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[edit]

Lie algebra
A vector space \mathfrak{g} over a field F with a binary operation [·, ·] (called the Lie bracket or abbr. bracket) , which satisfies the following conditions: \forall a,b \in F, x,y,z \in \mathfrak{g},
  1. [ax+by,z] = a[x,z] + b[y,z] (bilinearity)
  2. [x,x] = 0 (alternating)
  3. [[x,y], z ] + [[y,z],x] + [[z,x],y] = 0 (Jacobi identity)
associative algebra
An associative algebra A can be made to a Lie algebra by defining the bracket [x,y] = xy-yx (the commutator of x,y) \forall x,y \in A .
homomorphism 
A vector space homomorphism \phi : \mathfrak{g}_1 \to \mathfrak{g}_2 is said to be a Lie algebra homomorphism if \phi([x,y]) = [ \phi(x), \phi(y) ] \, \forall x,y \in \mathfrak{g}_1.
adjoint representation
Given x \in \mathfrak{g}, define map \textrm{ad}_x by
  \begin{align} \textrm{ad}_x : & \mathfrak{g} \to \mathfrak{g} \\ & y \mapsto [x,y]  \end{align}
\textrm{ad}_x is a Lie algebra derivation. The map
  \begin{align} \textrm{ad} : & \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) \\ & x \mapsto \mathrm{ad}_x  \end{align}
thus defined is a Lie algebra homomorphism.
\textrm{ad}: \mathfrak{g} \to \textrm{End}(\mathfrak{g}) 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 : \textrm{ad}_x([y,z]) = [ \textrm{ad}_x(y) , z] + [y , \textrm{ad}_x(z)] .

subalgebras[edit]

subalgebra
A subspace \mathfrak{g'} of a Lie algebra \mathfrak{g} is called the subalgebra of \mathfrak{g} if it is closed under bracket, i.e. [\mathfrak{g'}, \mathfrak{g'}] \subseteq \mathfrak{g'}.
ideal
A subspace \mathfrak{g'} of a Lie algebra \mathfrak{g} is the ideal of \mathfrak{g} if [\mathfrak{g'}, \mathfrak{g}] \subseteq \mathfrak{g'}.
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 \mathfrak{g} is [\mathfrak{g}, \mathfrak{g} ]. It is a subalgebra.
normalizer
The normalizer of a subspace K of a Lie algebra \mathfrak{g} is N_{\mathfrak{g}}(K) := \{x \in \mathfrak{g} | [x, K] \subseteq K \} .
centralizer
The centralizer of a subset X of a Lie algebra \mathfrak{g} is C_{\mathfrak{g}}(X) := \{x \in \mathfrak{g} | [x, X] = \{0\} \} .
center
The center of a Lie algebra is the centralizer of itself : Z(L) := \{x \in \mathfrak{g} | [x, \mathfrak{g}] = 0 \}
radical
The radical \textrm{Rad}(\mathfrak{g}) is the maximum solvable ideal of \mathfrak{g}.

Solvability, nilpotency, Jordan decomposition, semisimplicity[edit]

