Glossary of Lie groups and Lie algebras

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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.

Notations:

  • Throughout the glossary, denotes the inner product of a Euclidean space E and denotes the rescaled inner product
  • abelian
    1.  An abelian Lie group is a Lie group that is an abelian group.
    2.  An abelian Lie algebra is a Lie algebra such that   for every   in the algebra.
    adjoint
    1.  An adjoint representation of a Lie group:
     
    such that   is the differential at the identity element of the conjugation  .
    2.  An adjoint representation of a Lie algebra is a Lie algebra representation
      where  .
    Ado
    Ado's theorem: Any finite-dimensional Lie algebra is isomorphic to a subalgebra of   for some finite-dimensional vector space V.
    affine
    1.  An affine Lie algebra is a particular type of Kac–Moody algebra.
    2.  An affine Weyl group.
    analytic
    1.  An analytic subgroup
    automorphism
    1.  An automorphism of a Lie algebra is a linear automorphism preserving the bracket.
    B
    1.  (B, N) pair
    Borel
    1.  Armand Borel (1923 – 2003), a Swiss mathematician
    2.  A Borel subgroup.
    3.  A Borel subalgebra is a maximal solvable subalgebra.
    4.  Borel-Bott-Weil theorem
    Bruhat
    1.  Bruhat decomposition
    Cartan
    1.  Élie Cartan (1869 – 1951), a French mathematician
    2.  A Cartan subalgebra   of a Lie algebra   is a nilpotent subalgebra satisfying  .
    3.  Cartan criterion for solvability: A Lie algebra   is solvable iff  .
    4.  Cartan criterion for semisimplicity: (1) If   is nondegenerate, then   is semisimple. (2) If   is semisimple and the underlying field   has characteristic 0 , then   is nondegenerate.
    5.  The Cartan matrix of the root system   is the matrix  , where   is a set of simple roots of  .
    6.  Cartan subgroup
    7.  Cartan decomposition
    Casimir
    Casimir invariant, a distinguished element of a universal enveloping algebra.
    Clebsch–Gordan coefficients
    Clebsch–Gordan coefficients
    center
    2.  The centralizer of a subset   of a Lie algebra   is  .
    center
    1.  The center of a Lie group is the center of the group.
    2.  The center of a Lie algebra is the centralizer of itself :  
    central series
    1.  A descending central series (or lower central series) is a sequence of ideals of a Lie algebra   defined by  
    2.  An ascending central series (or upper central series) is a sequence of ideals of a Lie algebra   defined by   (center of L) ,  , where   is the natural homomorphism  
    Chevalley
    1.  Claude Chevalley (1909 – 1984), a French mathematician
    2.  A Chevalley basis is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups.
    complex reflection group
    complex reflection group
    coroot
    coroot
    Coxeter
    1.  H. S. M. Coxeter (1907 – 2003), a British-born Canadian geometer
    2.  Coxeter group
    3.  Coxeter number
    derived algebra
    1.  The derived algebra of a Lie algebra   is  . It is a subalgebra (in fact an ideal).
    2.  A derived series is a sequence of ideals of a Lie algebra   obtained by repeatedly taking derived algebras; i.e.,  .
    Dynkin
    1.  Eugene Borisovich Dynkin (1924 – 2014), a Soviet and American mathematician
    2.  
     
