space and itself . The eponymous definition is:

• A Lie algebra is called semi-simple if it is a direct sum of simple ideals.

An alternative description uses the radical of a Lie algebra , which can be defined as the greatest solvable ideal in . ${\displaystyle \mathrm {Rad} (L)}$ ${\displaystyle L}$ ${\displaystyle L}$

• A Lie algebra is semi-simple if and only if its radical is the null space.

It follows immediately

• A Lie algebra is semi-simple if and only if it does not contain any solvable ideals other than the null space.
• A Lie algebra is semisimple if and only if it does not contain any Abelian ideals other than the null space.

Is the adjoint representation of each of the through -defined linear operator on maps, as is a symmetric form on defined, named after William Killing Killing form . ${\displaystyle \mathrm {ad} :L\rightarrow {\mathfrak {gl}}(L)}$ ${\displaystyle x\in L}$ ${\displaystyle (\mathrm {ad} \,x)(y):=[x,y]}$ ${\displaystyle L}$ ${\displaystyle \kappa (x,y):=\mathrm {Spur} ((\mathrm {ad} \,x)(\mathrm {ad} \,y))}$ ${\displaystyle L}$

• A Lie algebra is semi-simple if and only if its killing form has not degenerated .

This Cartan criterion is in principle a method for checking semi-simplicity, even if this can be very laborious in individual cases. Determine the killing form, more precisely the representing matrix with respect to a basis . The Lie algebra is semi-simple if and only if the determinant of this matrix is ​​not 0.

Examples

The simplest example is the three-dimensional, special, linear Lie algebra

${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )=\{z\in \mathrm {Mat} (2,\mathbb {C} )|\,\mathrm {Spur} (z)=0\}}$

with the base

${\displaystyle x={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad h={\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\quad y={\begin{pmatrix}0&0\\1&0\end{pmatrix}}}$.

With regard to the specified base have the adjoint the basic elements of the following matrix representations which we will write as equality: . ${\displaystyle \mathrm {ad} \,x\,=\,{\begin{pmatrix}0&-2&0\\0&0&1\\0&0&0\end{pmatrix}},\quad \mathrm {ad} \,h\,=\,{\begin{pmatrix}2&0&0\\0&0&0\\0&0&-2\end{pmatrix}},\quad \mathrm {ad} \,y\,=\,{\begin{pmatrix}0&0&0\\-1&0&0\\0&2&0\end{pmatrix}}}$

This can be read from the commutator relationships of the basic elements. For example

${\displaystyle (\mathrm {ad} \,h)(x)=[h,x]={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}0&1\\0&0\end{pmatrix}}-{\begin{pmatrix}0&1\\0&0\end{pmatrix}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}=2{\begin{pmatrix}0&1\\0&0\end{pmatrix}}=2x}$,

this results in the first column of the matrix representation of , etc. The representing matrix of the Killing form consists by definition of the traces of all possible products of these 3-part matrices and after some calculation one gets with determinant −128. So according to the killing form criterion is semi-easy. It can easily be shown that it is even simple what the laborious calculation would save, but we will revisit this example below. ${\displaystyle \mathrm {ad} \,h}$ ${\displaystyle {\begin{pmatrix}0&0&4\\0&8&0\\4&0&0\end{pmatrix}}}$ ${\displaystyle {\mathfrak {sl}}(2,C)}$ ${\displaystyle {\mathfrak {sl}}(2,C)}$

Since we will state all semi-simple Lie algebras in the context of the classification, further examples are not necessary here.

The general linear Lie algebra is not semi-simple, because the multiples of the identity matrix form an Abelian ideal. This is equal to the radical of this algebra. ${\displaystyle {\mathfrak {gl}}(\mathbb {C} ^{n})}$

Basic properties

First of all, one can construct a semi-simple one from any Lie algebra:

• For any Lie algebra , the quotient algebra is semi- simple. ${\displaystyle L}$ ${\displaystyle L/\mathrm {Rad} (L)}$

The above list of equivalent characterizations also represents a list of properties of semi-simple Lie algebras. Further properties of a semi-simple Lie algebra are: ${\displaystyle L}$

