Lie groups |
---|

In mathematics, in particular the theory of Lie algebras, the **Weyl group** (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that *most* finite reflection groups are Weyl groups.^{ [1] } Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

- Definition and examples
- Weyl chambers
- Coxeter group structure
- Generating set
- Relations
- As a Coxeter group
- Weyl groups in algebraic, group-theoretic, and geometric settings
- The Weyl group of a connected compact Lie group
- In other settings
- Bruhat decomposition
- Analogy with algebraic groups
- Cohomology
- See also
- Footnotes
- Notes
- Citations
- References
- Further reading
- External links

The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.

Let be a root system in a Euclidean space . For each root , let denote the reflection about the hyperplane perpendicular to , which is given explicitly as

- ,

where is the inner product on . The Weyl group of is the subgroup of the orthogonal group generated by all the 's. By the definition of a root system, each preserves , from which it follows that is a finite group.

In the case of the root system, for example, the hyperplanes perpendicular to the roots are just lines, and the Weyl group is the symmetry group of an equilateral triangle, as indicated in the figure. As a group, is isomorphic to the permutation group on three elements, which we may think of as the vertices of the triangle. Note that in this case, is not the full symmetry group of the root system; a 60-degree rotation preserves but is not an element of .

We may consider also the root system. In this case, is the space of all vectors in whose entries sum to zero. The roots consist of the vectors of the form , where is the th standard basis element for . The reflection associated to such a root is the transformation of obtained by interchanging the th and th entries of each vector. The Weyl group for is then the permutation group on elements.

If is a root system, we may consider the hyperplane perpendicular to each root . Recall that denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of generated by all the 's. The complement of the set of hyperplanes is disconnected, and each connected component is called a **Weyl chamber**. If we have fixed a particular set Δ of simple roots, we may define the **fundamental Weyl chamber** associated to Δ as the set of points such that for all .

Since the reflections preserve , they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.

The figure illustrates the case of the A2 root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.

A basic general theorem about Weyl chambers is this:^{ [2] }

**Theorem**: The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl chambers.

A related result is this one:^{ [3] }

**Theorem**: Fix a Weyl chamber . Then for all , the Weyl-orbit of contains exactly one point in the closure of .

A key result about the Weyl group is this:^{ [4] }

**Theorem**: If is base for , then the Weyl group is generated by the reflections with in .

That is to say, the group generated by the reflections is the same as the group generated by the reflections .

Meanwhile, if and are in , then the Dynkin diagram for relative to the base tells us something about how the pair behaves. Specifically, suppose and are the corresponding vertices in the Dynkin diagram. Then we have the following results:

- If there is no bond between and , then and commute. Since and each have order two, this is equivalent to saying that .
- If there is one bond between and , then .
- If there are two bonds between and , then .
- If there are three bonds between and , then .

The preceding claim is not hard to verify, if we simply remember what the Dynkin diagram tells us about the angle between each pair of roots. If, for example, there is no bond between the two vertices, then and are orthogonal, from which it follows easily that the corresponding reflections commute. More generally, the number of bonds determines the angle between the roots. The product of the two reflections is then a rotation by angle in the plane spanned by and , as the reader may verify, from which the above claim follows easily.

Weyl groups are examples of finite reflection groups, as they are generated by reflections; the abstract groups (not considered as subgroups of a linear group) are accordingly finite Coxeter groups, which allows them to be classified by their Coxeter–Dynkin diagram. Being a Coxeter group means that a Weyl group has a special kind of presentation in which each generator *x _{i}* is of order two, and the relations other than

Weyl groups have a Bruhat order and length function in terms of this presentation: the * length * of a Weyl group element is the length of the shortest word representing that element in terms of these standard generators. There is a unique longest element of a Coxeter group, which is opposite to the identity in the Bruhat order.

Above, the Weyl group was defined as a subgroup of the isometry group of a root system. There are also various definitions of Weyl groups specific to various group-theoretic and geometric contexts (Lie algebra, Lie group, symmetric space, etc.). For each of these ways of defining Weyl groups, it is a (usually nontrivial) theorem that it is a Weyl group in the sense of the definition at the top of this article, namely the Weyl group of some root system associated with the object. A concrete realization of such a Weyl group usually depends on a choice – e.g. of Cartan subalgebra for a Lie algebra, of maximal torus for a Lie group.^{ [5] }

Let be a connected compact Lie group and let be a maximal torus in . We then introduce the **normalizer** of in , denoted and defined as

- .

We also define the **centralizer** of in , denoted and defined as

- .

The Weyl group of (relative to the given maximal torus ) is then defined initially as

- .

Eventually, one proves that ,^{ [6] } at which point one has an alternative description of the Weyl group as

- .

Now, one can define a root system associated to the pair ; the roots are the nonzero weights of the adjoint action of on the Lie algebra of . For each , one can construct an element of whose action on has the form of reflection.^{ [7] } With a bit more effort, one can show that these reflections generate all of .^{ [6] } Thus, in the end, the Weyl group as defined as or is isomorphic to the Weyl group of the root system .

