Affine Lie algebra

In mathematics, an affine Lie algebra is an infinite - dimensional Lie algebra that is canonically constructed from a finite-dimensional Lie algebra. This enables the construction of affine Kac-Moody algebras .

definition

Let be a finite-dimensional Lie algebra. Then the associated affine Lie algebra is defined as the vector space ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\hat {\mathfrak {g}}}}$

${\displaystyle {\hat {\mathfrak {g}}}={\mathfrak {g}}\otimes \mathbb {C} [t,t^{-1}]\oplus \mathbb {C} c}$

with the commutator relation

${\displaystyle \left[a\otimes t^{n}+\alpha c,b\otimes t^{m}+\beta c\right]=\left[a,b\right]\otimes t^{n+m}+\langle a,b\rangle n\delta _{m+n,0}c}$

for . Here the Lie bracket means in , the Killing form of and is the associative algebra of the Laurent polynomials . ${\displaystyle a,b\in {\mathfrak {g}},\alpha ,\beta \in \mathbb {C} ,n,m\in \mathbb {Z} }$ ${\displaystyle \left[a,b\right]}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \langle a,b\rangle }$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \mathbb {C} [t,t^{-1}]}$

${\displaystyle c}$is an element of the center of Lie algebra and is therefore a central ideal . You also have a short exact sequence ${\displaystyle \mathbb {C} c}$

${\displaystyle 0\rightarrow \mathbb {C} c\rightarrow {\hat {\mathfrak {g}}}\rightarrow {\mathfrak {g}}\otimes \mathbb {C} [t,t^{-1}]\rightarrow 0}$

of Lie algebras.

Extended affine Lie algebra

We define by the formulas

${\displaystyle d(c):=0}$

${\displaystyle d(a\otimes f):=a\otimes (t{\frac {\mathrm {d} }{\mathrm {d} t}})(f),\quad a\in {\mathfrak {g}},f\in \mathbb {C} [t,t^{-1}]}$

a Derivation on . Note that the differential operator explains a derivation on . The extended affine Lie algebra arises from by adjunction of this derivation , that is ${\displaystyle {\hat {\mathfrak {g}}}}$ ${\displaystyle t{\frac {\mathrm {d} }{\mathrm {d} t}}}$ ${\displaystyle \mathbb {C} [t,t^{-1}]}$ ${\displaystyle {\tilde {\mathfrak {g}}}}$ ${\displaystyle {\hat {\mathfrak {g}}}}$

${\displaystyle {\tilde {\mathfrak {g}}}:=\mathbb {C} d\ltimes {\hat {\mathfrak {g}}}}$,

where stands for the semi-direct sum . The algebra constructed in this way is called the associated extended affine Lie algebra or simply extended affine algebra. ${\displaystyle \ltimes }$ ${\displaystyle {\tilde {\mathfrak {g}}}}$ ${\displaystyle {\mathfrak {g}}}$

Affine type Kac-Moody algebras

From the known finite dimensional simple Lie algebras A _n , B _n , C _n , D _n , E ₆ , E ₇ , E ₈ , F ₄ , G ₂ one can by means of the above construction, the Kac-Moody algebras affine type construct , more precisely the untwisted Kac-Moody algebras of affine type. Others appear as fixed point algebras of certain automorphisms ; one then speaks of twisted Kac-Moody algebras of affine type. In the following we construct all Kac-Moody algebras of affine type that result from indecomposable generalized Cartan matrices . These are so-called realizations of the abstractly defined Kac-Moody algebras, i.e. for each Kac-Moody algebra affine type with indecomposable, generalized Cartan matrix, a concrete vector space with a Lie bracket as given above is obtained, so that these Lie- Algebra belongs to the corresponding isomorphism class.

Untwisted Kac-Moody algebras of affine type

For we select in the above construction, the endlichendimensionalen simple Lie algebra , , , , , , , , , and the types specified simple Lie algebra up to isomorphism characterize. Then refers to the extended affine Lie algebras with it , , , , , , , , . They are Kac-Moody algebras of affine type with indecomposable generalized Cartan matrices. In addition to these, we also provide the associated Dynkin diagrams . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle A_{n}}$ ${\displaystyle B_{n}}$ ${\displaystyle C_{n}}$ ${\displaystyle D_{n}}$ ${\displaystyle E_{6}}$ ${\displaystyle E_{7}}$ ${\displaystyle E_{8}}$ ${\displaystyle F_{4}}$ ${\displaystyle G_{2}}$ ${\displaystyle {\tilde {A}}_{n}}$ ${\displaystyle {\tilde {B}}_{n}}$ ${\displaystyle {\tilde {C}}_{n}}$ ${\displaystyle {\tilde {D}}_{n}}$ ${\displaystyle {\tilde {E}}_{6}}$ ${\displaystyle {\tilde {E}}_{7}}$ ${\displaystyle {\tilde {E}}_{8}}$ ${\displaystyle {\tilde {F}}_{4}}$ ${\displaystyle {\tilde {G}}_{2}}$

