Kac–Moody Groups

Introduction

A generalised Cartan matrix is an integer matrix $A = (a_{ij})$ with $a_{ii} = 2$, $a_{ij}\leq0$ for $i\neq j$ and $a_{ij} = 0$ exactly when $a_{ji} = 0$. No menu article owns the Kac–Moody algebras, so the object is defined here and marked as standard. To such a matrix one attaches a Kac–Moody algebra $\mathrm{G}(A)$, the Lie algebra generated by $3|I|$ generators $e_i, f_i, h_i$ subject to the Chevalley relations and the Serre relations $(\operatorname{ad}e_i)^{1-a_{ij}}e_j = 0$; when $A$ is a Cartan matrix of finite type this is a finite-dimensional simple Lie algebra, when $A$ is of affine type it is the affine algebra of the loop group, and for the remaining, indefinite, types it is an infinite-dimensional $\mathbb{Z}$-graded Lie algebra whose root system now contains imaginary roots, which were absent in the finite and affine cases. The Kac–Moody group $\mathcal{G}(A,R)$ is the group over a commutative ring $R$ attached to the same matrix by the Tits functor: the group generated by the root subgroups $U_i = \{x_i(t)\}$, their opposites and a torus, subject to the Chevalley commutator relations and the Steinberg-type relations; for the finite types it is the split simple algebraic group, and for the affine types it is the loop group or one of its forms.

The theory of the Kac–Moody groups is the theory of the infinite-dimensional groups with $BN$-pairs and buildings. The group $\mathcal{G}(A,R)$, for $R$ a field, carries a Tits system whose Weyl group is the Coxeter group $W$ determined by $A$, so that it acts on a building of type $(W,S)$: the building is spherical for the finite types, affine for the affine types and non-spherical and non-affine for the indefinite types, and it is here that the theory of the buildings of Buildings and Tits Systems finds its main family of examples beyond the classical ones. Over a ring, or for the group with both its positive and negative parabolics, one has the finer structure of a twin building: a pair of buildings of the same type with a codistance function, whose theory is due to Ronan and Tits and which characterises the Kac–Moody groups among the groups acting on the buildings. The maximal Kac–Moody groups, the topological completions of the minimal groups in the sense of Tits, Moody and Teo, carry a topology in which the root groups are topological groups, and in the affine cases the completion is the locally compact group of the corresponding $p$-adic or loop-theoretic description.

The article develops the generalised Cartan matrices and their classification, the Kac–Moody algebras with their root systems and the character formula, the Tits construction of the Kac–Moody groups, the $BN$-pairs and the buildings, the twin buildings and the Ronan–Tits theory, and the structure and classification of the groups with the linearity and the completions. The input from above is the theory of the buildings, apartments and Tits systems of Buildings and Tits Systems, the loop groups and the affine algebras of Loop Groups, the affine buildings and the parahorics of Bruhat–Tits Theory, the root systems and Weyl groups of Root Systems and Classification in Part I, and the Coxeter groups of Coxeter Groups in Part I. The generalised Cartan matrix, the Kac–Moody algebra, the Kac–Moody group and the twin building are defined in line, since no article above introduces them.

The boundary with Part III is the one fixed for this block. What is developed here is the algebraic and combinatorial structure: the Cartan matrices and their classification, the Kac–Moody algebras and their root systems, the integrable modules at the level of statement, the Tits construction of the groups, the $BN$-pairs, the buildings and the twin buildings, the linearity and the structure of the completions at the level of statement. What is deferred is the analytic theory: the representations of the Kac–Moody algebras on Hilbert spaces and their characters as analytic functions, the harmonic analysis on the Kac–Moody groups and their buildings, the measure theory and the ergodic theory of the actions, and the operator-algebraic theory of the completions — all of which belong to Analysis on Groups and to the infinite-dimensional representations of Part III. The character of an integrable module is a formal identity in the group algebra of the weight lattice, and no convergence is claimed. No physics is invoked.

