Matrix groups and Lie groups

Matrix groups and Lie groups

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

Last updates: 19 November 2010

Matrices

Mn= n×n  matrices with entries in  with product matrix multiplication
is not a group
because, if g= 1 0 1 0 0 0 0 0 0 then there does not exist g-1 with gg-1 =g-1g=1.

The general linear group is GLn = gMn there exists  g-1  with  gg-1 =g-1g=1 . The special linear group is is SLn =kerϕ where ϕ:GLn GL1 is the homomorphism given by ϕg=detg.

Orthogonal, Symplectic and Unitary groups

The orthogonal group is On= gGLn g1gt=1 . The symplectic group is Spn= gGLn gJgt=1 where J= 0 1 0 0 1 -1 0 0 -1 0 The conjugation involution on is -: given by x+yi =x-yi. The unitary group is Un = gGLn g1 g t , where g t = g11 g1n gn1 gnn t = g11 gn1 g1n gnn = g 11 g n1 g 1n g nn .

Linear transformations

Let V be a vector space. EndV= linear transformations  g:VV . If b1,,bn is a basis of V then EndV Mn g g11 g1n gn1 gnn where gbi =g1ib1++ gnibn. A  symmetric form skew-symmetric form Hermitian form   on V is a map 1 ,: VV v1,v2 v1,v2 such that
  1. If a1,a2 and v1,v2,v3V then a1v1+ a2v2,v3 = a1v1,v3 + a2v2,v3,
  2. If v1,v2V then v2,v1 = v1,v2. v2,v1 = -v1,v2. v2,v1 = v1,v2 .

Linear groups

The general linear group is GLV = gEndV there exists g-1  with  g-1g=gg-1 =1 . Let , be a symmetric form on V. The orthogonal group is OV gGLV gv1,gv2= v1,v2  for all  v1,v2V . Let , be a skew-symmetric form on V. The symplectic group is SpV gGLV gv1,gv2= v1,v2  for all  v1,v2V . Let , be a Hermitian form on V. The unitary group is UV gGLV gv1,gv2= v1,v2  for all  v1,v2V .

Exponential maps

e: >0 x ex 1

satisfies ex ey= ex+y and dex dx x=0 =1. If ex is a polynomial then ex= 1+x+ 12 x2+ 13! x3+. So we have an exponential map e: 0 r+iθ ereθ which is really a map e: M1 GL1 x ex which is a special case of e: Mn GLn x ex.

Example: e 1 0 0 2 = e 0 0 e2 and e 0 1 0 0 = 1 1 0 1 .

Lie groups

Conceptually, a Lie group is a group G with an exponential map.

Mn is a vector space with basis Eij 1ijn where Eij= jth ith 0 1 0 0

Example: 0 5 2 1 =5E12+ 2E21+E22.

Let G be a group of matrices (so GGLn). If it turns out that 𝔤=logG= xMn exG is a subspace of Mn, then e:𝔤G is an exponential map for G.

𝔤 is the Lie algebra of G.

One parameter subgroups

Let G be a Lie group. A one parameter subgroup of G is a map γ:G such that γt1+t2 =γt1 γt2.

Example: Let xMn. Then γx: GLn t etx is a one-parameter subgroup of GLn.

Example: Let x𝔤. Then γx: G t etx is a one-parameter subgroup of G. 𝔤 one parameter subgroups of G x γx

A one parameter subgroup of GLn is a homomorphism GLn.

SL2 embeddings

SL2 = gGL2 detg=1 . The group SL2 is generated by 1 t 0 1 and 1 0 s 1 where t,s. For each 1i<jn, ϕij: SL2 GL2 1 t 0 1 1 0 t 0 1 1 0 s 1 1 0 s 0 1 is an embedding of SL2 in GLn. A maximal torus in GLn is an embedding of the group T= * 0 0 * into  GLn. A Borel subgroup in GLn is an embedding of the group B= * * 0 * into  GLn.

Weyl groups

Let G be a Lie group. Let T be a maximul torus in G.

The normalizer of T in G is the

largest subgroup N of G such that T is normal in N.
The Weyl group of G is W=N/T.

Example: The normalizer of T in GLn is N= n×n matrices with (a) exactly one nonzero entry in each row and each column (b) the nonzero entries are in  GL1. The map that changes nonzero entries to 1 is a homomorphism ϕ:NSn with kerϕ=T.

Example:

ϕ 0 0 5 2 0 0 0 3 0 = 0 0 1 1 0 0 0 1 0 =

So the symmetric group Sn is the Weyl group of GLn.

Generators and relations for Weyl groups

Let

si = i i+1 = ith 1 0 1 0 1 1 0 1 0 1

Sn is generated by s1,s2, ,sn-1.

Proof.
Stretch w.

= = s2s3s2s1s2.

The symmetric group is given by generators s1,,sn-1 and relations

si2 =1, = sisj= sjsi, if  |i-j|>1 , = = si si+1 si= si+1 si si+1, = =

Elementary matrices

Eij 1i,jn is a basis of Mn and xij: GLn t etEij is a one parameter subgroup. xijt = 1+tEij + 12 t2 Eij2 + 13! t3 Eij3 + =1+tEij+0+0+ = jth ith 1 t 0 1 . The elementary matrices in GLn are xijt, 1i,jn, t. If gGLn then xijtg =g except 5 ( jth row )  is added to the ith  row, and gxijt=g  except 5 ( ith column )  is added to the jth column. So elementary matrices in GLn are row and column operations.

Generators and relations for Lie groups

GLn is generated by xijt, 1i,jn, t.

Proof.
Let gGLn. Finding g-1 by row reduction is equivalent to finding xiljl tl xi1j1 t1 g=1. Since xijt-1 =xij-t, g= xi1j1 -t1 xiljl -tl is a product of elementary matrices.

The relations are xijt1 xijt2 = xij t1+t2 , hit1 hit2 = hit1t2, xijt xkls = xkls xijt, xijt xjls = xjls xijt xilst, xijt xji -t-1 xijt = sij hit hj-t-1, wxijtw-1 = xwiwj t, for wSn, hxijt = xij hithj-1, for h= h1 0 0 hn T, where sij= ith jth 1 1 0 1 1 1 1 0 1 1 .

References

[Bou] N. Bourbaki, Groupes et Alg├Ębres de Lie, Masson, Paris, 1990.

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

page history