The group SU2U1() Spin3 and the Lie algebra 𝔰𝔲2

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

Last updates: 6 November 2011

The group SU2U1() Spin3 and the Lie algebra 𝔰𝔲2,

The maximal compact subgroup of SL2() is

SU2 = { g=( ab cd ) | ggt =1 and det(g)=1 } = { ( ab -ba ) | a,b, |a|2 +|b|2 =1 }
since
g-1 = ( d-b -ca ) and gt = ( a c b d ).
The Lie algebra 𝔰𝔲2 is
𝔰𝔲2 = { x𝔤𝔩2 | x+xt =0, trx=0 } , =-span{ iσx, iσy, iσz} ,
where
σx = ( 01 10 ) , σy = ( 0-i i0 ) , σz = ( 10 0-1 )
are the Pauli matrices and
[σx,σy] =2iσz, [σy,σz] =2iσx, [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 ), h=σz. (PtoC)

Let be the division algebra of Hamiltonians so that

× =GL1() and U1() ={x | |x|= xxt=1} .

The fundamental representation θ: U1() SU2

The action of × on the 2-dimensional -vector space by right multiplication provides

θ: × GL2() transpose GL2() x0+x1i +x2j+x3k ( x0+x1i -x2+x3i x2+x3i x0-x1i ) ( x0+x1i -x2+x3i x2+x3i x0-x1i )
This gives a group homomorphism
θ: × GL2() a+cj ( a c -c a ) for a=x0+x1i and c=x2+x3i in .
The Pauli matrices are
θ(i) = ( i 0 0 -i ) , θ(j) = ( 0 -1 1 0 ) , θ(k) = ( 0 i i 0 ) ,
and
θ: U1() SU2.

The Cartan subalgebra of 𝔲1() is 𝔥=-span{i} and the Cartan subgroup of U1() is the group H=U1(),

U1() U1()=H θ(H) SU2 a+bi (a+bi0 0a-bi) with |a+bi|=1 .

The isomorphism SU2 Spin3

The differential of the isomorphism θ: U1() SU2() is the Lie algebra isomorphism

θ:𝔲1() 𝔰𝔲2 ,
where
𝔲1() = {x𝔤𝔩1 () | x+xt =0} = {x1i +x2j +x3k | x1,x2, x3}
and
𝔰𝔲2 = {x𝔤𝔩2 () | x+xt =0and trx=0 } = -span{iσx , iσy, iσz}
and
θ(i)=iσz, θ(j)=iσy, θ(k)=iσx.
The adjoint representation is the 3-dimensional representation given by the action of G on 𝔤. If G GLn and 𝔤 𝔤𝔩n then the action is given by
gx=gxg-1, for gG and x𝔤.
In this case, 𝔤= 𝔰𝔲2=-span {x,y,h}, and the representation ρ:U1() SU2() GL3() coming from the adjoint action of SU2() has differential
dρ: 𝔰𝔲2 𝔰𝔩2() ad 𝔤𝔩3()
where ad is the adjoint representation of 𝔰𝔩2 given by
ad(x) = ( 0 -2 0 0 0 1 0 0 0 ) , ad(y) = ( 0 0 0 -1 0 0 0 2 0 ) , ad(h) = ( 2 0 0 0 0 0 0 0 -2 ) , (adsl2)
The change of basis formulas in (PtoC) allow any favourite element of 𝔰𝔲2 in any favourite basis of 𝔤 to be worked out from (adsl2).

The adjoint representation ρ:U1() SU2() GL3() gives rise to the exact sequence

{1} {±1} U1() ρ SO3() {1}
which realizes U1() as the 2-fold cover Spin3 U1() of SO3().

Notes and References

These notes were influenced by the Wikipedia articles ????. They were prepared for lectures and working seminars in Representation Theory at University of Melbourne in 2008-2011.

References

[St] R. Steinberg, Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968. iii+277 pp. MR0466335.

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