(a) Show thatB(x,y) =B0(x,y) +jB1(x,y), whereB0is aC-Hermitian form on C2nof signature(2p,2q)andB1is a nondegenerate skew-symmetricC-bilinear form onC2n.
(b) Use part (a) to prove thatSp(p,q) =Sp(C2n,B1)∩U(C2n,B0).
12. Use the identification ofHnwithC2nin the previous exercise to view the form C(x,y)from equation (1.7) as anH-valued function onC2n×C2n.
(a) Show that C(x,y) =C0(x,y) +jxty for x,y∈C2n, where C0(x,y) is a C- Hermitian form onC2nof signature(n,n).
(b) Use the result of part (a) to prove thatSO∗(2n) =SO(2n,C)∩U(C2n,C0).
13. Why can’t we just defineSL(n,H)by taking allg∈GL(n,H)such that the usual formula for the determinant ofgyields 1?
14. Consider the three embeddings ofCinHgiven by the subfields (1.4). These give three ways of writingX ∈Mn(H)as a 2n×2nmatrix overC. Show that these three matrices have the same determinant.
1.2 The Classical Lie Algebras
LetV be a vector space overF. Let End(V)denote the algebra (under composition) ofF-linear maps ofV toV. IfX,Y ∈End(V)then we set[X,Y] =XY−Y X. This defines a new product on End(V)that satisfies two properties:
(1) [X,Y] =−[Y,X]for allX,Y (skew symmetry).
(2) [X,[Y,Z]] = [[X,Y],Z] + [Y,[X,Z]]for allX,Y,Z (Jacobi identity).
Definition 1.2.1.A vector spacegoverFtogether with a bilinear mapX,Y7→[X,Y] ofg×gtogis said to be aLie algebraif conditions (1) and (2) are satisfied.
In particular, End(V) is a Lie algebra under the binary operation [X,Y] = XY−Y X. Condition (2) is a substitute for the associative rule for multiplication;
it says that for fixedX, the linear transformationY 7→[X,Y]is aderivationof the (nonassociative) algebra(g,[·,·]).
Ifgis a Lie algebra and ifhis a subspace such thatX,Y∈himplies that[X,Y]in h, thenhis a Lie algebra under the restriction of[·,·]. We will callhaLie subalgebra ofg(orsubalgebra, when the Lie algebra context is clear).
Suppose thatgandhare Lie algebras overF. ALie algebra homomorphismof gtohis anF-linear mapT:g //hsuch thatT[X,Y] = [T X,TY]for allX,Y∈g.
A Lie algebra homomorphism is anisomorphismif it is bijective.
1.2.1 General and Special Linear Lie Algebras
IfV is a vector space overF, we write gl(V) for End(V)looked upon as a Lie algebra under[X,Y] =XY−Y X. We writegl(n,F)to denoteMn(F)as a Lie algebra
under the matrix commutator bracket. If dimV =nand we fix a basis forV, then the correspondence between linear transformations and their matrices gives a Lie algebra isomorphismgl(V)∼=gl(n,R). These Lie algebras will be called thegeneral linear Lie algebras.
IfA= [ai j]∈Mn(F)then itstraceis tr(A) =∑iaii. We note that tr(AB) =tr(BA). This implies that ifAis the matrix ofT ∈End(V)with respect to some basis, then tr(A)is independent of the choice of basis. We will write tr(T) =tr(A). We define
sl(V) ={T∈End(V): tr(T) =0}.
Since tr([S,T]) =0 for allS,T ∈End(V), we see thatsl(V)is a Lie subalgebra of gl(V). Choosing a basis forV, we may identify this Lie algebra with
sl(n,F) ={A∈gl(n,F): tr(A) =0}. These Lie algebras will be called thespecial linear Lie algebras.
1.2.2 Lie Algebras Associated with Bilinear Forms
LetVbe a vector space overFand letB:V×V //Fbe a bilinear map. We define so(V,B) ={X∈End(V):B(X v,w) =−B(v,X w)}.
Thusso(V,B)consists of the linear transformations that areskew-symmetricrelative to the formB, and is obviously a linear subspace ofgl(V). IfX,Y∈so(V,B), then
B(XY v,w) =−B(Y v,X w) =B(v,Y X w).
It follows thatB([X,Y]v,w) =−B(v,[X,Y]w), and henceso(V,B)is a Lie subalgebra ofgl(V).