Generalised Cartan Matrices and Kac–Moody Algebras

Generalised Cartan Matrices and their Classification

Definition. A generalised Cartan matrix (GCM) is a matrix $A = (a_{ij})_{i,j\in I}$ with integer entries such that $a_{ii} = 2$ for all $i$, $a_{ij}\leq0$ for $i\neq j$, and $a_{ij} = 0$ if and only if $a_{ji} = 0$. The GCM is symmetrisable if there are positive rationals $d_i$ with $d_ia_{ij} = d_ja_{ji}$; a symmetrisable GCM has a realisation by a symmetrised form on the root lattice, and the theory of the algebras and groups is complete in that case. The Dynkin diagram of the GCM has a node for each $i$, with the edges and the labels recording the products $a_{ij}a_{ji}$: no edge when $a_{ij} = a_{ji} = 0$, a single edge when the product is $1$, a double edge with an arrow when the product is $2$, a triple edge when it is $3$, and a thick edge labelled by the pair when the product is at least $4$.

Theorem (classification). Let $A$ be an indecomposable symmetrisable GCM. Then exactly one of the following holds:

(a) $A$ is of finite type: the Dynkin diagram is one of the diagrams $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2$; the associated Kac–Moody algebra is finite-dimensional and simple;

(b) $A$ is of affine type: the Dynkin diagram is one of the affine diagrams $A_n^{(1)},\dots,G_2^{(1)}$ or the twisted diagrams $A_n^{(2)},\dots,$ of the tables; the associated Kac–Moody algebra is the affine algebra of the loop group of Loop Groups, its Weyl group is affine and it has a null root;

(c) $A$ is of indefinite type: the associated Kac–Moody algebra is infinite-dimensional of exponential growth, with imaginary roots of unbounded multiplicity. The hyperbolic GCMs are the indefinite ones for which every proper connected subdiagram is of finite or affine type; the Lorentzian ones are those whose symmetrised form has signature $(n,1)$.

Proof sketch. The classification of the finite and affine diagrams is the classification of the Cartan matrices of the simple Lie algebras and their affinisations: the finiteness of the Weyl group is equivalent to the positive definiteness of the symmetrised matrix, which selects the finite diagrams, and the positive semidefiniteness with a one-dimensional radical selects the affine ones. The remaining cases have an indefinite form, hence an infinite Weyl group of exponential growth, and the theory of the root system then produces the imaginary roots of unbounded multiplicity. The classification is due to Kac and Moody and is quoted from the literature.

The Kac–Moody Algebra and its Root System

Definition. Let $A$ be a GCM of size $|I| = n$ with a realisation $(\mathrm{H},\Pi,\Pi^\vee)$, where $\mathrm{H}$ is a complex vector space of dimension $2n-\operatorname{rank}A$, $\Pi = \{\alpha_i\}$ the simple roots in $\mathrm{H}^*$ and $\Pi^\vee = \{h_i\}$ the simple coroots with $\alpha_j(h_i) = a_{ij}$. The Kac–Moody algebra $\mathrm{G}(A)$ is the Lie algebra generated by $\mathrm{H}$ and the elements $e_i, f_i$ $(i\in I)$ subject to

$$ [h,h'] = 0,\quad [h,e_i] = \alpha_i(h)e_i,\quad [h,f_i] = -\alpha_i(h)f_i,\quad [e_i,f_j] = \delta_{ij}h_i, $$

and the Serre relations $(\operatorname{ad}e_i)^{1-a_{ij}}e_j = 0 = (\operatorname{ad}f_i)^{1-a_{ij}}f_j$ for $i\neq j$. It has the triangular decomposition $\mathrm{G}(A) = \mathrm{N}^-\oplus\mathrm{H}\oplus\mathrm{N}^+$, with $\mathrm{N}^\pm$ the subalgebras generated by the $e_i$ and the $f_i$, and it is graded by the root lattice $Q = \bigoplus_i\mathbb{Z}\alpha_i$: the root system $\Delta = \Delta_+\cup\Delta_-$ consists of the nonzero weights of $\mathrm{G}(A)$ under the adjoint action of $\mathrm{H}$, a root being real if it is $W$-conjugate to a simple root and imaginary otherwise; the multiplicities of the real roots are $1$ and those of the imaginary roots are the dimensions of the corresponding root spaces.