For a complex semisimple Lie algebra, the Weyl group is simply *defined* as the reflection group generated by reflections in the roots – the specific realization of the root system depending on a choice of Cartan subalgebra.

For a Lie group *G* satisfying certain conditions,^{ [note 1] } given a torus *T* < *G* (which need not be maximal), the Weyl group *with respect to* that torus is defined as the quotient of the normalizer of the torus *N* = *N*(*T*) = *N _{G}*(

The group *W* is finite – *Z* is of finite index in *N*. If *T* = *T*_{0} is a maximal torus (so it equals its own centralizer: ) then the resulting quotient *N*/*Z* = *N*/*T* is called *the* Weyl group of *G*, and denoted *W*(*G*). Note that the specific quotient set depends on a choice of maximal torus, but the resulting groups are all isomorphic (by an inner automorphism of *G*), since maximal tori are conjugate.

If *G* is compact and connected, and *T* is a *maximal* torus, then the Weyl group of *G* is isomorphic to the Weyl group of its Lie algebra, as discussed above.

For example, for the general linear group *GL,* a maximal torus is the subgroup *D* of invertible diagonal matrices, whose normalizer is the generalized permutation matrices (matrices in the form of permutation matrices, but with any non-zero numbers in place of the '1's), and whose Weyl group is the symmetric group. In this case the quotient map *N* → *N*/*T* splits (via the permutation matrices), so the normalizer *N* is a semidirect product of the torus and the Weyl group, and the Weyl group can be expressed as a subgroup of *G*. In general this is not always the case – the quotient does not always split, the normalizer *N* is not always the semidirect product of *W* and *Z,* and the Weyl group cannot always be realized as a subgroup of *G.*^{ [5] }

If *B* is a Borel subgroup of *G*, i.e., a maximal connected solvable subgroup and a maximal torus *T* = *T*_{0} is chosen to lie in *B*, then we obtain the Bruhat decomposition

which gives rise to the decomposition of the flag variety *G*/*B* into **Schubert cells** (see Grassmannian).

The structure of the Hasse diagram of the group is related geometrically to the cohomology of the manifold (rather, of the real and complex forms of the group), which is constrained by Poincaré duality. Thus algebraic properties of the Weyl group correspond to general topological properties of manifolds. For instance, Poincaré duality gives a pairing between cells in dimension *k* and in dimension *n* - *k* (where *n* is the dimension of a manifold): the bottom (0) dimensional cell corresponds to the identity element of the Weyl group, and the dual top-dimensional cell corresponds to the longest element of a Coxeter group.

There are a number of analogies between algebraic groups and Weyl groups – for instance, the number of elements of the symmetric group is *n*!, and the number of elements of the general linear group over a finite field is related to the *q*-factorial ; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Weyl groups to be simple algebraic groups over the field with one element.

For a non-abelian connected compact Lie group *G,* the first group cohomology of the Weyl group *W* with coefficients in the maximal torus *T* used to define it,^{ [note 2] } is related to the outer automorphism group of the normalizer as:^{ [8] }

The outer automorphisms of the group Out(*G*) are essentially the diagram automorphisms of the Dynkin diagram, while the group cohomology is computed in Hämmerli, Matthey & Suter 2004 and is a finite elementary abelian 2-group (); for simple Lie groups it has order 1, 2, or 4. The 0th and 2nd group cohomology are also closely related to the normalizer.^{ [8] }

- ↑ Different conditions are sufficient – most simply if
*G*is connected and either compact, or an affine algebraic group. The definition is simpler for a semisimple (or more generally reductive) Lie group over an algebraically closed field, but a*relative*Weyl group can be defined for a*split*Lie group. - ↑
*W*acts on*T*– that is how it is defined – and the group means "with respect to this action".

In mathematics, the **orthogonal group** in dimension *n*, denoted O(*n*), is the group of distance-preserving transformations of a Euclidean space of dimension *n* that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the **general orthogonal group**, by analogy with the general linear group. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

In mathematics, a **root system** is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in spectral graph theory.

In mathematics, a **Coxeter group**, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the **maximal torus** subgroups.

In mathematics, an **algebraic torus**, where a one dimensional torus is typically denoted by , , or , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups . These groups were named by analogy with the theory of *tori* in Lie group theory. For example, over the complex numbers the algebraic torus is isomorphic to the group scheme , which is the scheme theoretic analogue of the Lie group . In fact, any -action on a complex vector space can be pulled back to a -action from the inclusion as real manifolds.

In mathematics, a **compact** (**topological**) **group** is a topological group whose topology is compact. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

In mathematics, a **reductive group** is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group *G* over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group *GL*(*n*) of invertible matrices, the special orthogonal group *SO*(*n*), and the symplectic group *Sp*(2*n*). **Simple algebraic groups** and **semisimple algebraic groups** are reductive.

In mathematics, a **Cartan subalgebra**, often abbreviated as **CSA**, is a nilpotent subalgebra of a Lie algebra that is self-normalising. They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic .

In mathematics, a Lie algebra is **semisimple** if it is a direct sum of simple Lie algebras.

**Verma modules**, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.

In mathematical group theory, the **root datum** of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970.