SupposeV is finite-dimensional. Fix a basis{v1, . . . ,vn}forV and letΓ be the n×nmatrix with entriesΓi j=B(vi,vj). By a calculation analogous to that in Section 1.1.2, we see thatT∈so(V,B)if and only if its matrixArelative to this basis satisfies
AtΓ+ΓA=0. (1.8)
WhenBis nondegenerate thenΓ is invertible, and equation (1.8) can be written as At=−ΓAΓ−1. In particular, this implies that tr(T) =0 for allT∈so(V,B).
Orthogonal Lie Algebras
TakeV =Fn and the bilinear formBwith matrixΓ =Inrelative to the standard basis forFn. Define
1.2 The Classical Lie Algebras 15 so(n,F) ={X∈Mn(F):Xt=−X}.
SinceBis nondegenerate,so(n,F)is a Lie subalgebra ofsl(n,F).
WhenF=Rwe take integersp,q≥0 such thatp+q=nand letBbe the bilinear form onRnwhose matrix relative to the standard basis isIp,q(as in Section 1.1.2).
Define
so(p,q) ={X∈Mn(R):XtIp,q=−Ip,qX}. SinceBis nondegenerate,so(p,q)is a Lie subalgebra ofsl(n,R).
To obtain a basis-free definition of this family of Lie algebras, letBbe a non- degenerate symmetric bilinear form on an n-dimensional vector space V over F.
Let {v1, . . . ,vn} be a basis forV that is orthonormal (when F=C) or pseudo-
orthonormal (whenF=R) relative toB(see Lemma 1.1.2). Letµ(T)be the matrix ofT ∈End(V)relative to this basis . WhenF=C, thenµdefines a Lie algebra iso- morphism ofso(V,B)ontoso(n,C). WhenF=RandBhas signature(p,q), then µdefines a Lie algebra isomorphism ofso(V,B)ontoso(p,q).
Symplectic Lie Algebra
LetJbe the 2n×2nskew-symmetric matrix from Section 1.1.2. We define sp(n,F) ={X∈M2n(F):XtJ=−JX}.
This subspace ofgl(n,F)is a Lie subalgebra that we call thesymplectic Lie algebra of rank n.
To obtain a basis-free definition of this family of Lie algebras, letBbe a non- degenerate skew-symmetric bilinear form on a 2n-dimensional vector spaceV over F. Let{v1, . . . ,v2n}be aB-symplectic basis forV (see Lemma 1.1.5). The mapµ that assigns to an endomorphism ofV its matrix relative to this basis defines an isomorphism ofso(V,B)ontosp(n,F).
1.2.3 Unitary Lie Algebras
Let p,q≥0 be integers such that p+q=nand let Ip,qbe then×n matrix from Section 1.1.2. We define
u(p,q) ={X∈Mn(C):X∗Ip,q=−Ip,qX}
(notice that this space is arealsubspace ofMn(C)). One checks directly thatu(p,q) is a Lie subalgebra of gln(C) (considered as a Lie algebra over R). We define su(p,q) =u(p,q)∩sl(n,C).
To obtain a basis-free description of this family of Lie algebras, letV be ann- dimensional vector space overC, and letBbe a nondegenerate Hermitian form on
V. We define
u(V,B) ={T ∈EndC(V):B(T v,w) =−B(v,Tw)for allv,w∈V}. We setsu(V,B) =u(V,B)∩sl(V). IfBhas signature(p,q)and if{v1, . . . ,vn}is a pseudo-orthogonal basis ofV relative toB(see Lemma 1.1.7), then the assignment T7→µ(T)ofT to its matrix relative to this basis defines a Lie algebra isomorphism ofu(V,B)withu(p,q)and ofsu(V,B)withsu(p,q).
1.2.4 Quaternionic Lie Algebras
Quaternionic General and Special Linear Lie Algebras
We follow the notation of Section 1.1.4. Consider then×nmatrices over the quater- nions with the usual matrix commutator. We will denote this Lie algebra bygl(n,H), considered as a Lie algebra over R(we have not defined Lie algebras over skew fields). We can identify Hn with C2n using one of the isomorphic copies of C (R1+Ri,R1+Rj, orR1+Rk) inH. Define
sl(n,H) ={X∈gl(n,H): tr(X) =0}.