Definition. The Weyl group of $A$ is the subgroup $W\leq GL(\mathrm{H}^*)$ generated by the reflections $s_i(\lambda) = \lambda-\lambda(h_i)\alpha_i$; it is the Coxeter group with Coxeter matrix determined by the products $a_{ij}a_{ji}$, so that the finite types give the finite Weyl groups, the affine types the affine Weyl groups and the indefinite types the infinite Coxeter groups of exponential growth. The Tits cone is the $W$-invariant cone in $\mathrm{H}^*$ obtained as the union of the $W$-translates of the closed dominant chamber; a weight $\lambda$ is dominant integral if $\lambda(h_i)\in\mathbb{Z}_{\geq0}$ for all $i$, and the fundamental weights $\Lambda_i$ are defined by $\Lambda_i(h_j) = \delta_{ij}$ in the symmetrisable case.

Theorem (properties of the root system). Let $A$ be a symmetrisable GCM.

(a) The real roots are precisely the $W$-translates of the simple roots, each with multiplicity $1$, and the imaginary roots are the elements of $\Delta$ that are not real; the set of imaginary roots is $W$-invariant and consists of the nonzero elements of the Tits cone in the appropriate sense.

(b) The multiplicities of the imaginary roots are bounded if and only if $A$ is of finite or affine type; for the indefinite types the multiplicities grow exponentially and $\mathrm{G}(A)$ has exponential growth.

(c) The algebra $\mathrm{G}(A)$ is generated by the root spaces of the simple roots, and every ideal of $\mathrm{G}(A)$ intersects $\mathrm{H}$; the algebra is simple modulo its centre when $A$ is indecomposable, and the centre is the kernel of the realisation.

Proof sketch. The statements are the standard theory of the Kac–Moody algebras: the real roots are the $W$-translates of the simple roots by the definition of the reflection and the multiplicity-one statement, and the imaginary roots are governed by the Tits cone and the Kostant formula; the growth statement is the dichotomy of the symmetrisable GCMs between the finite, affine and indefinite types, and the constancy of the multiplicities in the affine case is the classical fact of the affine root systems. The theorem is from the book of Kac and is quoted from the literature.

Integrable Modules and the Character Formula

Definition. A $\mathrm{G}(A)$-module $V$ is a highest weight module of highest weight $\lambda\in\mathrm{H}^*$ if it is generated by a vector $v$ with $e_iv = 0$ and $hv = \lambda(h)v$, and it is integrable if the $e_i$ and the $f_i$ act locally nilpotently; for a dominant integral $\lambda$ there is a unique irreducible integrable module $L(\lambda)$, and it is the quotient of the Verma module by its maximal submodule.

Theorem (Gabber–Kac). Let $A$ be a symmetrisable GCM. An irreducible highest weight $\mathrm{G}(A)$-module with highest weight $\lambda$ is integrable if and only if $\lambda$ is dominant integral.

Theorem (Weyl–Kac character formula). For a dominant integral $\lambda$ the character of $L(\lambda)$, as a formal series in the group algebra of the weight lattice, is

$$ \operatorname{ch}L(\lambda) = \frac{\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda+\rho)-\rho}}{\prod_{\alpha\in\Delta_+}(1-e^{-\alpha})^{\mathrm{mult}\,\alpha}} , $$

where $\rho$ is any element with $\rho(h_i) = 1$ for all $i$ and the product runs over the positive roots with their multiplicities; for the finite types this is the classical Weyl character formula, and for the affine types it is the formula stated in Loop Groups. The formula is a formal identity in the group algebra; its analytic convergence at special values of the formal variables is Part III.