abelian
A Lie algebra is said to be abelian if its derived algebra is zero.
nilpotent Lie algebra
A Lie algebra L is said to be nilpotent if C^N(L) = \{0\} for some positive integer N .
The following conditions are equivalent:
  • C^N(L) = \{0\} for some positive integer N, i.e. the descending central series eventually terminates to \{0\}.
  • C_N(L) = L for some positive integer N, i.e. the ascending central series eventually terminates to L.
  • There exists a chain of ideals of L, L = I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots \supseteq I_n = \{0\}, such that [L, I_k] \subseteq I_{k+1} .
  • There exists chain of ideals of L, L = I_1 \supseteq I_2 \supseteq I_3 \cdots \supseteq I_n = \{0\}, such that I_k / I_{k+1} \subseteq Z( L/ I_{k+1} ).
  • \textrm{ad}\, x is nilpotent \forall x \in L . (Engel's theorem)
  • \textrm{ad}\, L is a nilpotent Lie algebra.
In particular, every nilpotent Lie algebra is solvable.
If L is nilpotent, any subalgebra and quotient of L are nilpotent.
nilpotent element in a Lie algebra
An element x \in L is said to be nilpotent in L if ad_x is a nilpotent endomorphism, i.e. viewing \textrm{ad}_x as a matrix in \mathfrak{gl}_\mathfrak{g}, \exists N \in \mathbb{Z}^+, (\textrm{ad}_x)^N = 0. It is equivalent to (\textrm{ad}_x)^N y = [x [x \ldots [x [x, y] \ldots] = 0\ \forall y \in L
descending central series
a sequence of ideals of a Lie algebra L defined by C^0(L) = L , \, C^1(L) = [L,L] , \, C^{n+1}(L) = [L, C^n(L)]
ascending central series
a sequence of ideals of a Lie algebra L defined by C_0(L) = L , \, C_1(L) = Z(L) (center of L) , C_{n+1}(L) = \pi_n^{-1} ( Z ( L / C_{n}(L) ) ), where \pi_i is the natural homomorphism L \to L/C_n(L)
solvable Lie algebra
A Lie algebra L is said to be solvable if L^{(N)} = 0 for some positive integer N , i.e. the derived series eventually terminates to \{0\}.
The following condition is equivalent to solvability:
* There exists chain of ideals of L, L = I_1 \supseteq I_2 \supseteq I_3 \cdots \supseteq I_n = \{ 0 \} , such that [I_k, I_k] \subseteq I_{k+1} .
If L is solvable, any subalgebra and quotient of L are solvable.
Let I is an ideal of a Lie algebra L. If L/I, I are solvable, L is solvable.
derived series
a sequence of ideals of a Lie algebra L defined by L^{(0)} = L , \, L^{(1)} = [L,L], \, L^{(n+1)} = [L^{(n)}, L^{(n)}]
simple
A Lie algebra is said to be simple if it is non-abelian and has only two ideals, itself and \{0\}.
semisimple Lie algebra
A Lie algebra is said to be semisimple if its radical is \{0\}.
semisimple element in a Lie algebra
split Lie algebra
free Lie algebra
toral Lie algebra
Lie's theorem
Let \mathfrak{g} be a finite-dimensional complex solvable Lie algebra over algebraically closed field of characteristic 0, and let V be a nonzero finite dimensional representation of \mathfrak{g}. Then there exists an element of V which is a simultaneous eigenvector for all elements of \mathfrak{g}.
Corollary: There exists a basis of V with respect to which all elements of \mathfrak{g} are upper triangular.
Killing form
The Killing form on a Lie algebra \mathfrak{g} is a symmetric, associative, bilinear form defined by \kappa(x, y) := \textrm{Tr}( \textrm{ad}\,x\, \textrm{ad}\, y )\  \forall x,y \in \mathfrak{g}.
Cartan criterion for solvability
A Lie algebra \mathfrak{g} is solvable iff \kappa( \mathfrak{g}, [\mathfrak{g},\mathfrak{g}] ) = 0.
Cartan criterion for semisimplity
If \kappa( \cdot, \cdot) is nondegenerate, then \mathfrak{g} is semisimple.
If \mathfrak{g} is semisimple and the underlying field F has characteristic 0 , then \kappa( \cdot, \cdot) is nondegenerate.
lower central series
synonymous to "descending central series".
upper central series
synonymous to "ascending central series".

Semisimple Lie algebra[edit]

Root System (for classification of semisimple Lie algebra)[edit]