Thensl(n,H)is the real Lie algebra that is usually denoted bysu∗(2n).
Quaternionic Unitary Lie Algebras
Forn=p+qwithp,qnonnegative integers, we define
sp(p,q) ={X∈gl(n,H):X∗Ip,q=−Ip,qX}
(the quaternionic adjointX∗was defined in Section 1.1.4). We leave it as an exer- cise to check thatsp(p,q)is a real Lie subalgebra ofgl(n,H). Let the quaternionic Hermitian formB(x,y)be defined as in (1.6). Thensp(p,q)consists of the matrices X∈Mn(H)that satisfy
B(X x,y) =−B(x,X∗y) for all x,y∈Hn.
The Lie Algebraso∗(2n)
Let the automorphismθ ofM2n(C)be as defined in Section 1.1.4 (θ(A) =−JAJ).
Define
so∗(2n) ={X∈so(2n,C): θ(X) =X}.
1.2 The Classical Lie Algebras 17
This real vector subspace ofso(2n,C)is a real Lie subalgebra ofso(2n,C)(con- sidered as a Lie algebra overR). IdentifyC2nwithHnas in Section 1.2.4 and let the quaternionic skew-Hermitian formC(x,y)be defined as in (1.7). Thenso∗(2n) corresponds to the matricesX∈Mn(H)that satisfy
C(X x,y) =−C(x,X∗y) for all x,y∈Hn.
1.2.5 Lie Algebras Associated with Classical Groups
The Lie algebrasgdescribed in the preceding sections constitute the list ofclassical Lie algebrasover RandC. These Lie algebras will be a major subject of study throughout the remainder of this book. We will find, however, that the given matrix form ofgis not always the most convenient; other choices of bases will be needed to determine the structure ofg. This is one of the reasons that we have stressed the intrinsic basis-free characterizations.
Following the standard convention, we have labeled each classical Lie algebra with a fraktur-font version of the name of a corresponding classical group. This passage from a Lie group to a Lie algebra, which is fundamental to Lie theory, arises bydifferentiatingthe defining equations for the group. In brief, each classical group Gis a subgroup ofGL(V)(whereVis a real vector space) that is defined by a setR of algebraic equations. The corresponding Lie subalgebragofgl(V)is determined by taking differentiable curvesσ :(−ε,ε)→GL(V)such thatσ(0) =I andσ(t) satisfies the equations inR. Thenσ0(0)∈g, and all elements ofgare obtained in this way. This is the reason whygis called theinfinitesimal formofG.
For example, ifGis the subgroupO(V,B)ofGL(V)defined by a bilinear form B, then the curve σ must satisfy B(σ(t)v,σ(t)w) =B(v,w) for all v,w∈V and t∈(−ε,ε). If we differentiate these relations we have
0= d
dtB(σ(t)v,σ(t)w)
t=0=B(σ0(0)v,σ(0)w) +B(σ(0)v,σ0(0)w) for allv,w∈V. Sinceσ(0) =I, we see thatσ0(0)∈so(V,B), as asserted.
We will return to these ideas in Section 1.3.4 after developing some basic aspects of Lie group theory.
1.2.6 Exercises
1. Prove that the Jacobi identity (2) holds for End(V).
2. Prove that the inverse of a bijective Lie algebra homomorphism is a Lie algebra homomorphism.
3. LetBbe a bilinear form on a finite-dimensional vector spaceVoverF. (a) Prove thatso(V,B)is a Lie subalgebra ofgl(V).
(b) Suppose thatBis nondegenerate. Prove that tr(X) =0 for allX∈so(V,B).
4. Prove thatu(p,q),sp(p,q), andso∗(2n)are real Lie algebras.
5. LetB0(x,y)be the Hermitian form andB1(x,y)the skew-symmetric form onC2n in Exercises 1.1.5 #11.
(a) Show thatsp(p,q) =su(C2n,B0)∩sp(C2n,B1)whenMn(H)is identified with a real subspace ofM2n(C)as in Exercises 1.1.5 #10.
(b) Use part (a) to show thatsp(p,q)⊂sl(p+q,H).
6. Let X ∈Mn(H). For each of the three choices of a copy of CinHgiven by (1.4) write out the corresponding matrix ofXas an element ofM2n(C). Use this formula to show that the trace ofXis independent of the choice.