Proof sketch. The character formula is proved by reducing the computation to the finite-dimensional case in the category of the integrable modules and using the inversion in the Weyl group, the multiplicities of the imaginary roots being accounted for by the product; the integrability criterion of Gabber–Kac is proved by the local nilpotence of the action of the root elements and the classification of the modules with a dominant integral weight. The results are quoted from the literature.

Kac–Moody Groups

The Tits Construction

Definition. Let $A$ be a GCM and let $R$ be a commutative ring with identity. The Kac–Moody group $\mathcal{G}(A,R)$ is the group generated by the root subgroups $U_i = \{x_i(t) : t\in R\}$, $U_i^- = \{y_i(t) : t\in R\}$, each isomorphic to $(R,+)$, together with a torus $H$, subject to the relations

(KMG1) $x_i(t)x_i(t') = x_i(t+t')$ and $y_i(t)y_i(t') = y_i(t+t')$, and $x_i(0) = y_i(0) = 1$;

(KMG2) $h\,x_i(t)\,h^{-1} = x_i(\chi_i(h)t)$ and $h\,y_i(t)\,h^{-1} = y_i(\chi_i(h)^{-1}t)$ for $h\in H$, where $\chi_i$ is the character of $H$ corresponding to the simple root $\alpha_i$;

(KMG3) (Chevalley commutator formula) for $i\neq j$ the commutator $[x_i(t),x_j(t')]$ is a finite product of the elements $x_{m\alpha_i+n\alpha_j}(c_{mn}t^mt'^n)$ with $m,n>0$ and $m\alpha_i+n\alpha_j$ a root, the constants $c_{mn}$ being the structure constants of the Kac–Moody algebra over $\mathbb{Z}$;

(KMG4) (Steinberg relations) for each pair $i\neq j$ the subgroup generated by the four root groups $U_i,U_i^-,U_j,U_j^-$ together with $H$ is the rank-two group prescribed by the root datum of the pair $(i,j)$, as given by the Steinberg presentation of the (possibly infinite) rank-two root system.

The group $\mathcal{G}(A,R)$ is the minimal Kac–Moody group; it is the value on $R$ of the Tits functor $\mathcal{G}(A,-)$, a functor from the category of commutative rings to the category of groups, uniquely determined up to canonical isomorphism by the above presentation, and it is a group scheme in the appropriate sense.

Theorem (Tits). The functor $\mathcal{G}(A,-)$ is well defined and independent of the choices of the root datum and the structure constants; for $A$ of finite type it is the split simply connected simple algebraic group of the corresponding type; for $A$ of affine type it is the loop group of Loop Groups (or the appropriate twist); and for $A$ of indefinite type it is an infinite-dimensional group. For $R = K$ a field, the group $G = \mathcal{G}(A,K)$ has a Tits system $(G,B,N)$ with Weyl group $W$ and acts on a thick building of type $(W,S)$: a panel of the building is the set of the $|K|+1$ chambers of a projective line over $K$ through the panels of the rank-two residues, so the building is thick for every field.

Proof sketch. The independence of the presentation is the theorem of Tits on the uniqueness of the Steinberg presentation: two groups generated by the same root data with the same commutator and Steinberg relations are canonically isomorphic, because the relations determine the multiplication via the normal form of the free product with amalgamation implicit in the root groups. The identification in the finite and affine cases is the classical Chevalley and Steinberg theory, and the Tits system is obtained from the subgroup $B = H\ltimes\prod_{i}U_i$ and the subgroup $N$ generated by the $H$ and the elements $n_i = x_i(1)y_i(-1)x_i(1)$ mapping to the simple reflections; the building axioms are verified by the identification $G/B$ with the set of chambers and the Bruhat decomposition $G = \bigsqcup_{w\in W}BwB$. The theorem is Tits's and is quoted from the literature.

