Lie Algebra
January 25, 2017
Jakelic
Announcements: Meeting Friday at 2
1 For
(Lie) Subalgebras of gl(n, F) :1
Ex 1:
D(n, F) = the set of all diagonal n × n matrices (abelian subalgebra)
Ex 2:
T (n, F) = set of all upper triangular matrices
Ex 3:
St(n, F) = strictly upper triangular2
Theorem 1.
Let
{xi |i ∈ I} be a basis of a vector space L
(L, [, ]) is a Lie algebra if and only if:
general linear groups, this
is the commutator.
2 Upper
and
[, ] : L × L → L
triangular with diago-
nals set to zero.
be a
bilinear map. Then
1.
[xi , xi ] = 0 ∀i ∈ I
2.
[xi , xj ] = −[xj , xi ]3; ∀i, j ∈ I.
3.
[xi , [xj , xk ]] + [xi , [xj , xk ]] + [xi , [xj , xk ]] = 0 ∀i, j, k, ∈ F
(⇒) Incorporated in denition.
⇐ Use bilinearity of bracket and linear combinations of basis elements..
Proof.
In particular, the bracket structure on L is completely determined by the bracket
structure on {xi |i ∈ I}
Note:
3 [x , x ] ∈ L, [x , x ] = P ak x
i
j
i
j
ij k
k∈I
Ex 4: (Vector cross product.)
3 ak
ij
are
stants.
L = R, {e1 , e2 , e3 }.
Dene: [e1 , e2 ] = e3 [e2 , e3 ] = e1 [e3 , e1 ] = e2 and extend by linearity.4
Properties 1 and 2 are trivial, should show Jacobi, but obvious. Note
similarities to vector cross product.
4
Ex 5: (Heisenberg Lie Algebra.) Let V = F[x] and dene the following
operators on V :
I(f (x)) = f (x)
Dx (f (x)) = d/dxf (x)
Lx (f (x)) = xf (x)
Letting L = span {Ix , Dx , Lx } and [A, B] = A ◦ B − B ◦ A we can show
that [Dx , Lx ] = Ix and the others are identically zero. We can also show
the Jacobi identity holds, so L is a Lie Algebra known as the Heisenberg
Lie Algebra.
4
called
you only need
4 Really,
structure
con-
By anticommutativity,
i < j.
bilinearity
commutativity.
and
anti-
Lie Algebra
January 25, 2017
Jakelic
Ex 6: Let V be a vector space and L = {T
: V → V | T is a linear transformation}5
5 The set of
Let [T1 , T2 ] be the commutator bracket. Then, L, [, ] is a Lie Algebra de∼
phisms on V .
noted by gl(V ). If Dim V = n, then gl(V ) = gl(n, F).
4
Denition 0.1. An associative algebra
A over a eld F is a vector space over F
with an associative bilinear map: A × A → A, (a, b) 7→ ab.
Ex 7:
Associative Algebras can be made into Lie Algebras under the
commutator.
4
Ex 8: (Special Linear Lie Algebra) sln (F) = sl(n, F) = {A ∈ gl(n, F) | tr(A) = 0}
Claim: sln (F) < gln (F) Show vector subspace and closed under bracket.
In particular, show the bracket of gl(n, F) ⊆ sl(n, F).
4
vector
endomor-