In mathematics, a **zonal spherical function** or often just **spherical function** is a function on a locally compact group *G* with compact subgroup *K* that arises as the matrix coefficient of a *K*-invariant vector in an irreducible representation of *G*. The key examples are the matrix coefficients of the *spherical principal series*, the irreducible representations appearing in the decomposition of the unitary representation of *G* on *L*^{2}(*G*/*K*). In this case the commutant of *G* is generated by the algebra of biinvariant functions on *G* with respect to *K* acting by right convolution. It is commutative if in addition *G*/*K* is a symmetric space, for example when *G* is a connected semisimple Lie group with finite centre and *K* is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant *L*^{1} functions is larger; when *G* is a semisimple Lie group with maximal compact subgroup *K*, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

In mathematics, the **Plancherel theorem for spherical functions** is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space *X*; it also gives the direct integral decomposition into irreducible representations of the regular representation on L^{2}(*X*). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.

**Representation theory** is a branch of mathematics that studies abstract algebraic structures by *representing* their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.

In mathematics, the **Kostant polynomials**, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the finite reflection group of a root system.

In mathematics, a **regular element** of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible.

In algebra, the **Nichols algebra** of a braided vector space is a braided Hopf algebra which is denoted by and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed Hopf algebra such as a quantum groups and their well known finite-dimensional truncations. Nichols algebras can immediately be used to write down new such quantum groups by using the Radford biproduct.

In mathematics, **Borel–de Siebenthal theory** describes the closed connected subgroups of a compact Lie group that have *maximal rank*, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer of its center. They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.

In mathematics, the **complexification** or **universal complexification** of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

- Hall, Brian C. (2015),
*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*, Graduate Texts in Mathematics,**222**(2nd ed.), Springer, ISBN 978-3-319-13466-6 - Knapp, Anthony W. (2002),
*Lie Groups: Beyond an Introduction*, Progress in Mathematics,**140**(2nd ed.), Birkhaeuser, ISBN 978-0-8176-4259-4 - Popov, V.L.; Fedenko, A.S. (2001) [1994], "Weyl group",
*Encyclopedia of Mathematics*, EMS Press - Hämmerli, J.-F.; Matthey, M.; Suter, U. (2004), "Automorphisms of Normalizers of Maximal Tori and First Cohomology of Weyl Groups" (PDF),
*Journal of Lie Theory*, Heldermann Verlag,**14**: 583–617, Zbl 1092.22004

- Bourbaki, Nicolas (2002),
*Lie Groups and Lie Algebras: Chapters 4-6*, Elements of Mathematics, Springer, ISBN 978-3-540-42650-9, Zbl 0983.17001 - Björner, Anders; Brenti, Francesco (2005),
*Combinatorics of Coxeter Groups*, Graduate Texts in Mathematics,**231**, Springer, ISBN 978-3-540-27596-1, Zbl 1110.05001 - Coxeter, H. S. M. (1934), "Discrete groups generated by reflections",
*Ann. of Math.*,**35**(3): 588–621, CiteSeerX 10.1.1.128.471 , doi:10.2307/1968753, JSTOR 1968753 - Coxeter, H. S. M. (1935), "The complete enumeration of finite groups of the form ",
*J. London Math. Soc.*, 1,**10**(1): 21–25, doi:10.1112/jlms/s1-10.37.21 - Davis, Michael W. (2007),
*The Geometry and Topology of Coxeter Groups*(PDF), ISBN 978-0-691-13138-2, Zbl 1142.20020 - Grove, Larry C.; Benson, Clark T. (1985),
*Finite Reflection Groups*, Graduate texts in mathematics,**99**, Springer, ISBN 978-0-387-96082-1 - Hiller, Howard (1982),
*Geometry of Coxeter groups*, Research Notes in Mathematics,**54**, Pitman, ISBN 978-0-273-08517-1, Zbl 0483.57002 - Howlett, Robert B. (1988), "On the Schur Multipliers of Coxeter Groups",
*J. London Math. Soc.*, 2,**38**(2): 263–276, doi:10.1112/jlms/s2-38.2.263, Zbl 0627.20019 - Humphreys, James E. (1992) [1990],
*Reflection Groups and Coxeter Groups*, Cambridge Studies in Advanced Mathematics,**29**, Cambridge University Press, ISBN 978-0-521-43613-7, Zbl 0725.20028 - Ihara, S.; Yokonuma, Takeo (1965), "On the second cohomology groups (Schur-multipliers) of finite reflection groups" (PDF),
*J. Fac. Sci. Univ. Tokyo, Sect. 1*,**11**: 155–171, Zbl 0136.28802 - Kane, Richard (2001),
*Reflection Groups and Invariant Theory*, CMS Books in Mathematics, Springer, ISBN 978-0-387-98979-2, Zbl 0986.20038 - Vinberg, E. B. (1984), "Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension",
*Trudy Moskov. Mat. Obshch.*,**47** - Yokonuma, Takeo (1965), "On the second cohomology groups (Schur-multipliers) of infinite discrete reflection groups",
*J. Fac. Sci. Univ. Tokyo, Sect. 1*,**11**: 173–186, hdl:2261/6049, Zbl 0136.28803

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.