    Dynkin diagrams
    Dynkin diagrams.
    extension
    An exact sequence   or   is called a Lie algebra extension of   by  .
    exponential map
    The exponential map for a Lie group G with   is a map   which is not necessarily a homomorphism but satisfies a certain universal property.
    exponential
    E6, E7, E7½, E8, En, Exceptional Lie algebra
    free Lie algebra
    F
    F4
    fundamental
    For "fundamental Weyl chamber", see #Weyl.
    G
    G2
    generalized
    1.  For "Generalized Cartan matrix", see #Cartan.
    2.  For "Generalized Kac–Moody algebra", see #Kac–Moody algebra.
    3.  For "Generalized Verma module", see #Verma.
    group
    Group analysis of differential equations.
    homomorphism
    1.  A Lie group homomorphism is a group homomorphism that is also a smooth map.
    2.  A Lie algebra homomorphism is a linear map   such that  
    Harish-Chandra
    1.  Harish-Chandra, (1923 – 1983), an Indian American mathematician and physicist
    2.  Harish-Chandra homomorphism
    3.  Harish-Chandra isomorphism
    highest
    1.  The theorem of the highest weight, stating the highest weights classify the irreducible representations.
    2.  highest weight
    3.  highest weight module
    ideal
    An ideal of a Lie algebra   is a subspace   such that   Unlike in ring theory, there is no distinguishability of left ideal and right ideal.
    index
    Index of a Lie algebra
    invariant convex cone
    An invariant convex cone is a closed convex cone in the Lie algebra of a connected Lie group that is invariant under inner automorphisms.
    Iwasawa decomposition
    Iwasawa decomposition
    Jacobi identity
    1.  
     
    Carl Gustav Jacob Jacobi
    Carl Gustav Jacob Jacobi (1804 – 1851), a German mathematician.
    2.  Given a binary operation  , the Jacobi identity states: [[x, y], z] + [[y, z], x] + [[z, x], y] = 0.
    Kac–Moody algebra
    Kac–Moody algebra
    Killing
    1.  Wilhelm Killing (1847 – 1923), a German mathematician.
    2.  The Killing form on a Lie algebra   is a symmetric, associative, bilinear form defined by  .
    Kirillov
    Kirillov character formula
    Langlands
    Langlands decomposition
    Langlands dual
    Lie
    1.  
     
    Sophus Lie
    Sophus Lie (1842 – 1899), a Norwegian mathematician
    2.  A Lie group is a group that has a compatible structure of a smooth manifold.
    3.  A Lie algebra is a vector space   over a field   with a binary operation [·, ·] (called the Lie bracket or abbr. bracket) , which satisfies the following conditions:  ,
    1.   (bilinearity)
    2.   (alternating)
    3.   (Jacobi identity)
    4.  Lie group–Lie algebra correspondence
    5.  Lie's theorem
    Let   be a finite-dimensional complex solvable Lie algebra over algebraically closed field of characteristic  , and let   be a nonzero finite dimensional representation of  . Then there exists an element of   which is a simultaneous eigenvector for all elements of  .
    6.  Compact Lie group.
    7.  Semisimple Lie group; see #semisimple.
    Levi
    Levi decomposition
    nilpotent
    1.  A nilpotent Lie group.
    2.  A nilpotent Lie algebra is a Lie algebra that is nilpotent as an ideal; i.e., some power is zero:  .
    3.  A nilpotent element of a semisimple Lie algebra[1] is an element x such that the adjoint endomorphism   is a nilpotent endomorphism.
    4.  A nilpotent cone
    normalizer
    The normalizer of a subspace   of a Lie algebra   is  .
    maximal
    1.  For "maximal compact subgroup", see #compact.
    2.  For "maximal torus", see #torus.
    parabolic
    1.  Parabolic subgroup
    2.  Parabolic subalgebra.
    positive
    For "positive root", see #positive.
    quantum
    quantum group.
    quantized
    quantized enveloping algebra.
    radical
    1.  The radical of a Lie group.
    2.  The radical of a Lie algebra   is the largest (i.e., unique maximal) solvable ideal of  .
    real
    real form.
    reductive
    1.  A reductive group.
    2.  A reductive Lie algebra.
    reflection
    A reflection group, a group generated by reflections.
    regular
    1.  A regular element of a Lie algebra.
    2.  A regular element with respect to a root system.
    Let   be a root system.   is called regular if  .
    For each set of simple roots   of  , there exists a regular element   such that  , conversely for each regular   there exist a unique set of base roots   such that the previous condition holds for  . It can be determined in following way: let  . Call an element   of   decomposable if   where  , then   is the set of all indecomposable elements of  
    root
    1.  root of a semisimple Lie algebra:
    Let   be a semisimple Lie algebra,   be a Cartan subalgebra of  . For  , let  .   is called a root of   if it is nonzero and  
    The set of all roots is denoted by   ; it forms a root system.
    2.  Root system
    A subset   of the Euclidean space   is called a root system if it satisfies the following conditions:
    •   is finite,   and  .
    • For all   and  ,   iff  .
    • For all  ,   is an integer.
    • For all  ,  , where   is the reflection through the hyperplane normal to  , i.e.  .
    3.  Root datum
    4.  Positive root of root system   with respect to a set of simple roots   is a root of   which is a linear combination of elements of   with nonnegative coefficients.
    5.  Negative root of root system   with respect to a set of simple roots   is a root of   which is a linear combination of elements of   with nonpositive coefficients.
    6.  long root
    7.  short root
    8.  inverse of a root system: Given a root system  . Define  ,   is called the inverse of a root system.
      is again a root system and have the identical Weyl group as  .
    9.  base of a root system: synonymous to "set of simple roots"
    10.  dual of a root system: synonymous to "inverse of a root system"
    Serre
    Serre's theorem states that, given a (finite reduced) root system  , there exists a unique (up to a choice of a base) semisimple Lie algebra whose root system is  .
    simple
    1.  A simple Lie group is a connected Lie group that is not abelian which does not have nontrivial connected normal subgroups.
    2.  A simple Lie algebra is a Lie algebra that is non abelian and has only two ideals, itself and  .
    3.  simply laced group (a simple Lie group is simply laced when its Dynkin diagram is without multiple edges).
    4.  simple root. A subset   of a root system   is called a set of simple roots if it satisfies the following conditions:
    •   is a linear basis of  .
    • Each element of   is a linear combination of elements of   with coefficients that are either all nonnegative or all nonpositive.
    5.  Classification of simple Lie algebras