The Building and the Twin Building

Definition. Let $G = \mathcal{G}(A,K)$ for a field $K$ with a Tits system $(G,B,N)$ and Weyl group $(W,S)$. The building of $G$ is the simplicial complex of Buildings and Tits Systems whose chambers are the cosets $G/B$ and whose apartments are the translates of the standard apartment; it is thick when $|K|\geq2$, a panel having $|K|+1\geq3$ chambers. The standard parabolic subgroups are the conjugates of the subgroups $P_J = BW_JB$, and the building is the flag complex of the parabolics. The Weyl group $W$ may be finite, affine or of exponential growth; accordingly the building is spherical, affine or of non-spherical non-affine type, the latter case having no analogue among the classical groups.

Definition. The twin building of a Kac–Moody group is the pair $(\Delta_+,\Delta_-)$ of buildings of type $(W,S)$ associated with the two Tits systems $(G,B^+,N)$ and $(G,B^-,N)$ of the group, where $B^+$ and $B^-$ are the positive and the negative Borel-like subgroups; the chambers of $\Delta_+$ are the cosets $G/B^+$, those of $\Delta_-$ the cosets $G/B^-$, and the codistance is the $W$-valued function $d^*(x,y)$ defined on the pairs $(\text{chamber of }\Delta_+,\text{chamber of }\Delta_-)$ by the double cosets $B^+wB^-$. A twin building is a pair of buildings with such a codistance satisfying the Ronan–Tits axioms, and it encodes the opposition relation between the two buildings.

Theorem (Ronan–Tits). Let $(\Delta_+,\Delta_-)$ be a thick irreducible twin building of type $(W,S)$ with $|S|\geq2$. Then there is a group $G$ acting on the twin building by type-preserving automorphisms preserving the codistance, generated by the root groups of the local twin buildings, and $G$ is a Kac–Moody group; conversely, the twin building of a Kac–Moody group is a thick twin building of the given type, and the group is determined by the twin building together with the root data. A group acting on a thick twin building preserves the codistance, and the twin building recovers the $BN$-pair data.

Proof sketch. The Ronan–Tits theory constructs the group from the twin building by the root group functors and the local structure of the panels, showing that the axioms of the twin building force the Steinberg relations; the reconstruction is the twin form of the classification of the buildings of spherical type. The statements are from the work of Ronan and Tits and are quoted from the literature.

Structure and Classification

The Cases of Finite and Affine Type

Theorem (the classical cases). Let $A$ be a GCM and let $\mathcal{G}(A,K)$ be the Kac–Moody group over a field $K$.

(a) If $A$ is of finite type, the group is the split simply connected simple algebraic group over $K$ and its building is the spherical building of the flag complex of the corresponding vector space or polar space of Buildings and Tits Systems; the Tits system is the classical $BN$-pair with $B$ a Borel subgroup, and the Weyl group is the finite Weyl group.

(b) If $A$ is of affine type, the group is the loop group (the polynomial loop group $G(K[t,t^{-1}])$ or the appropriate form) of Loop Groups, and its building is the affine building of Bruhat–Tits Theory: the spherical building at infinity is the building of the underlying finite type, and the affine Weyl group acts on the apartment. Over a local field this is the Bruhat–Tits building of the corresponding $p$-adic group.

(c) Consequently the Kac–Moody groups of finite and affine type are the classical reductive and loop groups, and the genuinely new objects of the theory are the groups of indefinite type.

Proof sketch. The identification in the finite case is the Chevalley construction of the simple groups from the root data, and in the affine case it is the realisation of the affine algebra as a loop algebra together with the identification of the $K[t,t^{-1}]$-points of the algebraic group with the loop group; the buildings are identified with the corresponding flag complexes and affine buildings of the earlier articles. The statements are standard and are quoted from the literature.

Indefinite Type, Linearity and Completions

Theorem (properties of the indefinite case). Let $A$ be an indecomposable GCM of indefinite type and let $G = \mathcal{G}(A,K)$ be the minimal Kac–Moody group over a field $K$.

(a) The group $G$ is infinite-dimensional, and the minimal Kac–Moody groups of indefinite type are not linear: they are not isomorphic to a subgroup of $GL_n(K')$ for any field $K'$ and any $n$, a theorem of the theory proved through the structure of the twin building and the root groups. Its building is of non-spherical non-affine type.