Type	Generalized Cartan Matrix	Dynkin diagram
${\displaystyle {\tilde {A}}_{n},\,n\geq 1}$	${\displaystyle {\begin{pmatrix}2&-1&&&&&-1\\-1&2&-1&&&&\\&-1&.&.&&&\\&&.&.&.&&\\&&&.&.&-1&\\&&&&-1&2&-1\\-1&&&&&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {B}}_{n},\,n\geq 3}$	${\displaystyle {\begin{pmatrix}2&&-1&&&&\\&2&-1&&&&\\-1&-1&2&-1&&&\\&&-1&.&.&&\\&&&.&.&-1&\\&&&&-1&2&-1\\&&&&&-2&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {C}}_{n},\,n\geq 2}$	${\displaystyle {\begin{pmatrix}2&-1&&&&&\\-2&2&-1&&&&\\&-1&.&.&&&\\&&-1&.&-1&&\\&&&.&.&-1&\\&&&&-1&2&-2\\&&&&&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {D}}_{n},\,n\geq 4}$	${\displaystyle {\begin{pmatrix}2&&-1&&&&\\&2&-1&&&&\\-1&-1&.&.&&&\\&&-1&.&-1&&\\&&&.&.&-1&-1\\&&&&-1&2&\\&&&&-1&&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {E}}_{6}}$	${\displaystyle {\begin{pmatrix}2&&&&-1&&\\&2&-1&&&&\\&-1&2&-1&&&\\&&-1&2&-1&-1&\\-1&&&-1&2&&\\&&&-1&&2&-1\\&&&&&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {E}}_{7}}$	${\displaystyle {\begin{pmatrix}2&&&&&&&-1\\&2&-1&&&&&\\&-1&2&-1&&&&\\&&-1&2&-1&&&\\&&&-1&2&-1&-1&\\&&&&-1&2&&\\&&&&-1&&2&-1\\-1&&&&&&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {E}}_{8}}$	${\displaystyle {\begin{pmatrix}2&-1&&&&&&&\\-1&2&-1&&&&&&\\&-1&2&-1&&&&&\\&&-1&2&-1&&&&\\&&&-1&2&-1&&&\\&&&&-1&2&-1&-1&\\&&&&&-1&2&&\\&&&&&-1&&2&-1\\&&&&&&&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {F}}_{4}}$	${\displaystyle {\begin{pmatrix}2&-1&&&\\-1&2&-1&&\\&-1&2&-1&\\&&-2&2&-1\\&&&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {G}}_{2}}$	${\displaystyle {\begin{pmatrix}2&-1&\\-1&2&-1\\&-3&2\\\end{pmatrix}}}$

Note that the matrix size, i.e. the number of rows or columns, is one larger than the index of the type name. It is the same with the number of nodes in the associated Dynkin diagram.

Affine-type twisted Kac-Moody algebras

The non-trivial automorphisms of the Dynkin diagrams

The other Kac-Moody algebras of affine type can be constructed as subalgebras of untwisted Kac-Moody algebras, more precisely as a fixed point algebra of an automorphism. Below is the automorphism which to become one of the shown opposite Graphautomorphismen the Dynkin diagrams , and heard. A peculiarity occurs with , here there is an additional automorphism of order 3, all other automorphisms obviously have order 2. For this we now construct automorphisms on the extended affine Lie algebras by defining: ${\displaystyle \sigma :{\mathfrak {g}}\rightarrow {\mathfrak {g}}}$ ${\displaystyle A_{n}}$ ${\displaystyle D_{n}}$ ${\displaystyle E_{6}}$ ${\displaystyle D_{4}}$ ${\displaystyle \tau }$

${\displaystyle \tau (a\otimes t^{m})=e^{2\pi \mathrm {i} m/\mathrm {ord} (\sigma )}\cdot \sigma (a)\otimes t^{m}}$

