Glossary of Lie algebras
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This is a glossary for the terminology applied in the mathematical theories of Lie algebras. The statements in this glossary mainly focus on the algebraic sides of the concepts, without referring to Lie groups or other related subjects.
Definition
- Lie algebra
- A vector space over a field with a binary operation [·, ·] (called the Lie bracket or abbr. bracket) , which satisfies the following conditions: ,
- associative algebra
- An associative algebra can be made to a Lie algebra by defining the bracket (the commutator of ) .
- homomorphism
- A vector space homomorphism is said to be a Lie algebra homomorphism if
- adjoint representation
- Given , define map by
- is a Lie algebra derivation. The map
- thus defined is a Lie algebra homomorphism.
- is called adjoint representation.
- Jacobi identity
- The identity [[x, y], z] + [[y, z], x] + [[z, x], y] = 0.
- To say Jacobi identity holds in a vector space is equivalent to say adjoint of all elements are derivations : .
subalgebras
- subalgebra
- A subspace of a Lie algebra is called the subalgebra of if it is closed under bracket, i.e.
- ideal
- A subspace of a Lie algebra is the ideal of if
- In particular, every ideal is also a subalgebra. Every kernel of a Lie algebra homomorphism is an ideal. Unlike in ring theory, there is no distinguishability of left ideal and right ideal.
- derived algebra
- The derived algebra of a Lie algebra is . It is a subalgebra.
- normalizer
- The normalizer of a subspace of a Lie algebra is .
- centralizer
- The centralizer of a subset of a Lie algebra is .
- radical
- The radical is the maximum solvable ideal of .
Solvability, nilpotency, Jordan decomposition, semisimplicity
- abelian
- A Lie algebra is said to be abelian if its derived algebra is zero.
- nilpotent Lie algebra
- A Lie algebra is said to be nilpotent if for some positive integer .
- The following conditions are equivalent:
- for some positive integer , i.e. the descending central series eventually terminates to .
- for some positive integer N, i.e. the ascending central series eventually terminates to L.
- There exists a chain of ideals of , , such that .
- There exists chain of ideals of ,, such that .
- is nilpotent . (Engel's theorem)
- is a nilpotent Lie algebra.
- In particular, every nilpotent Lie algebra is solvable.
- solvable Lie algebra
- A Lie algebra is said to be solvable if for some positive integer , i.e. the derived series eventually terminates to .
- The following condition is equivalent to solvability:
- * There exists chain of ideals of , , such that .
- If is solvable, any subalgebra and quotient of are solvable.
- Let is an ideal of a Lie algebra . If are solvable, is solvable.
- simple
- A Lie algebra is said to be simple if it is non-abelian and has only two ideals, itself and .
- semisimple Lie algebra
- A Lie algebra is said to be semisimple if its radical is .
- semisimple element in a Lie algebra
- split Lie algebra
- free Lie algebra
- toral Lie algebra
- 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 .
- Corollary: There exists a basis of with respect to which all elements of are upper triangular.
- Killing form
- The Killing form on a Lie algebra is a symmetric, associative, bilinear form defined by .
- Cartan criterion for solvability
- A Lie algebra is solvable iff .
- Cartan criterion for semisimplity
- If is nondegenerate, then is semisimple.
- If is semisimple and the underlying field has characteristic 0 , then is nondegenerate.
- lower central series
- synonymous to "descending central series".
- upper central series
- synonymous to "ascending central series".
Semisimple Lie algebra
Root System (for classification of semisimple Lie algebra)
- In the below section, denote as the inner product of a Euclidean space E.
- In the below section, denoted the function defined as .
- Cartan subalgebra
- A Cartan subalgebra of a Lie algebra is a nilpotent subalgebra satisfying .
- maximal toral subalgebra
- root of a semisimple Lie algebra
- Let be a semisimple Lie algebra, be a Cartan subalgebra of . For , let . \alpha is called a root of if it is nonzero and
- The set of all roots is denoted by ; it forms a root system.