(b) The group $G$ is generated by the root groups of the simple roots; it has a Bruhat decomposition $G = \bigsqcup_{w\in W}BwB$ with the Weyl group infinite, and the parabolic subgroups are the stabilisers of the residues of its building; the subgroups of $G$ are governed by the theory of the buildings of non-spherical type.

(c) The group $G$ admits a completion in the sense of the theory: there is a topology on $G$, the topology of the root datum, in which the root groups are topological groups and $G$ becomes a topological group; the completion is independent of the choices and behaves well with respect to the $BN$-pair. The completions are the objects on which the harmonic analysis of Part III operates, and for the affine types over a local field they are the locally compact $p$-adic groups of $p$-adic Lie Groups.

Proof sketch. (a) The non-linearity is the theorem of the theory that a building of non-spherical non-affine type with the twin structure does not admit a faithful finite-dimensional linear action of the associated group; the proof uses the structure of the twin building and the root groups. (b) is the Tits system and the Bruhat decomposition, valid for the infinite Weyl group with the usual conventions on the infinite unions. (c) The completion is constructed by completing the group with respect to the root group topologies of the root datum, and the completion theorem asserts that the result is independent of the choices and behaves well with respect to the $BN$-pair. The statements are quoted from the literature.

Remark (Borcherds algebras and the wider frame). The theory extends to the Borcherds superalgebras, for which the diagonal entries of the Cartan matrix are allowed to be at most $2$ and the sign conditions are relaxed; the resulting algebras have imaginary simple roots, and their vertex-algebra constructions are the setting of Borcherds' proof of the moonshine conjectures. The classification of the indefinite types is not known and is not expected to be finite: the hyperbolic matrices of rank at least three are numerous, and the class of the Kac–Moody algebras of indefinite type is the richest part of the theory.

The Boundary with Analysis

The theory of this article is algebraic and combinatorial; the analysis on the Kac–Moody algebras and groups is Part III.

The characters of the integrable modules as analytic functions, the Kac–Moody modular forms, the theta correspondences and the representations of the affine algebras on Hilbert spaces are Part III; the character formula above is a formal identity.
The harmonic analysis on the Kac–Moody groups and on their buildings, the Hecke algebras of the parabolics, the spherical functions and the representations of the completions are Analysis on Groups in Part III.
The operator algebra aspects: the group von Neumann algebras, the crossed products and the Kac–Moody groups as quantum groups, are Part III.
The topological completions of the Kac–Moody groups, the maximal Kac–Moody groups, and their locally compact structure are stated above at the level of the existence; their analysis belongs to the locally compact group theory of this Part and to the analysis of Part III.
What is not deferred: the generalised Cartan matrices and their classification; the Kac–Moody algebras and their root systems; the definition of the integrable modules and the character identity; the Tits construction of the Kac–Moody groups and the Tits functor; the $BN$-pairs and the buildings; the twin buildings and the Ronan–Tits theory; and the structure, linearity and completions at the level of statement.

Summary

A generalised Cartan matrix $A$ has $a_{ii} = 2$, $a_{ij}\leq0$ off the diagonal and $a_{ij} = 0\Leftrightarrow a_{ji} = 0$; it is of finite, affine or indefinite type, and the finite and affine types reproduce the Cartan matrices of the simple Lie algebras and their affinisations. The Kac–Moody algebra $\mathrm{G}(A)$ is generated by $e_i,f_i,h_i$ subject to the Chevalley relations and the Serre relations, with the triangular decomposition $\mathrm{N}^-\oplus\mathrm{H}\oplus\mathrm{N}^+$, the root system of real and imaginary roots, the Weyl group $W$ generated by the reflections $s_i$ and the Tits cone. The irreducible highest weight module $L(\lambda)$ is integrable exactly when $\lambda$ is dominant integral (Gabber–Kac), and its character is the Weyl–Kac formula

$$ \operatorname{ch}L(\lambda) = \frac{\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda+\rho)-\rho}}{\prod_{\alpha\in\Delta_+}(1-e^{-\alpha})^{\mathrm{mult}\,\alpha}} . $$