• Ideal and homomorphic images are again semi-simple.
• The center of is the null space because the center is an Abelian ideal. ${\displaystyle L}$
• ${\displaystyle L\,=\,[L,L]}$, that is, the sub-algebra generated by all products coincides with the algebra itself.
• According to Weyl's theorem , every finite-dimensional representation of is completely reducible . ${\displaystyle L}$
• ${\displaystyle \mathrm {ad} \,L\,=\,\mathrm {Der} (L)}$, that is, the Lie algebra of the derivatives on agrees with the picture of the adjoint representation ; in short, all derivatives on are inner. ${\displaystyle L}$ ${\displaystyle L}$
• The Cartan subalgebras of are exactly the maximum subalgebras of diagonalizable elements. These sub-algebras are Abelian. ${\displaystyle L}$

classification

introduction

In the following we only consider Lie algebras over an algebraically closed field of characteristic 0. The finite-dimensional, semi-simple among them can be fully classified. To this end, a geometric object, a so-called reduced root system , is assigned to each such algebra . This is a finite generating system of a Euclidean vector space with restricting conditions for angles and length ratios among the generating vectors. Then one shows that the isomorphism class of the semi-simple Lie algebra is uniquely determined by this root system and that there is an associated semi-simple Lie algebra for each such root system. The simple Lie algebras, which according to the above definition form the building blocks of the semi-simple, can all be given; there are four infinite series of simple Lie algebras and five more, so-called exceptional, Lie algebras. Every finite-dimensional, semi-simple Lie algebra is isomorphic to a finite direct sum of such simple Lie algebras.

Construction of the root system

For a finite-dimensional, semi-simple Lie algebra , we construct a reduced root system as follows . Choose a Cartan subalgebra of . All elements are simultaneously diagonalizable , that is, there are a finite number of linear functionals on , for is not the null space, and is the vector sum of these room . The functionals different from the zero functional form a finite generating system in the dual space of . The product of carries the bilinear form, which is dual to the non-degenerate killing form . Note that the field of characteristic 0 contains the prime field . This bilinear form can be extended to a positively definite symmetric bilinear form on the -vector space , just like all those whose extensions are denoted by the same name. One can show that form a reduced root system. ${\displaystyle L}$ ${\displaystyle H}$ ${\displaystyle L}$ ${\displaystyle \mathrm {ad} \,H}$ ${\displaystyle \alpha }$ ${\displaystyle H}$ ${\displaystyle L_{\alpha }:=\{x\in L|[hx]=\alpha (h)x\,\forall h\in H\}}$ ${\displaystyle L}$ ${\displaystyle L_{\alpha }}$ ${\displaystyle L_{\alpha }}$ ${\displaystyle \Phi \subset H^{*}}$ ${\displaystyle H}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle E_{\mathbb {Q} }}$ ${\displaystyle \Phi }$ ${\displaystyle H}$ ${\displaystyle (\cdot ,\cdot )}$ ${\displaystyle K}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle E=\mathbb {R} \otimes _{\mathbb {Q} }E_{\mathbb {Q} }}$ ${\displaystyle \alpha \in \Phi }$ ${\displaystyle E,\,(\cdot ,\cdot ),\,\Phi }$

example

To clarify the specified construction, we take the above example again. is a Cartan sub-algebra, the diagonalisability of can be easily read from the above matrix representation. As is one-dimensional and the eigenvalues 2, 0 and -2 has are accurate for Functional , and different from the null space, so it is . Thus the root system consists of a vector together with its negative. This is already clear when you know that this is the only one-dimensional, reduced root system apart from isomorphism. ${\displaystyle L={\mathfrak {sl}}(2,C)}$ ${\displaystyle H:=\mathbb {C} {\begin{pmatrix}1&0\\0&-1\end{pmatrix}}=\{{\begin{pmatrix}z&0\\0&-z\end{pmatrix}}|\,z\in \mathbb {C} \}}$ ${\displaystyle \mathrm {ad} \,h}$ ${\displaystyle H}$ ${\displaystyle \mathrm {ad} \,h}$ ${\displaystyle L_{\alpha }\neq \{0\}}$ ${\displaystyle \alpha _{2}:\,zh\mapsto 2z}$ ${\displaystyle \alpha _{0}:\,zh\mapsto 0}$ ${\displaystyle \alpha _{-2}:\,zh\mapsto -2z}$ ${\displaystyle \Phi =\{\alpha _{2},\alpha _{-2}\}}$

