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Root Systems in Lie Theory: From the Classic Definition to Nowadays
Communicated by Notices Associate Editor Han-Bom Moon
1. Root Systems: The Origin
The purpose of this article is to discuss the role played by root systems in the theory of Lie algebras and related objects in representation theory, with focus on the combinatorial description and properties.
1.1. Semisimple Lie algebras
The study of Lie algebras began toward the end of the 19th century. They emerged as the algebraic counterpart of a purely geometric object: Lie groups, which we can briefly define as groups that admit a differentiable structure such that multiplication and the function that computes inverses are differentiable. Lie algebras appeared as some algebraic structure attached to the tangent space of the unit of this group.
Initially Lie algebras were only considered over complex or real numbers, but the abstraction of the definition led to Lie algebras over arbitrary fields.
There is a subtle difference when the field is of characteristic two: the antisymmetry is replaced by for all (which implies the former one). From now on all Lie algebras considered here are assumed to be finite-dimensional.
An easy example is to pick a vector space together with trivial bracket for all these Lie algebras are called abelian. ;
There is a general way to move from an associative algebra to a Lie algebra: take as vector space and set for each pair A prominent example of this construction is the general linear algebra . which is the set of linear endomorphisms of a finite-dimensional vector space , Other classical examples appear as Lie subalgebras (that is, subspaces closed under the bracket) of . :
- •
those endomorphisms whose trace is zero; if , then we simply denote , by or , when the field is clear from the context.
- •
The orthogonal and symplectic Lie subalgebras respectively , of those endomorphisms , such that
where is a symmetric, respectively antisymmetric, nondegenerate bilinear form on .
Analogously, we may start with the algebra of , matrices, and take some subalgebras, as the subspaces of upper triangular matrices, those of trace 0, the orthogonal matrices, between others.
Once we have a notion of algebra, it is natural to ask for ideals: in the case of Lie algebras, these are subspaces such that This leads to consider simple Lie algebras, those Lie algebras . such that and the unique ideals are the trivial ones: and In addition, we say that a Lie algebra . is semisimple if is isomorphic to the direct sum of simple Lie algebras.
For the rest of this section we fix We know that a Lie algebra . is simple if and only if is (isomorphic to) , , and a few exceptional examples , , , , That is, up to 5 exceptions, all the complex simple Lie algebras are subalgebras of matrices. Thus one may wonder if some properties of the algebras of matrices still hold for simple Lie algebras. We will recall some of them by the end of this section, following .Hum78.
As for associative algebras, we can study modules over Lie algebras. A module is a pair - where , is a space and -vector is a linear map such that
For example, the bracket gives an action of over itself, called the adjoint action.
For each we look at the inner derivation
associated to the adjoint action. These endomorphisms induce a symmetric bilinear form on called the Killing form: ,
The Killing form and the give other characterizations of semisimplicity: -modules is semisimple if and only if is nondegenerate if and only if every module is semisimple, i.e., every admits a complement which is a -submodule -submodule.
When is one of the Lie algebras of matrices above, the action of diagonal matrices is, in fact, diagonalizable. Mimicking this fact we look for subalgebras such that the action of their elements is diagonalizable, called toral subalgebras.
From now on assume that is also semisimple. It can be shown that toral algebras are abelian, and we pick a maximal one Thus . decomposes as the direct sum of the -eigenspaces:
As is abelian, we have that one can show that we have an equality, : Thus, if we set . then , a finite set called the root system of , gives a decomposition of , into as follows: -eigenspaces
This decomposition is compatible with the bracket,
and the Killing form
We can derive that is nondegenerate, thus it induces a symmetric nondegenerate bilinear form .
1.2. Root systems for Lie algebras
We may derive strong properties of the root system using the representation theory of we refer to ,Bou02Hum78 for more details.
- (i)
is spanned by .
- (ii)
If then , Moreover, for each . , .
- (iii)
For each the eigenspace , is one-dimensional. Moreover, is a subalgebra isomorphic to Notice that . .
- (iv)
If then , .
