some writing

src/doc/en/thematic_tutorials/lie/affine.rst

@@ -26,6 +26,61 @@ explain what facilities there are in Sage for computing
 with these. We will often restrict ourselves to the case
 of affine Lie algebras.
 
+-------------
+Cartan Matrix
+-------------
+
+See [Kac]_ Chapter 1 for this topic.
+
+The basic data defining a Kac-Moody Lie algebra is a
+*generalized Cartan matrix*. This is a square matrix `A=(a_{ij})`
+with diagonal entries equal to 2 and nonpositive off
+diagonal entries such that `a_{ij}=0` if and only if
+`a_{ji}=0`. It is useful to assume that it is *indecomposable*
+and *symmetrizable*. Indecomposable means that it cannot
+be put arranged into two diagonal blocks by permuting
+the rows and columns; and symmetrizable means that
+`DA` is diagonal for a diagonal matrix `D`.
+
+Given a generalized Cartan matrix a vector space `\mathfrak{h}` containing
+vectors `\alpha_i^\vee` (called *simple coroots*) and vectors `\alpha_i` in
+`\mathfrak{h}^*` (called *simple roots*) such that `\alpha_i^\vee(\alpha_j)=a_{ij}`.
+Moreover there exists a Kac-Moody Lie algebra `\mathfrak{g}`
+containing `\mathfrak{h}` as an abelian subalgebra
+that is generated by `\mathfrak{h}` and elements `e_i` and
+`f_i` such that `[e_i,f_i]=\delta_{ij}\alpha_i^\vee`,
+`[h,e_i]=\alpha_i(h)e_i`, `[h,f_i]=-\alpha_i(h)f_i`.
+(These conditions do not quite characterize `\mathfrak{g}`,
+but they do if supplemented by the Serre relations, which
+we will not need or state.)
+
+The significance of the diagonalizability assumption
+is that `\mathfrak{g}` admits an invariant symmetric
+bilinear form, and hence has a Casimir operator and
+a good representation theory.
+
+The transpose of `A` is also a symmetrizable indecomposable
+generalized Cartan matrix, so there is a *dual Cartan type*
+in which the roots and coroots are interchanged.
+
+In Sage, we may recover the Cartan matrix as follows::
+
+   sage: RootSystem(['B',2]).cartan_matrix()
+   [ 2 -1]
+   [-2  2]
+   sage: RootSystem(['B',2,1]).cartan_matrix()
+   [ 2  0 -1]
+   [ 0  2 -1]
+   [-2 -2  2]
+
+If `\det(A)=0` and its nullspace is one-dimensional,
+then `\mathfrak{g}` is an *affine Lie algebra*, as in
+the second example above. 
+
+-----------------------------
+Affine Kac-Moody Lie Algebras
+-----------------------------
+
 One realization of affine Lie algebras, described in Chapter 7
 of [Kac]_ begins with a
 finite-dimensional isimple Lie algebra `\mathfrak{g}^\circ`,
@@ -72,6 +127,13 @@ We can infer some Levi subalgebras of `\mathfrak{g}`, obtained by
 omitting one node from the Dynkin diagram; particularly omitting
 the "affine node" `0` gives `E_6`, that is `\mathfrak{g}^\circ`.
 
+The subset of `\lambda\in\mathfrak{h}^*` characterized by `\lambda(\alpha_i^\vee)\in\ZZ`
+for the coroots `\alpha_i` is called the *weight lattice* `P`.
+The term lattice is a slight misnomer because `P` is not
+discrete; it contains all multiples of `\delta`, which is
+orthogonal to the coroots. However
+
+
 There are two versions of the weight lattice, depending on
 whether we are working with `\mathfrak{g}` or `\mathfrak{g}'`.
 The larger Lie algebra of `\mathfrak{g}` is called the
@@ -144,7 +206,7 @@ the Cartan type as the dual of the corresponding untwisted type::
 Roots and Weights
 -----------------
 
-The Lie algebra `\mathfrak{g}` has a triangular decomposition
+A Kac-Moody Lie algebra `\mathfrak{g}` has a triangular decomposition
 
 .. MATH::
 
@@ -169,8 +231,8 @@ As a special case, `\mathfrak{g}` is a module over itself
 under the adjoint representation, and it has a weight
 decomposition.
 