Independence from the Cartan sub-algebra

If a Lie algebra isomorphism is as above, then it is also a Cartan subalgebra and the above construction for yields an isomorphic root system, essentially because one can pull through the entire construction. ${\displaystyle \varphi :L\rightarrow L'}$ ${\displaystyle H}$ ${\displaystyle H':=\varphi (H)\subset L'}$ ${\displaystyle L',H'}$ ${\displaystyle \varphi }$

However, the construction of a root system uses the choice of a Cartan sub-algebra . In order to obtain an isomorphic invariant that is only dependent on, one has to show that every other Cartan subalgebra leads to an isomorphic root system. The so-called conjugation theorem helps here, for which no semi-simplicity is required: ${\displaystyle H}$ ${\displaystyle L}$

• For a finite-dimensional Lie algebra over an algebraically closed field, all Cartan subalgebras are conjugated to one another.

So is next to another Cartan subalgebra, then there is an isomorphism , even conjugation with , and initially made remark shows that the election or lead to isomorphic root systems. ${\displaystyle H'}$ ${\displaystyle H}$ ${\displaystyle \varphi :L\rightarrow L}$ ${\displaystyle H'=\varphi (H)}$ ${\displaystyle H}$ ${\displaystyle H'}$

Conclusion: The construction of a reduced root system described above is an isomorphism invariant of Lie algebra , that is, isomorphic, finite-dimensional, semi-simple Lie algebras have isomorphic root systems. ${\displaystyle L}$

The isomorphism theorem

The Dynkin diagrams are graphs assigned to the root systems.

So far we know that isomorphic, finite-dimensional, semi-simple Lie algebras have isomorphic reduced root systems. The so-called isomorphism theorem states that, conversely, two finite-dimensional, semi-simple Lie algebras with isomorphic reduced root systems are in turn isomorphic (note the assumptions about the basic field).

But we all know the reduced root systems. They fall into irreducible components, ie connected components of the corresponding Dynkin diagrams , and these correspond to simple direct summands of the Lie algebra. The irreducible, reduced root systems can be enumerated. As stated in the article on root systems , these are

${\displaystyle A_{n},n\geq 1,\quad B_{n},n\geq 2,\quad C_{n},n\geq 3,\quad D_{n},n\geq 4,\quad E_{6},E_{7},E_{8},\quad F_{4},\quad G_{2}}$.

The associated Dynkin diagrams are shown in the sketch opposite. Isomorphism classes of root systems are defined more precisely by the abbreviations; one therefore also says that a root system is of the specified type. A Lie algebra with a corresponding root system is also called a Lie algebra of this type. Thus every finite-dimensional, semi-simple Lie algebra is isomorphic to a direct sum of simple ideals whose types appear in the above list.

The existence proposition

After what has been said so far, we know that the reduced root systems of finite-dimensional, simple Lie algebras must be of the types listed above. Conversely, the question naturally arises whether there actually is a suitable simple Lie algebra for every type of reduced root system. This question is answered positively by the so-called existence theorem.

According to a method going back to Serre , one can use free Lie algebras , in which infinite dimensional algebras also occur, and relations explained on them, i.e. equations existing between the generators of free algebra, to prove the existence of the Lie algebras sought. The relations result from the root systems, they generate an ideal in a certain free Lie algebra and one finally has to show that the quotient algebra is a finite-dimensional, semi-simple Lie algebra with the given root system.