- (v)
Let be such that Then there exist . such that
Moreover, That is, the root string over . in the direction of has no holes.
By (i) there exists a basis of contained in We can check that all the coefficients of any . written in terms of , are rational numbers, so we may consider the , subspace -linear generated by and take the extension to we get a finite-dimensional : space -vector which contains all the information and the geometry of .
2. Classical Root Systems
From the information above one may wonder if there exists an abstract notion of root system. The answer is yes, and we will recall it following Bou02, see also Hum78. We can classify all finite root systems in terms of so-called finite Cartan matrices. We will also recall a way to come back from (abstract) root systems to complex Lie algebras.
2.1. Abstract definition
In Hum78 one also requires that for each , In other references, root systems with this extra property are called reduced. .
The reflections , are univocally determined and there exists a symmetric invariant nondegenerate bilinear form which is moreover invariant by , and positive definite. Now, the elements are recovered using this form:
Also, the set is a root system of with , There are four examples of reduced root systems in rank 2: . , , and with , , , and , roots, respectively. The third one is depicted in Figure 1.
Let be such that One may check that . (respectively, if ) (respectively, This is the starting point, together with ).(RS2) and (RS3), to check that an analogue of (v) holds for (abstract) root systems.
Another key point is the existence of a base of a root system. It means a subset such that is a basis of (as a vector space), and every is written, in terms of as a linear combination whose coefficients are all nonnegative integers, or all nonpositive integers. ,
The proof of existence of bases gives the geometric flavor behind root systems. We take a vector such that the orthogonal hyperplane to does not contain any root. Indeed belongs to where , is the kernel of i.e., the hyperplane orthogonal to , the connected components of : are called the Weyl chambers. Thus where ,
A base is made by those indecomposable roots in those : which cannot be written as a sum with , Moreover every base can be constructed in this way. .
For example, in Figure 1 we take the green hyperplane: the positive roots are the red ones, the negative are the blue ones, and is a base.
The Weyl group permutes bases (and Weyl chambers as well), and the action is simply transitive. We check then that any root belongs to a base, and for each base , is generated by , (we reduce the number of generators of to the rank of the root system). This leads to the study of groups generated by reflections and Coxeter groups considered in Bou02, which became an important subject of research on its own, and remains active until now.
2.2. The classification
As for algebraic objects, we may ask for irreducible root systems: those which cannot split into two orthogonal subsets (otherwise each subset is itself a root system). Every root system of decomposes uniquely as a union of irreducible root systems corresponding to the subspaces of spanned by Thus, in order to classify root systems, we can restrict to the irreducible ones. .
Assume now that is an irreducible root system of rank Set . as the matrix with entries
where is a base. One can check that is well-defined; i.e., it does not depend on the chosen base. In addition, is indecomposable: for all there exist such that Moreover, .
- (GCM1)
for all ,
- (GCM2)
if and only if ,
- (GCM3)
for all , .
Any satisfying (GCM1)–(GCM3) is called a generalized Cartan matrix (GCM) Kac90. The information of GCM is encoded in a graph called the Dynkin diagram: it has vertices, labelled with and for each pair , ,
- •
if then we add , edges between vertices and with an arrow from , to (respectively to if ) (respectively, in particular, if ); (so as well) then we draw no edges between and and if , then we draw just a line; ,
- •
if then we draw a thick line between , and labelled with .
For example, the Dynkin diagrams of and and are, respectively
One reason to differentiate between and is all finite and affine Dynkin diagrams satisfy the first condition, and these are probably the most studied cases. We refer to Bou02Hum78 for the definition of affine Dynkin diagrams while finite ones are depicted in Figure 2, in connection with finite-dimensional complex Lie algebras.
One may define the Weyl group of a GCM as the subgroup of generated by reflections , where , is the canonical basis of if : is the Cartan matrix of a Lie algebra as above, then the Weyl group of is generated by these Analogously, we can define ’s.
Then one can prove that is finite if and only if is finite, which is equivalent to the notion of finite GCM. Finite GCM are parametrized by finite Dynkin diagrams, i.e., those in Figure 2.
