# Affine Kac–Moody Lie Algebras For affine Lie algebras there exists a well-known explicit construction of these in terms of an underlying finite-dimensional Lie algebra and a central extension (see [[Kac, 1990](../../references.md#cite-kac90)] (Chapters 7,8)). We briefly reiterate the construction here. Suppose $A$ is an affine Cartan matrix, so that $\frak{g}(A)$ is an affine Lie algebra; then $A$ is of affine Cartan type X̃$_n$ for X$=$A,B,C,D,E,F, or G, and some $n$. If we let $\frak{g}_0$ be the finite variant (i.e., a Lie algebra of Cartan type X$_n$) then $$ \frak{g}(A) \cong \frak{g}_0 \otimes {\bf C}[t, t^{-1}] \oplus {\bf C}c \oplus {\bf C}d $$ for some formal basis elements $c$ and $d$, where ${\bf C}[t, t^{-1}]$ is the ring of Laurent polynomials over ${\bf C}$. In Magma we represent affine Lie algebras and their elements using the form on the right hand side. Multiplication is given by $[t^k \otimes x \oplus \lambda c \oplus \mu d, t^{k_1} \otimes y \oplus \lambda_1 c \oplus \mu_1 d] =$$$(t^{k+k_1} \otimes [x,y] + \mu k_1 t^{k_1} \otimes y - \mu_1 k t^k \otimes x ) \oplus k \delta_{k,-k_1} (x|y) c,$$ where $(x|y)$ denotes a fixed non-degenerate invariant symmetric bilinear ${\bf C}$-valued form on $\frak{g}_0$. If we fix $E_i$, $F_i$ to be canonical generators of $g_0$, then the canonical generators of $\frak{g}(A)$, as described in the introduction ([Introduction](introduction.md#sectalgliekmintro)) are given by $e_0 = t \otimes E_0$, $f_0 = t^{-1} \otimes F_0$, and $e_i = 1 \otimes E_i$, $f_i = 1 \otimes F_i$, for $i = 1, \ldots, l$, where $l$ is the rank of the Cartan matrix. Affine Lie algebras and their elements are of type `AlgKac` and `AlgKacElt` respectively. ## Constructing Affine Kac–Moody Lie Algebras ### `AffineLieAlgebra(N, F): MonStgElt, Fld -> AlgKac` Construct the affine Kac–Moody Lie algebra of type N over the field $F$. N should be a string describing an affine Cartan type (e.g. `A~3`). See Section [Finite and Affine Coxeter Groups](../CoxeterSystems/finiteaffine.md#sectcartanfinaff) for more information on the conventions, syntax, and functions for creating and working with affine Cartan matrices. ### `AffineLieAlgebra(C, F): AlgMatElt, Fld -> AlgKac` Construct the affine Kac–Moody Lie algebra with affine Cartan matrix $C$ over the field $F$. ### `Example: construct (ex-055fd7)` We demonstrate the construction functions. ```magma > L := AffineLieAlgebra("G~2", Rationals()); > L; Affine Kac--Moody Lie algebra over Rational Field > C := Matrix(Integers(),3,3,[2,-1,-1,-1,2,-1,-1,-1,2]); > CartanName(C); A~2 > L := AffineLieAlgebra(C, Rationals()); > L; Affine Kac--Moody Lie algebra over Rational Field ``` ## Properties of Affine Kac–Moody Lie Algebras ### `CartanMatrix(L): AlgKac -> AlgMatElt` The Cartan matrix of $L$. ### `CartanName(L): AlgKac -> MonStgElt` The Cartan type of $L$. ### `Dimension(L): AlgKac -> Infty` Infinity. ### `CoefficientRing(L): AlgKac -> Rng` ### `BaseRing(L): AlgKac -> Rng` The coefficient ring of $L$. ### `FiniteLieAlgebra(L): AlgKac -> AlgLie` The Lie algebra $\frak{g}_0$ underlying $L$ (see the Introduction, Section [Introduction](introduction.md#sectalgliekmintro)). ### `LaurentSeriesRing(L): AlgKac -> RngSerLaur` The Laurent series ring ${\bf C}[t, t^{-1}]$ underlying $L$ (see the Introduction, Section [Introduction](introduction.md#sectalgliekmintro)). ### `StandardGenerators(L): AlgKac -> SeqEnum[AlgKacElt], SeqEnum[AlgKacElt], SeqEnum[AlgKacElt]` The standard generators of $L$. These are returned as three sequences, the first containing the $e_i$, the second containing the $f_i$, and the last containing the $h_i$. Note that the root usually labeled “0” occurs as the last element of each of these sequences. ### `Example: construct (ex-17cbd6)` We demonstrate some properties of affine Lie algebras. ```magma > L := AffineLieAlgebra("A~2", Rationals()); > L; Affine Kac--Moody Lie algebra over Rational Field > Lf := FiniteLieAlgebra(L); > Lf; Lie Algebra of dimension 8 with base ring Rational Field > SemisimpleType(Lf); A2 > e,f,h := StandardGenerators(L); > e; [ (0 0 0 0 0 1 0 0), (0 0 0 0 0 0 1 0), (t)*(1 0 0 0 0 0 0 0) ] > F := FreeLieAlgebra(Rationals(), 6); > phi := hom L | e cat f>; > phi(e1); (0 0 0 0 0 1 0 0) > phi(e1*e0) eq phi(e1)*phi(e0); true ``` ## Constructing Elements of Affine Kac–Moody Lie Algebras ### `L . i: AlgKac, RngIntElt -> AlgKacElt` The $i$-th basis element of the finite dimensional Lie algebra underlying $L$, as an element of $L$. ### `HasAttribute(L, "c"): AlgKac, MonStgElt -> BoolElt, AlgKacElt` ### `HasAttribute(L, "d"): AlgKac, MonStgElt -> BoolElt, AlgKacElt` Return `true` and the basis element $c$ or $d$ of $L$, according to the second argument of `HasAttribute`. ### `elt}, ... ], λ, μ {>} >: AlgKac, Tup -> AlgKacElt` For a 3-tuple $t$ such that $t_1$ is a sequence of elements of ${\bf C}[t, t^{-1}] \times \frak{g}_0$, and $t_2$ and $t_3$ are elements of the coefficient ring of $L$, construct $$ \sum_{(p,y) \in t_1} p \otimes y \oplus t_2 c \oplus t_3 d \in L. $$ See [`EltTup`](#function-kmaff-elttup) below for the converse function. ## Properties of Elements of Affine Kac–Moody Lie Algebras ### `EltTup(x): AlgKacElt -> Tup` The element $x$ of the affine Lie algebra $L$ as a three-tuple $t$ such that $$ x = \sum_{(p,y) \in t_1} p \otimes y \oplus t_2 c \oplus t_3 d. $$ The first entry, $t_1$, is a sequence of pairs $(p, y) \in {\bf C}[t, t^{-1}] \times \frak{g}_0$, $t_2$ is the coefficient of $c$ and $t_3$ is the coefficient of $d$. ### `IsZero(x): AlgKacElt -> BoolElt` Whether $x$ is zero. ### `x eq y: AlgKacElt, AlgKacElt -> BoolElt` Whether $x$ and $y$ are equal. ### `x + y: AlgKacElt, AlgKacElt -> AlgKacElt` ### `x - y: AlgKacElt, AlgKacElt -> AlgKacElt` ### `x * y: AlgKacElt, AlgKacElt -> AlgKacElt` ### `c * y: RngElt, AlgKacElt -> AlgKacElt` Respectively the sum, difference, and multiplication of $x$ and $y$. ### `-x: AlgKacElt -> AlgKacElt` The negation of $x$. ### `Example: construct (ex-1599ed)` We perform various computations with elements of an affine Lie algebra. ```magma > L := AffineLieAlgebra("B~3", Rationals()); > Lf := FiniteLieAlgebra(L); > e,f,h := StandardGenerators(L); > E,F,H := StandardBasis(Lf); > e[1] eq L!E[1]; true > x := e[4]; > x; (t)*(1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) > EltTup(x); <[ ], 0, 0> > elt eq x; true > y := elt,<-2,Lf.3>], -1/3, 1> >; > y; (-2)*(0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) + (-t^-2 + t^2)*(0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0) -1/3*c + d > z := t^3*L.2 - 1/5*h[1] + 1/7*L`c-L`d; > z; (t^3)*(0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) + (2/5)*(0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0) (-1/5)*(0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0) + 1/7*c -1*d > x*(y*z) + y*(z*x) + z*(x*y); 0 ```