What is still missing is a concrete realization of these simple Lie algebras, the complete specification of which of course also proves the existence theorem. This is very simple for the four series , the exceptional Lie algebras result from algebras of derivatives on other exceptional, non-associative algebras, more precisely on certain Jordan algebras and on the Cayley algebra . After specifying this list, one can write down all finite-dimensional, semi-simple Lie algebras except for isomorphism. ${\displaystyle A_{n},B_{n},C_{n},D_{n}}$ ${\displaystyle E_{6},E_{7},E_{8},F_{4},G_{2}}$

The classical algebras

The simple Lie algebras for the root systems are called the classical algebras . These can easily be specified. ${\displaystyle A_{n},B_{n},C_{n},D_{n}}$

${\displaystyle {\mathfrak {sl}}(n+1,K):=\{x\in \mathrm {Mat} (n+1,K)|\,\mathrm {Spur} (x)=0\}}$

is with the commutator bracket a -dimensional, simple Lie algebra of the type , that is, the associated reduced root system is of this type. It is called the special linear algebra because it is the Lie algebra for the special linear group . The case is the example presented above. ${\displaystyle n(n+2)}$ ${\displaystyle A_{n}}$ ${\displaystyle n=1}$

Let it be the -unit matrix, 0 denotes a null matrix of the appropriate size and stands for the transpose of a matrix . ${\displaystyle I_{n}}$ ${\displaystyle n\times n}$ ${\displaystyle x^{t}}$ ${\displaystyle x}$

${\displaystyle {\mathfrak {o}}(2n+1,K):=\left\{x\in \mathrm {Mat} (2n+1,K){\Big |}\,{\begin{pmatrix}1&0&0\\0&0&I_{n}\\0&I_{n}&0\end{pmatrix}}x=-x^{t}{\begin{pmatrix}1&0&0\\0&0&I_{n}\\0&I_{n}&0\end{pmatrix}}\right\}}$

is called orthogonal algebra and, with the commutator bracket, is a simple -dimensional, simple Lie algebra of the type . ${\displaystyle n(2n+1)}$ ${\displaystyle B_{n}}$

${\displaystyle {\mathfrak {sp}}(2n,K):=\left\{x\in \mathrm {Mat} (2n,K){\Big |}\,{\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}}x=-x^{t}{\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}}\right\}}$

is called symplectic algebra and with the commutator bracket is a simple -dimensional, simple Lie algebra of the type . ${\displaystyle n(2n+1)}$ ${\displaystyle C_{n}}$

${\displaystyle {\mathfrak {o}}(2n,K):=\left\{x\in \mathrm {Mat} (2n,K){\Big |}\,{\begin{pmatrix}0&I_{n}\\I_{n}&0\end{pmatrix}}x=-x^{t}{\begin{pmatrix}0&I_{n}\\I_{n}&0\end{pmatrix}}\right\}}$

is called orthogonal algebra and, with the commutator bracket, is a simple -dimensional, simple Lie algebra of the type . The term orthogonal algebra , which is used again here , does not entail any risk of confusion, since the sizes of the matrices that occur are each odd or even. ${\displaystyle n(2n-1)}$ ${\displaystyle D_{n}}$

The exceptional algebras

We start with the simpler case of Lie type algebra . ${\displaystyle G_{2}}$

• If the Cayley algebra denotes over , the algebra of the derivatives on a 14-dimensional simple Lie algebra is of the type . ${\displaystyle {\mathfrak {C}}}$ ${\displaystyle K}$ ${\displaystyle \mathrm {Der} ({\mathfrak {C}})}$ ${\displaystyle {\mathfrak {C}}}$ ${\displaystyle G_{2}}$

The specification of the exceptional Lie algebras is more complex, since exceptional Jordan algebras come into play here. Let the involution on Cayley algebra be denoted by a slash. Then define ${\displaystyle E_{6},E_{7},E_{8},F_{4}}$ ${\displaystyle {\mathfrak {C}}}$

${\displaystyle {\mathfrak {H}}({\mathfrak {C}}_{3}):=\left\{{\begin{pmatrix}\xi _{1}&c&{\overline {b}}\\{\overline {c}}&\xi _{2}&a\\b&{\overline {a}}&\xi _{3}\end{pmatrix}}{\Big |}\,\xi _{1},\xi _{2},\xi _{3}\in K,\,a,b,c\in {\mathfrak {C}}\right\}}$,

this is the 27-dimensional space of the “ Hermitian ” 3-part matrices above . The Jordan product ${\displaystyle {\mathfrak {C}}}$

