Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 2 October 2014

Lecture 10

Lie algebras

A Lie group is a group that is also a manifold, i.e. a topological group that is locally isomorphic to n image/svg+xml 1 U G U is an open neighbourhood of 1 in G 0 V R n V is an open neighbourhood of 0 in Rn G n U V 0 1 φ : U V U is an open neighborhood of 1 in G V is an open neighborhood of 0 in n If G is connected then G is generated by the elements of U.

The exponential map is a smooth homomorphism 𝔤 G 0 1 which is a homeomorphism on a neighborhood of 0. The Lie algebra 𝔤 contains the structure of G in a neighbourhood of the identity.

A one parameter subgroup of G is a smooth group homomorphism γ:G. image/svg+xml 1 γ ( t ) 0 t γ

(1) γ: GLn() t 1+tEij = xij(t) for ij. Note that xij(t) xij(s)= xij(s+t) since (1t01) (1s01)= (1t+s01).
(2) γ: GLn() t ( 1 1 et 1 1 ) = hi(et). Note that hi(et) hi(es)= hi(et+s).

Let G be a Lie group. The ring of functions on G is C(G)= { f:G| fis smooth atgfor all gG } where fis smooth atgif dkfdxk |x=g exists for allk>0.

Let gG. A tangent vector to G at g is a linear map η:C(G) such that η(f1f2)= f1(g) η(f2)+ η(f1) f2(g), for all f1,f2C(G). A vector field is a linear map :C(G)C(G) such that (f1f2)= f1(f2)+ (f1)f2, for f1,f2C(G). A left invariant vector field on G is a vector field :C(G)C(G) such that Lgξ=ξLg, for all gG, where Lg:C(G) C(G) is given by (Lgf)(x)= f(g-1x), for fC(G), g,xG.

The Lie algebra of G is the vector space 𝔤 of left invariant vector fields on G with bracket [1,2]= 12- 21. A one-parameter subgroup of G is a smooth group homomorphism γ:G. If γ is a one-parameter subgroup of G define df(γ(t))dt= limh0 f(γ(t+h))-f(γ(t))h and let γ1 be the tangent vector at 1 given by γ1(f)= df(γ(t))dt |t=0. Identify the vector spaces {left invariant vector fields onG},
{one parameter subgroups ofG}, and
{tangent vectors at1},
by the vector space isomorphisms {one parameter subgroups} {tangent vectors at1} γ γ1 and {left invariant vector fields} {tangent vectors at1} ξ ξ1 where ξ1(f)= (ξf)(1).

The exponential map is 𝔤 G tX etX where etX=γ(t), where γ is the 1-parameter subgroup corresponding to X.

(a) The Lie algebra 𝔤𝔩n is 𝔤𝔩n={xMn()} with bracket [x1,x2]= x1x2-x2x1. Our favourite basis of 𝔤𝔩n is {Eij|1i,jn}. The exponential map is 𝔤𝔩n GLn tX etX, where eA=1+A+ A22!+ A33!+ for a matrix A. In fact etEij= 1+tEij= ( 1 t 1 ) for ij, where t is the (i,j) matrix entry, and etEii= ( 1 1 et 1 1 ) =hi(et).
(b) If n=1 the exponential map × tx etx is a homeomorphism from a neighborhood of 0 to a neighborhood of 1. In fact, if e(t)=a0+a1t+a2t2+ and e(s+t)= e(s)e(t), then e(s+t) = a0+ a1(s+t)+ a2(s+t)2+ a3(s+t)3+ = a0+ a1s+a1t+ a2s2+2a2st+a2t2 a3s3+3a3s2t+3a3st2+a3t3+ and e(s)e(t) = (a0+a1s+a2s2+a3s3+) (a0+a1t+a2t2+a3t3+) = a02+ a0a1s+a0a1t+ a0a2s2+a12st+a0a2t2 a0a3s3+a2a1s2t+a1a2st2+a0a3t3+ Hence e(s+t)=e(s)e(t) only if a0a1=a1, 2a2=a12, 3a3=a1a2, 4a3=a1a3, so that a0=1, a2=a122, a3=a133!, a4=a144!, and e(t)=1+a1t+ a122t2+ a13t33!+ =ea1t. So × z ez is the "unique" smooth homomorphism ×.

The Lie groups SL2 and SU2

SL2= { (abcd) |ad-bc=1 } One parameter subgroups are x12(t)= (1t01), x21(t)= (1t01), hα1(et)= (et00e-t) and the Lie algebra 𝔰𝔩2= {x𝔤𝔩2|trx=0} has basis {x,y,h} where x=(0100), y=(0010), h=(100-1). The Lie algebra 𝔰𝔩2 is presented by generators x,y,h and relations [x,y]=h, [h,x]=2xand [h,y]=-2y. The group SL2 is presented by generators xα(t)= (1t01), x-α(t)= (10t1),t𝔽 with relations xα(s+t) = xα(s) xα(t), hα(c1c2) = hα(c1) hα(c2),and nα(t) xα(u) nα(-t) = x-α(-t-2n) where nα(t) = xα(t) x-α(-t-1) xα(t)and hα(t) = nα(t) nα(-1). Note that nα(t) = (1t01) (10-t-11) (1t01) = (0t-t-11) (1t01) = (0t-t-10) and hα(t)= (0t-t-10) (0-110)= (t00t-1). The maximal compact subgroup of SL2() is SU2 = { g=(abcd) |ggt=1, det(g)=1 } = { (ab-ba) |a,b, a2+ b2=1 } since g-1= (d-b-ca) andgt= (acbd). The Lie algebra 𝔰𝔲2 is 𝔰𝔲2 = { x𝔤𝔩2() |trx=0 andx+xt =0 } = -span{iσx,iσy,iσz}, where σx=(0110), σy=(0-ii0), σz=(100-1) are the Pauli matrices and [σz,σy]=2iσz, [σy,σz]=2iσz, [σz,σx]=2iσy. Then 𝔰𝔩2() is the complexification of 𝔰𝔲2, 𝔰𝔩2()= 𝔰𝔲2 and the change of basis is given by σx=x+y, σy=-ix+iy, σz=h, x=12(σx+iσy), y=12(σx-iσy). Note that GL1()=× has maximal compact subgroup U(1)=S1= {z×|zz=1} and SL1()={±1} has maximal compact subgroup SU(1)={1}.

Notes and References

These are a typed copy of Lecture 10 from a series of handwritten lecture notes for the class Representation Theory given on October 14, 2008.

page history