    Classical Lie algebras:

    Special linear algebra       (traceless matrices)
    Orthogonal algebra      
    Symplectic algebra      
    Orthogonal algebra      

    Exceptional Lie algebras:

    Root System dimension
    G2 14
    F4 52
    E6 78
    E7 133
    E8 248
    semisimple
    1.  A semisimple Lie group
    2.  A semisimple Lie algebra is a nonzero Lie algebra that has no nonzero abelian ideal.
    3.  In a semisimple Lie algebra, an element is semisimple if its image under the adjoint representation is semisimple; see Semisimple Lie algebra#Jordan decomposition.
    solvable
    1.  A solvable Lie group
    2.  A solvable Lie algebra is a Lie algebra   such that   for some  ; where   denotes the derived algebra of  .
    split
    Stiefel
    Stiefel diagram of a compact connected Lie group.
    subalgebra
    A subspace   of a Lie algebra   is called the subalgebra of   if it is closed under bracket, i.e.  
    Tits
    Tits cone.
    toral
    1.  toral Lie algebra
    2.  maximal toral subalgebra
    Weyl
    1.  Hermann Weyl (1885 – 1955), a German mathematician
    2.  A Weyl chamber is one of the connected components of the complement in V, a real vector space on which a root system is defined, when the hyperplanes orthogonal to the root vectors are removed.
    3.  The Weyl character formula gives in closed form the characters of the irreducible complex representations of the simple Lie groups.
    4.  Weyl group: Weyl group of a root system   is a (necessarily finite) group of orthogonal linear transformations of   which is generated by reflections through hyperplanes normal to roots of  

    References

    edit
    1. ^ Editorial note: the definition of a nilpotent element in a general Lie algebra seems unclear.
    • Bourbaki, N. (1981), Groupes et Algèbres de Lie, Éléments de Mathématique, Hermann
    • Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
    • Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5
    • Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
    • Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8.
    • Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.
    • Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4.
    • J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French)