- Root system
- A subset of the Euclidean space is called a root system if it satisfies the following conditions:
- Cartan matrix
- Cartan matrix of root system is matrix where is a set of simple roots of .
- Dynkin diagrams
- Simple Roots
- A subset of a root system is called a set of simple roots if it satisfies the following conditions:
- 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
- positive roots
- 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.
- negative roots
- 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.
- long root
- short root
- 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
- 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 .
- base of a root system
- synonymous to "set of simple roots"
- dual of a root system
- synonymous to "inverse of a root system"
theory of weights
<--
- weight in a root system
- is called a weight if . -->
- weight lattice
- weight space
- fundamental dominant weight
- Given a set of simple roots , it is a basis of . is a basis of too; the dual basis defined by , is called the fundamental dominant weights.
- highest weight
- minimal weight
- multiplicity (of weight)
- radical weight
- strongly dominant weight
Representation theory
- module
- Define an action of on a vector space ( i.e. an operation ) such that: satisfy
- #
- #
- #
- Then is called a -module. (Remark: have the same underlying field .)
- Each -module corresponds to a representation .
- A subspace W is a submodule (more precisely, sub -module) of if -module .
- representation
- For a vector space , if there is a Lie algebra homomorphism , then is called a representation of .
- Each representation corresponds to a -module .
- A subrepresentation is the representation corresponding to a submodule.
- homomorphism
- Given two -module V, W, a -module homomorphism is a vector space homomorphism satisfying .
- trivial representation
- A representation is said to be trivial if the image of is the zero vector space. It corresponds to the action of on module by .
- tautology representation
- If a Lie algebra is defined as a subalgebra of , like (the upper triangular matrices), the tautology representation is the imbedding . It corresponds to the action on module by the matrix multipation.
- adjoint representation
- The representation . It corresponds viewing as a -module - the action on the module is given by the adjoint endomorphism.
- irreducible modules
- A module is said to be irreducible if it has only two submodules, itself and zero.
- indecomposable module
- A module is said to be indecomposable if it cannot be written as direct sum of two non-zero submodules.
- An irreducible module need not be indecmoposable but the converse is not true.
- completely reducible module
- A module is said to be completely reducible if it can be written as direct sum of irreducible modules.
- simple module
- Synonymous as irreduible module.
- quotient module / quotient representation
- Given a -module V and its submodule W, an action on V/W can be defined by . V/W is said to be a quotient module in this case.
- Schur's lemma
- Statement in the language of module theory: Given V an irreducible -module, is a -module homomorphism iff for some .
- Statement in the language of representation theory: Given an irreducible representation , for , iff for some .
- simple module
- synonymous to "irreduible module".
- factor module
- synonymous to "quotient module".
Universal enveloping algebras
PBW theorem (Poincaré–Birkhoff–Witt theorem)
Verma modules
BGG category \mathcal{O}
cohomology
Chevalley basis
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.
The generators of a Lie group are split into the generators H and E such that:
where p = m if β + γ is a root and m is the greatest positive integer such that γ − mβ is a root.
Examples of Lie algebra
- Ado's theorem
- Any finite-dimensional Lie algebra is isomorphic to a subalgebra of for some finite-dimensional vector space V.
complex Lie algebras of 1D, 2D, 3D
Simple Algebras
Classical Lie algebras:
Name | Root System | dimension | construction as subalgebra of |
---|---|---|---|
Special linear algebra | (traceless matrices) | ||
Orthogonal algebra | |||
Symplectic algebra | |||
Orthogonal algebra |
Exceptional Lie algebras:
Root System | dimension |
---|---|
G_{2} | 14 |
F_{4} | 52 |
E_{6} | 78 |
E_{7} | 133 |
E_{8} | 248 |
Miscellaneous
- Lie group
- Glossary of semisimple groups
- Linear algebraic group
- Particle physics and representation theory
References
- 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