Finite connected Dynkin diagrams.
, | |
Up to now we deal with three notions:
- (i)
Simple Lie algebras over ,
- (ii)
Irreducible root systems,
- (iii)
Finite Cartan matrices, or the corresponding Dynkin diagrams.
We moved first from (i) to (ii), and then state a correspondence (ii) (iii). Now we need to come back to (i). We can check that has Cartan matrix of type (see Example 1.2), while matrices of types , and appear for orthogonal and symplectic Lie algebras. For each one of the exceptional finite Cartan matrices in Figure 2 we can construct by hand a simple Lie algebra with Cartan matrix The natural question is if there exists a systematic way to build these Lie algebras. We will recall it in the next subsection, i.e., a correspondence .(iii)(i).
2.3. Back to Lie algebras: Kac-Moody construction
Looking at Example 1.2, the Cartan matrix of can be recovered from the action of the Cartan subalgebra on eigenvectors of a base of the root system In addition the decomposition . into positive and negative roots for the chosen base corresponds in this case to the upper and lower triangular matrices of (recall that is spanned by the set of all the diagonal matrices in ).
As for associative algebras, we have a notion of a Lie algebra presented by generators and relations as the appropiate quotient of a free Lie algebra. We will attach a Lie algebra to each matrix these algebras were introduced by Serre in 1966 for finite matrices ; and by Kac and Moody in two independent and simultaneous works in the late sixties, see ,Kac90 and the references therein. For the sake of simplicity of the exposition we assume that .
Let be the Lie algebra presented by generators , , , and relations ,
1Let be the subspace spanned by , the subalgebra generated by respectively , We have the following facts: .
- (a)
is a free Lie algebra in generators.
- (b)
As a vector space, .
- (c)
The adjoint action of on is diagonalizable.
- (d)
Among all the ideals of intersecting trivially there exists a maximal one , which satisfies ,
Because of the definition of , is generated by , , , has a triangular decomposition ,
where is the image of under the projection i.e., the subalgebra generated by (the image of) , respectively , and any other Lie algebra with a triangular decomposition as above, generated by the same set of generators satisfying ,1, projects onto .
When the generalized Cartan matrix is not of finite type, the associated Kac-Moody Lie algebra is infinite-dimensional. Although for the purposes of this exposition we are interested in the finite-dimensional examples, the infinite-dimensional Lie algebras (or at least some of them, mainly the affine ones) are quite important since they have appeared in connection either with other areas of mathematics, especially representation theory, or theoretical physics, for example in conformal field theory.
3. Root Systems for Other Kinds of Lie Algebras
Next we deal with contragredient Lie algebras over fields of positive characteristic and later with Lie superalgebras over any field. We will recall the main differences with the picture of Lie algebras over which leads to a more general notion of root system. This root system still captures the combinatorics of these Lie theoretic objects.
3.1. Lie algebras over fields of positive characteristic
Let be an algebraically closed field of characteristic The study of simple Lie algebras becomes more and more complicated as far as . is smaller, see, e.g., Str04. A main difference with the case of complex numbers is that not all simple Lie algebras have a triangular decomposition as above, and the Cartan subalgebra plays a weaker role in the structure of the whole Lie algebra.
On the other hand, Definition 2.3 still holds over so we may ask about the classification of finite-dimensional contragredient Lie algebras. A subtle difference is that we restrict to , ideals intersecting trivially -homogeneous where each , has degree 1, each has degree -1 and each has degree 0. Thus, there exists a finer grading of the Lie algebra by where , (the element of the canonical basis), -th and as well. Let be the subset of all nonzero degrees whose homogeneous components are nontrivial.