${\displaystyle \textstyle x\cdot y:={\frac {1}{2}}(xy+yx),\quad x,y\in {\mathfrak {H}}({\mathfrak {C}}_{3})}$

makes this space a Jordan algebra labeled with . This cannot be taken for granted, since the space of the 3-part matrices is not associative. (One can show that is exceptional, that is, is not isomorphic to a Jordan algebra that is derived from an associative algebra.) With this we can realize the next exceptional Lie type: ${\displaystyle {\mathfrak {J}}}$ ${\displaystyle {\mathfrak {C}}}$ ${\displaystyle {\mathfrak {J}}}$

• With the commutator bracket, the algebra of the derivatives is a 52-dimensional, simple Lie algebra of the type . ${\displaystyle \mathrm {Der} ({\mathfrak {J}})}$ ${\displaystyle {\mathfrak {J}}}$ ${\displaystyle F_{4}}$

We are now enlarging . For denote the right multiplication with , that is , where the Jordan product is used here. Furthermore, let the set of all elements from with track 0, that is, for those in the definition used above . ${\displaystyle \mathrm {Der} ({\mathfrak {J}})}$ ${\displaystyle x\in {\mathfrak {J}}}$ ${\displaystyle R_{x}\in \mathrm {End} ({\mathfrak {J}})}$ ${\displaystyle x}$ ${\displaystyle R_{x}(y):=y\cdot x}$ ${\displaystyle {\mathfrak {J}}_{0}}$ ${\displaystyle {\mathfrak {J}}}$ ${\displaystyle \xi _{1}+\xi _{2}+\xi _{3}=0}$

• The sum in with the commutator bracket is a 78-dimensional, simple Lie algebra of the type . ${\displaystyle \mathrm {Der} ({\mathfrak {J}})+\{R_{x}|x\in {\mathfrak {J}}_{0}\}}$ ${\displaystyle \mathrm {End} ({\mathfrak {J}})}$ ${\displaystyle E_{6}}$

For the 133-dimensional, simple Lie algebra of the type and the 248-dimensional, simple Lie algebra of the type , reference is made to the textbook by Richard D. Schafer given below or to the literature given there. ${\displaystyle E_{7}}$ ${\displaystyle E_{8}}$

See also

• Cartan matrix
• Lie-type group
• Semi-simple lie group
• Kac-Moody algebra
• Symmetrical space

Web links

Wikibooks: Full Proof of Classification  - Learning and Teaching Materials

Individual evidence

1. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 .
2. ^ Richard D. Schafer: An Introduction to Nonassociative Algebras . Courier Dover Publications 1966, ISBN 0-486-68813-5 ( freely available in Project Gutenberg ).
3. ^ Roger Carter: Lie Algebras of Finite and Affine Type , Cambridge University Press (2005), ISBN 978-0-521-85138-1
4. ↑ ^{a b} James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Chapter II: Semisimple Lie Algebras .
5. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Chapter III, 15.3: Cartan subalgebras .
6. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Chapter II, 8: Root space decomposition .
7. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Chapter IV, 16: Conjugacy theorems .
8. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Chapter IV, 14.2: Isomorphism theorem .
9. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Chapter V, 18.3: Serre's Theorem .
10. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Chapter V, 19.2: The classical algebras .
11. ↑ James E. Humphreys: Introduction to Lie Algebras and Representation Theory . Springer, Berlin / New York 1972, ISBN 0-387-90053-5 , Chapter V, 19.3: The algebra G ₂ .
12. ^ Richard D. Schafer: An Introduction to Nonassociative Algebras . Courier Dover Publications, 1966, ISBN 0-486-68813-5 , Chapter III, 8: Derivations; Simple Lie Algebras of Type G .
13. ^ Richard D. Schafer: An Introduction to Nonassociative Algebras . Courier Dover Publications, 1966, ISBN 0-486-68813-5 , Chapter IV, Theorem 4.9.
14. ^ Richard D. Schafer: An Introduction to Nonassociative Algebras . Courier Dover Publications, 1966, ISBN 0-486-68813-5 , Chapter IV, 4: Simple Lie Algebras of Type E ₆ .