The Kac–Moody group $\mathcal{G}(A,R)$ is the group generated by the root subgroups $U_i$, $U_i^-$ and a torus subject to the Chevalley commutator and Steinberg relations; it is the value of the Tits functor, it is the split simple algebraic group for the finite types and the loop group for the affine types, and for the indefinite types it is the genuinely infinite-dimensional group, non-linear and generated by its root groups. Over a field it has a Tits system with Weyl group $W$ and acts on a thick building of type $(W,S)$, spherical, affine or non-spherical non-affine according to the type of $A$; over a ring or with both Borel-like subgroups one has the twin building $(\Delta_+,\Delta_-)$ with the codistance, characterised by the Ronan–Tits theory, and the maximal groups are the topological completions of the minimal ones. The analysis of the characters, of the harmonic analysis on the buildings and of the completions belongs to Part III.

Summary of Notation

Symbol	Meaning
$A = (a_{ij})$	Generalised Cartan matrix: $a_{ii}=2$, $a_{ij}\leq0$, $a_{ij}=0\Leftrightarrow a_{ji}=0$
$I$, $\vert I\vert = n$	Index set of the simple roots
symmetrisable	$d_ia_{ij} = d_ja_{ji}$ for positive rationals $d_i$
Dynkin diagram of $A$	Nodes $i$, edges labelled by $a_{ij}a_{ji}$
finite / affine / indefinite type	The three cases of the classification
hyperbolic	Indefinite with every proper subdiagram finite or affine
$\mathrm{G}(A)$	Kac–Moody algebra
$\mathrm{H}$, $\Pi=\{\alpha_i\}$, $\Pi^\vee=\{h_i\}$	Cartan subalgebra, simple roots, simple coroots
$e_i,f_i,h_i$	Chevalley generators
Serre relations	$(\operatorname{ad}e_i)^{1-a_{ij}}e_j=0$, $(\operatorname{ad}f_i)^{1-a_{ij}}f_j=0$
$\mathrm{N}^\pm$	Positive and negative maximal nilpotent subalgebras
$\Delta$, $\Delta_+$, mult $\alpha$	Root system, positive roots, multiplicities
real / imaginary root	$W$-conjugate to a simple root / otherwise
$W$, $s_i$, Tits cone	Weyl group, simple reflections, the invariant cone
dominant integral $\lambda$, $\Lambda_i$	$\lambda(h_i)\in\mathbb{Z}_{\geq0}$; fundamental weights
$L(\lambda)$	Irreducible integrable module of highest weight $\lambda$
$\operatorname{ch}L(\lambda)$	Character; Weyl–Kac formula
$\mathcal{G}(A,R)$	Kac–Moody group over $R$ (Tits functor)
$U_i, U_i^-, x_i(t), y_i(t)$	Root subgroups and their parametrisations
$H$	Torus of the Kac–Moody group
$B$, $N$, $P_J=BW_JB$	Borel-like, monomial, standard parabolic subgroups
Bruhat decomposition	$G = \bigsqcup_{w\in W}BwB$
$\Delta_+,\Delta_-$, codistance $d^*$	The twin building and its codistance
maximal Kac–Moody group	Topological completion of the minimal Kac–Moody group
Borcherds algebra	Generalised Kac–Moody algebra with $a_{ii}\leq2$

Kac–Moody Groups

Introduction

Generalised Cartan Matrices and Kac–Moody Algebras

Generalised Cartan Matrices and their Classification

The Kac–Moody Algebra and its Root System

Integrable Modules and the Character Formula

Kac–Moody Groups

The Tits Construction

The Building and the Twin Building

Structure and Classification

The Cases of Finite and Affine Type

Indefinite Type, Linearity and Completions

The Boundary with Analysis

Summary

Summary of Notation

Further Reading