${\displaystyle \tau (c)=c,\quad \tau (d)=d}$

The scalar factor is a -th root of unity . If one were to simply use 1 instead of this, one would also get automorphisms, which, however, do not achieve the desired result. Because of the unity root one speaks of twisted automorphisms and therefore also calls the fixed point algebras ${\displaystyle \mathrm {ord} (\sigma )}$

${\displaystyle {\mathfrak {g}}^{\tau }:=\{x\in {\mathfrak {g}}\mid \tau (x)=x\}}$

twisted algebras. With this construction, the remaining Kac-Moody algebras affine type to the standard designations may be , , , , , are constructed. The same names are used for the associated generalized Cartan matrices and for the associated Dynkin diagrams. The superscript t does not stand for an automorphism, but for matrix transposition . ${\displaystyle {\tilde {B}}_{n}^{t}}$ ${\displaystyle {\tilde {C}}_{n}^{t}}$ ${\displaystyle {\tilde {F}}_{4}^{t}}$ ${\displaystyle {\tilde {G}}_{2}^{t}}$ ${\displaystyle {\tilde {A}}_{1}'}$ ${\displaystyle {\tilde {C}}_{n}'}$

The following list is obtained:

Type	Fixed point algebra	Generalized Cartan Matrix	Dynkin diagram
${\displaystyle {\tilde {B}}_{n}^{t},\,n\geq 3}$	${\displaystyle {\tilde {A}}_{2n-1}^{\tau }}$	${\displaystyle {\begin{pmatrix}2&&-1&&&&\\&2&-1&&&&\\-1&-1&.&.&&&\\&&-1&.&-1&&\\&&&.&.&-1&\\&&&&-1&2&-2\\&&&&&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {C}}_{n}^{t},\,n\geq 2}$	${\displaystyle {\tilde {D}}_{n+1}^{\tau }}$	${\displaystyle {\begin{pmatrix}2&-2&&&&&\\-1&2&-1&&&&\\&-1&.&-1&&&\\&&.&.&.&&\\&&&-1&.&-1&\\&&&&-1&2&-1\\&&&&&-2&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {F}}_{4}^{t}}$	${\displaystyle {\tilde {E}}_{6}^{\tau }}$	${\displaystyle {\begin{pmatrix}2&-1&&&\\-1&2&-1&&\\&-1&2&-2&\\&&-1&2&-1\\&&&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {G}}_{2}^{t}}$	${\displaystyle {\tilde {D}}_{4}^{\tau }}$	${\displaystyle {\begin{pmatrix}2&-1&\\-1&2&-3\\&-1&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {A}}_{1}'}$	${\displaystyle {\tilde {A}}_{2}^{\tau }}$	${\displaystyle {\begin{pmatrix}2&-1\\-4&2\\\end{pmatrix}}}$
${\displaystyle {\tilde {C}}_{n}',\,n\geq 2}$	${\displaystyle {\tilde {A}}_{2n}^{\tau }}$	${\displaystyle {\begin{pmatrix}2&-2&&&&&\\-1&2&-1&&&&\\&-1&.&-1&&&\\&&.&.&.&&\\&&&.&.&-1&\\&&&&-1&2&-2\\&&&&&-1&2\\\end{pmatrix}}}$

What has been said for untwisted algebras applies to the sizes of the matrices and Dynkin diagrams. Also note that appears twice in the above list. The two cases are different automorphisms , for the autorphism of the special case mentioned above is used for, for with it is the automorphism which only exchanges the two ends of the Dynkin diagram. ${\displaystyle {\tilde {D}}_{4}^{\tau }}$ ${\displaystyle \tau }$ ${\displaystyle {\tilde {G}}_{2}^{t}}$ ${\displaystyle D_{4}}$ ${\displaystyle {\tilde {D}}_{n+1}^{\tau }}$ ${\displaystyle n=3}$

Individual evidence

1. ^ Igor Frenkel , James Lepowsky , Arne Meurman : Vertex Operator Algebras and the Monster , Academic Press, New York (1989) ISBN 0-12-267065-5 , Chapter 1.6: Affine Lie Algebras
2. ↑ Roger Carter : Lie Algebras of Finite and Affine Type , Cambridge studies in advanced mathematics 96 (2005), ISBN 978-0-521-85138-1 , Chapter 18: Realizations of affine Kac-Moody algebras