For example we can consider the finite Cartan matrices over since the entries of these Cartan matrices are integer numbers, and show that the associated Lie algebras are finite-dimensional. But, even for contragredient Lie algebras, there are significant differences with the case of complex numbers. As shown in ,VK71, there are examples of finite-dimensional Lie algebras with diagonal entries and two different matrices can give place to isomorphic contragredient Lie algebras. The classification shown in ,VK71 was incomplete: there was a missing example for the 29-dimensional Brown algebra , discovered by Brown in the eighties, whose realization as contragredient Lie algebra with two different matrices was shown in ,Skr93:
The expression of is close to that for the action of reflections of the Weyl group on complex Lie algebras, but here relates two “different” contragredient data.
3.2. Lie superalgebras
Recall that a Lie superalgebra is a vector space -graded ( is the even part and is the odd part) together with a linear map -graded satisfying analogous versions of antisymmetry and Jacobi identity:
for all homogeneous elements see, e.g., ,Kac77. Here, denotes the degree of We have examples from associative algebras, analogous to those of Lie algebras: given . a associative algebra, set -graded
In particular we have, for the Lie superalgebra , with ,
For each set the super trace of , We can consider the subalgebra .
Here we consider contragredient data where , is still the matrix of scalars determining the action of the generators on the remaining generators, and gives the -grading: for all Notice that all . are necessarily even.
As observed in Kac77 when different contragredient data can give isomorphic Lie superalgebras. But, similar to Lie algebras in positive characteristic, we can give isomorphisms between some pairs , , with formulas close to the action of the Weyl group for complex simple Lie algebras. In this direction, Serganova Ser96 introduced the notion of odd reflection relating two different pairs by a kind of reflection but on a simple odd root such that (called isotropic). This is consistent with one of the differences with Lie algebras: there exists a symmetric bilinear form on but either the bilinear form can have isotropic roots , (i.e., or else the matrix ) can take nonintegral values. We have to distinguish the matrix from the GCM responsible for the odd reflections.
If we study Lie superalgebras over fields of positive characteristic, then we can have more and more exceptional examples. Finite-dimensional contragredient Lie algebras over fields of prime characteristic were classified in BGL09. The picture is the same: several pairs of contragredient data give isomorphic Lie superalgebras.
The question is then how to handle uniformly all possible pairs giving isomorphic Lie superalgebras, and their corresponding roots (i.e., the of the nontrivial components). This will be done with a groupoid, i.e., a category where all the morphisms are invertible. As we look for a generalization of the Weyl group, we will consider a groupoid generated by reflections. -degrees
3.3. Generalized root systems
There exist different notions of generalized root systems in the literature. They try to capture different situations, as for example the one by Serganova in Ser96 for complex finite-dimensional Lie superalgebras. A nice axiomatic version was given in HY08, see also HS20 for a refined version of these ideas.
Fix , Let . a set (which will correspond to the different contragredient data). A semi-Cartan graph ( for short) of rank over consist of
- •
functions , such that , ,
- •
GCM , ,
such that for all , .
As for Lie algebras, we set as the reflection .
Let be a monoid. There exists a small category whose set of objects is and the set of morphisms between any two objects is Given . and we write for viewed as an element of so the composition becomes ,
for any , .
We are interested in the case the group of automorphisms of , .
Notice that is indeed a groupoid, since
Fix Then . is the set of all elements of the form where , , , This is the set of real roots of . As for roots of Lie (super)algebras, we consider the subsets .
of positive and negative real roots, and set
If is finite for all (equivalently, for some then we say that ), is finite.
A semi-Cartan graph is a Cartan graph if the following conditions hold for all :
- •
;
- •
for all such that , .
Mimicking what happens for Lie algebras, see Remarks 3.1 and 3.3, we introduce the following notion:
Cuntz and Heckenberger obtained the classification of finite root systems CH15. The proof involves the bijection between root systems of rank and crystallographic arrangements (certain subsets of hyperplanes) in About the list of finite root systems, in rank . there are infinitely many examples, in bijection with triangulations of for any -gons For . we only have families corresponding to Lie superalgebras and Lie algebras of types while for , we have members of the families of Lie (super)algebras and several exceptions.
The definition of a root system seems to carry the possibility to have several examples attached to the same Cartan graph But this is not the case when . is finite. Indeed, by HS20, 10.4.7, if is a finite Cartan graph, then is the only reduced root system over .