In the below section, denote ( \cdot, \cdot ) as the inner product of a Euclidean space E.
In the below section, < \cdot, \cdot > denoted the function defined as < \beta, \alpha > = \frac{(\beta, \alpha)}{(\alpha, \alpha)} \, \forall \alpha, \beta \in E .
Cartan subalgebra
A Cartan subalgebra \mathfrak{h} of a Lie algebra \mathfrak{g} is a nilpotent subalgebra satisfying N_\mathfrak{g}(\mathfrak{h}) = \mathfrak{h} .
regular element of a Lie algebra
maximal toral subalgebra
Borel subalgebra
root of a semisimple Lie algebra
Let \mathfrak{g} be a semisimple Lie algebra, \mathfrak{h} be a Cartan subalgebra of \mathfrak{g}. For \alpha \in \mathfrak{h}^*, let \mathfrak{g_\alpha} := \{ x \in \mathfrak{g} | [h,x] = \alpha(h) x \, \forall h \in \mathfrak{h} \} . \alpha is called a root of \mathfrak{g} if it is nonzero and \mathfrak{g_\alpha} \ne \{0\}
The set of all roots is denoted by \Phi ; it forms a root system.
Root system
A subset \Phi of the Euclidean space E is called a root system if it satisfies the following conditions:
  • \Phi is finite, \textrm{span} (\Phi) = E and 0 \notin \Phi.
  • For all \alpha \in \Phi and c \in \mathbb{R}, c \alpha \in \Phi iff c = \pm 1.
  • For all \alpha,\beta \in \Phi, < \alpha, \beta > is an integer.
  • For all \alpha,\beta \in \Phi, S_\alpha(\beta)\in \Phi, where S_\alpha is reflection through hyperplane normal to \alpha i.e. S_\alpha(x) = x - <x , \alpha > \alpha.
Cartan matrix
Cartan matrix of root system \Phi is matrix ( < \alpha_i , \alpha_j > )_{i,j=1}^n where \Delta = \{\alpha_1 \ldots \alpha_n\} is a set of simple roots of \Phi.
Dynkin diagrams
Simple Roots
A subset \Delta of a root system \Phi is called a set of simple roots if it satisfies the following conditions:
  • \Delta is linear basis of E.
  • Each element of \Phi is a linear combination of elements of \Delta with coefficients which are either all nonnegative or all nonpositive.
a partial order on the Eucliean space E defined by the set of simple root
\forall \lambda, \mu \in E,  \,\, \lambda > \mu \iff \lambda - \mu > 0 \iff \,\, \, \exists k_1, k_2, ..., k_n \in \mathbb{Z}^+, \, \alpha_1, \alpha_2, ..., \alpha_n \in \Delta, \,\, \lambda - \mu = \sum_i k_i \alpha_i
regular element with respect to a root system
Let \Phi be a root system. \gamma \in E is called regular if (\gamma, \alpha) \ne 0 \, \forall \gamma \in \Phi.
For each set of simple roots \Delta of \Phi, there exists a regular element \gamma \in E such that  ( \gamma, \alpha ) > 0 \, \forall \gamma \in \Delta, conversely for each regular \gamma there exist a unique set of base roots \Delta(\gamma) such that the previous condition holds for \Delta = \Delta(\gamma). It can be determined in following way: let \Phi^+(\gamma) = \{\alpha \in \Phi | (\alpha, \gamma) > 0\}. Call an element \alpha of \Phi^+(\gamma) decomposable if \alpha = \alpha' + \alpha'' where \alpha', \alpha'' \in \Phi^+(\gamma), then \Delta(\gamma) is the set of all indecomposable elements of \Phi^+(\gamma)
positive roots
Positive root of root system \Phi with respect to a set of simple roots \Delta is a root of \Phi which is a linear combination of elements of \Delta with nonnegative coefficients.
negative roots
Negative root of root system \Phi with respect to a set of simple roots \Delta is a root of \Phi which is a linear combination of elements of \Delta with nonpositive coefficients.
long root
short root
Weyl group
Weyl group of a root system \Phi is a (necessarily finite) group of orthogonal linear transformations of E which is generated by reflections through hyperplanes normal to roots of \Phi
inverse of a root system
Given a root system \Phi. Define \alpha^v = \frac{2 \alpha}{ ( \alpha, \alpha) }, \Phi^v = \{ \alpha^v | \alpha \in \Phi \} is called the inverse of a root system.
\Phi^v is again a root system and have the identical Weyl group as  \Phi .
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[edit]

<--

weight in a root system
\lambda \in E is called a weight if <\lambda, \alpha> \in \mathbb{Z} \, \forall \alpha \in \Phi . -->
weight lattice
weight space
dominant weight
A weight \lambda is dominant if <\lambda, \alpha> \in \mathbb{Z}^+ for some \alpha \in \Phi
fundamental dominant weight
Given a set of simple roots \Delta = \{ \alpha_1 , \alpha_2 , ... , \alpha_n \}, it is a basis of E. \alpha_1^v , \alpha_2^v , ... , \alpha_n^v \in \Phi^v is a basis of E too; the dual basis \lambda_1 , \lambda_2 , ..., \lambda_n defined by (\lambda_i, \alpha_j^v) = \delta_{ij} , is called the fundamental dominant weights.
highest weight
minimal weight
multiplicity (of weight)
radical weight
strongly dominant weight

Representation theory[edit]