-The nonzero weights in the adjoint representation of `\mathcal{g}`
-on itself are called *roots*. In contrast with the finite-dimensional
+The roots are the nonzero weights in the adjoint representation of `\mathcal{g}`
+on itself. In contrast with the finite-dimensional
 case, if `\mathcal{g}` is an infinte Kac-Moody Lie algebra there are two
 types of roots, called *real* and imaginary. The real roots have
 multiplicity 1, while the imaginary roots can have multiplicity
@@ -206,10 +268,11 @@ We now specialize to affine Kac-Moody Lie algebras and their
 root systems. The basic reference for the affine root system and Weyl
 group is [Kac]_ Chapter 6.
 
-There is a minimal imaginary root `\delta`. The imaginary roots
+There is a minimal imaginary root `\delta`, sometimes called
+the *nullroot*. The imaginary roots
 are the vectors `n\delta` where `n` is a nonzero integer. This
 root is positive if and only if `n>0`. The root system `\Delta`
-contains a copy of the finite root system `Delta^\circ` of
+contains a copy of the finite root system `\Delta^\circ` of
 `\mathfrak{g}^\circ`. In the untwisted case, the real roots
 are `\alpha+n\delta` where `n` is an integer; the root is
 positive if `n>0` or if `n=0` and `\alpha` is positive. For
@@ -251,14 +314,6 @@ option ``extended=True``::
     sage: WL.basic_imaginary_roots()[0]
     delta
 
-Certain constants `a_i` label the vertices `i=0,\cdots,\ell` in
-the tables Aff1, Aff2 and Aff3 in [Kac]_ Chapter 4. They
-play an important role in the theory. In Sage they are available
-as follows::
-
-    sage: CartanType(['B',5,1]).a()
-    Finite family {0: 1, 1: 1, 2: 2, 3: 2, 4: 2, 5: 2}
-
 Be aware that for the exceptional groups, the ordering of the indices
 are different from those in [Kac]_. This is because Sage uses the Bourbaki
 ordering of the roots, and Kac does not. Thus in Bourbaki (and in Sage)
@@ -272,6 +327,48 @@ the `G_2` short root is `\alpha_1`::
   
 By contrast in Kac, `\alpha_2` is the short root.
 
+Column annihilator of the Cartan matrix
+---------------------------------------
+
+Certain constants `a_i` label the vertices `i=0,\cdots,\ell` in
+the tables Aff1, Aff2 and Aff3 in [Kac]_ Chapter 4. They
+play an important role in the theory. In Sage they are available
+as follows::
+
+    sage: CartanType(['B',5,1]).a()
+    Finite family {0: 1, 1: 1, 2: 2, 3: 2, 4: 2, 5: 2}
+
+The column vector `a` with these entries spans the
+nullspace of `A`::
+
+    sage: RS = RootSystem(['E',6,2]); RS
+    Root system of type ['F', 4, 1]^*
+    sage: A=RS.cartan_matrix(); A
+    [ 2 -1  0  0  0]
+    [-1  2 -1  0  0]
+    [ 0 -1  2 -2  0]
+    [ 0  0 -1  2 -1]
+    [ 0  0  0 -1  2]
+    sage: ann = Matrix([[v] for v in RS.cartan_type().a()]); ann
+    [1]
+    [2]
+    [3]
+    [2]
+    [1]
+    sage: A*ann
+    [0]
+    [0]
+    [0]
+    [0]
+    [0]
+
+The nullroot `\delta` equals `\sum a_i\alpha_i`::
+
+    sage: WL = RootSystem('C3~').weight_lattice(extended=True); WL
+    Extended weight lattice of the Root system of type ['C', 3, 1]
+    sage: sum(WL.cartan_type().a()[i]*WL.simple_root(i) for i in WL.cartan_type().index_set())
+    delta
+
 The Weyl Group and extended Affine Weyl Group
 ---------------------------------------------
 
@@ -374,4 +471,5 @@ dominant weights are the nonnegative linear combinations of the
 algebra of Cartan type `X_\ell^{(1)}`` then `r=\ell+1`. The labels correspond
 to the nodes in the Dynkin diagram.
 
-:class:`~sage.combinat.root_system.integrable_representations.IntegrableRepresentation`
+:class:`~sage.combinat.root_system.integrable_representations.IntegrableRepresentation`
+