Once we show the existence of a finite root system for a Lie superalgebra, there are many strong properties derived from the combinatorics of the Weyl groupoid. For example:
- •
for all .
- •
There might exist roots such that which are the odd nonisotropic roots. All of them are the image of simple odd nonisotropic roots of some pair , obtained up to odd reflections, and as shown by Andruskiewitsch-Angiono. ,
- •
The whole set is obtained up to reflections of the simple roots, attaching for each odd nonisotropic root.
In the same line we may wonder if there exists a geometric-combinatoric side on these Lie superalgebras (Weyl chambers and so on) coming from the associated crystallographic arrangements.
4. Other Contexts and Problems
We finish by recalling other algebraic structures where these generalized root systems appear, as Nichols algebras, and posing some related problems where they could play a key role: Lie algebras in a broad sense and their representations. We are not going to introduce all the involved concepts, we refer to the corresponding papers for more information.
4.1. Nichols algebras
Quantized enveloping algebras are certain deformations of enveloping algebras of semisimple Lie algebras introduced in the eighties by Drinfeld and Jimbo, depending on a parameter Later on, Lusztig considered Hopf algebras obtained by evaluation of . at a root of unity, which lead to some finite-dimensional examples, usually called Frobenius-Lusztig kernels. These examples have a triangular decomposition whose zero part is a group algebra of copies of finite cyclic groups, and the positive (also, the negative) part is a kind of Hopf algebra.
In the denomination currently used, these positive parts are examples of Nichols algebras. Nichols algebras are Hopf algebras in the category of Yetter-Drinfeld modules over a group algebra (or more precisely, over a Hopf algebra), which play a fundamental role in the classification of finite-dimensional Hopf algebras. Following the line of work of Andruskiewitsch and Schneider, joined by Heckenberger, one can define Hopf algebras with triangular decomposition, whose positive part is a Nichols algebra and which have generalised root systems, see the book HS20, and also AHS10.
The list of all finite-dimensional Nichols algebras is known when the group is finite and abelian, thanks to the work of Heckenberger, and almost complete when the group is finite but nonabelian, by Heckenberger-Vendramin. Both works explode the existence of the generalized root system.
We can see that the list of all generalized root systems appearing for some Nichols algebras contains properly the list of all those appearing for Lie superalgebras, but there are some root systems not attached to any Nichols algebras.
4.2. Lie algebras in symmetric tensor categories and representations
One can extend the definition of Lie algebra to symmetric tensor categories. Indeed Lie superalgebras are essentially Lie algebras in the category of super vector spaces. When is of characteristic zero, Deligne proved that any symmetric tensor category (under a mild condition) fibers over the category of supervector spaces, so any Lie algebra over these symmetric tensor categories can be considered as a Lie superalgebra. When is of characteristic Coulembier-Etingof-Ostrik proved recently ,CEO23 that any symmetric tensor category (under a mild condition) fibers over the Verlinde category This category is the semisimplification of the category of representations of . over and contains properly the category of super vector spaces. Thus, in this case, the consideration of Lie algebras in symmetric tensor categories essentially reduces to Lie algebras in One may ask about the existence of contragredient Lie algebras in . and root systems. ,
In the classical case (that is, over the root system controls the representation theory of simple Lie algebras, or more precisely a quite interesting subcategory called the category ), For example, finite-dimensional modules are parametrized by nonnegative weights associated to the root system, and the Weyl group describes a character formula for these simple modules. The situation is a bit more complicated for Lie superalgebras, and a character formula exists for certain weights. Recently Sergeev and Veselov used what they called a Weyl groupoid (which is not clearly related to the one considered here) to describe strong properties on the representations. Also, Yamane described very recently character formulas for the so-called atypical weights of quantized enveloping Lie superalgebras by means of the Weyl groupoid. So one may wonder if the Weyl groupoid plays a key role in the description of the representations of Lie algebras in a broad sense. .
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Credits
Figures 1 and 2 and author photo are courtesy of Iván Angiono.