module
Define an action of \mathfrak{g} on a vector space V ( i.e. an operation \mathfrak{g} \times V \to V , \, (x,v) \mapsto xv) such that: \forall a,b \in F, x,y \in \mathfrak{g}, v,w \in V satisfy
# (ax+by) v = a (xv) + b(yv)
# x (av+bw) = a (xv) + b (xw)
# [x,y] v = x (yv) - y (xv)
Then V is called a \mathfrak{g}-module. (Remark: V, \mathfrak{g} have the same underlying field F.)
Each \mathfrak{g}-module corresponds to a representation \mathfrak{g} \to \mathfrak{gl}_V.
A subspace W is a submodule (more precisely, sub \mathfrak{g}-module) of V if \mathfrak{g}-module W \subset V .
representation
For a vector space V, if there is a Lie algebra homomorphism \pi : \mathfrak{g} \to \mathfrak{gl}_V, then \pi is called a representation of \mathfrak{g}.
Each representation \mathfrak{g} \to \mathfrak{gl}_V corresponds to a \mathfrak{g}-module V.
A subrepresentation is the representation corresponding to a submodule.
homomorphism
Given two \mathfrak{g}-module V, W, a \mathfrak{g}-module homomorphism \phi is a vector space homomorphism satisfying \phi(xv) = x\phi(v) \forall x \in \mathfrak{g}, v \in V.
trivial representation
A representation is said to be trivial if the image of \mathfrak{g} is the zero vector space. It corresponds to the action of \mathfrak{g} on module V by xv = 0 \forall x \in \mathfrak{g}, v \in V .
faithful representation
If the representation \mathfrak{g} \to \mathfrak{gl}_V is injective, it is said to be faithful.
tautology representation
If a Lie algebra \mathfrak{g} is defined as a subalgebra of \mathfrak{gl}(n,F) , like \mathfrak{sl}(n,F), \mathfrak{o}(2l,F), \mathfrak{t}(n,F) (the upper triangular matrices), the tautology representation is the imbedding \mathfrak{g} \to \mathfrak{gl}(n,F) . It corresponds to the action on module F^n by the matrix multipation.
adjoint representation
The representation   \begin{align} \textrm{ad} : & \mathfrak{g} \to \textrm{gl}_{\mathfrak{g}} \\ & x \mapsto \textrm{ad}_x  \end{align}. It corresponds viewing \mathfrak{g} as a \mathfrak{g}-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 \mathfrak{g}-module V and its submodule W, an action \mathfrak{g} on V/W can be defined by x(v+W) = xv + W \, \forall x \in \mathfrak{g}, v \in V . 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 \mathfrak{g}-module, \phi V \to V is a \mathfrak{g}-module homomorphism iff \phi = \lambda 1_V for some \lambda \in F .
Statement in the language of representation theory: Given an irreducible representation \phi\colon L \to \mathfrak{gl}(V), for \theta \in \textrm{End}(V), \theta \phi(x) = \phi(x) \theta iff \theta = \lambda 1_V for some \lambda \in F.
simple module
synonymous to "irreduible module".
factor module
synonymous to "quotient module".

Universal enveloping algebras[edit]

PBW theorem (Poincaré–Birkhoff–Witt theorem)

Verma modules[edit]

BGG category \mathcal{O}

cohomology[edit]

Chevalley basis[edit]

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:

[H_{\alpha_i},H_{\alpha_j}]=0
[H_{\alpha_i},E_{\alpha_j}]=A_{ij}E_{\alpha_j}
[E_{\alpha_i},E_{\alpha_j}]=H_{\alpha_j}
[E_{\beta},E_{\gamma}]=\pm(p+1)E_{\beta+\gamma}

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[edit]

general linear algebra gl(n, F)
Ado's theorem
Any finite-dimensional Lie algebra is isomorphic to a subalgebra of \mathfrak{gl}_V for some finite-dimensional vector space V.

complex Lie algebras of 1D, 2D, 3D[edit]

Simple Algebras[edit]

Classical Lie algebras:

Name Root System dimension construction as subalgebra of \mathfrak{gl}(n,F)
Special linear algebra A_l \  (l \ge 1) l^2 + 2l \mathfrak{sl}(l+1, F) = \{ x \in \mathfrak{gl}(l+1,F) | Tr(x) = 0 \} (traceless matrices)
Orthogonal algebra B_l \  (l \ge 1) 2 l^2 + l \mathfrak{o}(2l+1, F) = \{ x \in \mathfrak{gl}(2l+1,F) | s x = - x^t s , s = \begin{pmatrix}   1 & 0 & 0 \\ 0 & 0 & I_l \\ 0 & I_l & 0  \end{pmatrix}\}
Symplectic algebra C_l \  (l \ge 2) 2 l^2 - l \mathfrak{sp}(2l, F) = \{ x \in \mathfrak{gl}(2l,F) | s x = - x^t s, s = \begin{pmatrix}   0 & I_l \\  -I_l & 0  \end{pmatrix}\}
Orthogonal algebra D_l (l \ge 1) 2 l^2 + l \mathfrak{o}(2l, F) = \{ x \in \mathfrak{gl}(2l,F) | s x = - x^t s, s = \begin{pmatrix}   0 & I_l \\   I_l & 0  \end{pmatrix}\}

Exceptional Lie algebras:

Root System dimension
G2 14
F4 52
E6 78
E7 133
E8 248

Miscellaneous[edit]

Other discipline related[edit]

References[edit]

  • 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