Download Lecture 8 - Universal Enveloping Algebras and Related Concepts, II

Lecture 8 - Universal Enveloping Algebras and Related
Concepts, II
February 8, 2013
1
Filtrations and Graded Algebras
1.1
The basics
An algebra U has a filtration if there are subsets {U (i) }i∈Z of U so that
i) · · · ⊆ U (i) ⊆ U (i+1) . . .
ii) U(i) U (j) ⊆ U (i+j)
S
iii) U = i U (i) .
A stronger notion is that of a grading. An algebra U is called a graded algebra if it has
subsets {U i } so that
i) U i U j ⊆ U i+j
L
ii) U = i U (i) .
We have the following main points:
Homogeneous Elements. An element x of a graded algebra U is called homogeneous if x ∈ U i for some i. The
decomposition of a general element x ∈ U is
X
x =
xi
(1)
i
where xi = π i (x) is the vector-space projection onto the ith summand.
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Canonical Filtration. Given a grading on U , its canonical filtration is given by
M
U (i) =
Uk
(2)
k≤i
Associated Graded Algebra. Given a filtered algebra U , we can define its associated graded algebra (which is usually
a different algebra), by first setting
i
U = U (i) / U (i−1) .
i
j
(3)
i+j
After noting that U U ⊆ U , we see that the product on U passes to a new product
L i
on i U , which we take to be the associated graded algebra:
M i
(4)
U ,
U .
i
Graded Ideals. If U is graded and I is an ideal, then I isLcalled a graded ideal provided that if x ∈ I,
then also each π i (x) ∈ I. That is, I = i I ∩ U i . If U is graded and I is a graded
ideal, then U/I is graded.
Inherited Filtrations. If π : A → B is a surjection of algebras and A is filtered by {A(i) }, then B is filtered
by {B (i) , π(A(i) )}.
1.2
1.2.1
Examples
Tensor, symmetric, and antisymmetric algebras
Given a vector space V the tensor algebra T V is a graded algebra T V =
T i V = V ⊗i . The ideal IS generated by elements of the form
L
i≥0
T i V with
x⊕y − y ⊕x
(5)
x⊕y + y ⊕x
(6)
and IA generated by elements
are both graded ideals, so, as we have seen, generate the graded algebras
M
SV , T V / IS =
SiV
i
^
M ^i
V , T V / IA =
V
(7)
i
where
S i V = spanF x1 · · · xi xl ∈ V
^i
V = spanF x1 ∧ · · · ∧ xi xl ∈ V
2
(8)
1.2.2
The Clifford Algebras
The Clifford algebras, on the other hand, are quotients by ideas generated by the nonhomogeneous elements x ⊗ y + y ⊗ x + 2 hx, yi, and so do not inherit a grading in the usual
sense.
Nevertheless they inherit what is called a Z2 -grading. We can decompose the tensor
algebra into two parts
T V = T 0V ⊕ T 1V
(9)
where
T 0V =
M
T 1V =
M
n≥0
n≥0
T 2n V
(10)
T 2n+1 V
are termed the even and odd tensors. Note that T 0 V is itself a subalgebra. The observation
that xa ∈ T a V , xb ∈ T b V implies xa ⊗ xb ∈ T a+b (mod 2) V justifies calling this a Z2 -grading.
One observes that the ideal generated by
x ⊗ y + y ⊗ x + 2 hx, yi
s.t.
x, y ∈ V
(11)
is a Z2 -graded ideal. Therefore the algebras Clp,q and Cln are Z2 -graded, where we have
0
1
Clp,q = Clp,q
⊕ Clp,q
,
Cln = Cln0 ⊕ Cln1 .
(12)
Further, the Cl0 components are subalgebras. We call the subalgebras
0
Clp,q
and
Cln0
(13)
the even Clifford algebras.
We note one further important fact. Let Vp,q be the vector space with inner product
of signature (p, q), where we take
x1 , . . . , x p , y 1 , . . . , y q
(14)
to be an orthonormal basis, meaning
hxi , xj i = δij
hyi , yj i = −δij
(15)
hxi , yj i = 0.
0
(and Cln → Cln0 ) induced on
Proposition 1.1 There is an isomorphism Clr,s → Clr+1,s
elements v ∈ Vp,q by
v 7→ v · xp+1 .
3
(16)
Pf. Essentially trivial: one easily sees the kernel is zero, and the map is surjective.
For a concrete example, consider the algebra
H ≈ Cl2,0 .
(17)
We have two canonical structures on this algebra, a filtration and a Z2 -grading, both inherited from T R2 , where we take {i, j} to be an orthonormal basis of R2 . As for the filtration,
we have
(Cl2,0 )
(0)
≈ R
(Cl2,0 )
(1)
≈ spanR { 1, i, j }
(Cl2,0 )
(2)
= Cl2,0 ≈ spanR { 1, i, j, ij }.
(18)
As for the Z2 -grading, we have
0
Cl2,0
≈ spanR { 1, ij }
1
Cl2,0
≈ spanR { i, j }
(19)
and we see immediately the isomorphism
Cl1,0 ≈ span{1, i}
0
Cl2,0
≈ span{1, ij}
7−→
(20)
induced precisely as specified above. Finally note that a generic quaternion a + bk + cj + di
(where k = ij) can be written, in its Z2 -graded form, as
(a + bk) + (c − dk) j
(21)
so that we should regard H as C ⊕ C with a twisted product.
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Statement of Poincare-Birkhoff-Witt and Its Corollaries
Given a Lie algebra g, we have seen the construction of its universal enveloping algebra
U (g) ≈ T g / Ig
(22)
where Ig ⊂ T g is the ideal generated by x ⊗ y − y ⊗ x − [x, y]. We know that U (g) inherits
a filtration from T g; let
U (g)
(23)
be the associated graded algebra. As usual let Sg is the symmetric algebra on the underlying
vector space g.
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We have vector space homomorphisms
Tg
πT
π
U (g)
U (g)
(24)
to which we give the name
π T : T g → U (g)
(25)
Lemma 2.1 The map The map π T it factors through the canonical homomorphism T g →
Sg. Thus we have a commutative diagram of vector space homomorphisms
Tg
πT
can.
U (g)
Sg
πT
(26)
πS
π
U (g)
and a commutative diagram of graded algebra homomorphisms
Tg
can.
Sg
πT
(27)
πS
U (g)
Pf. Given x ∈ T m g, y ∈ T s g be homogeneous tensors, and let x̄ = π T (x), ȳ = π T (y) be
their images in U (g). Their multiplication is given by
π T (x)π T (y) = πT (x)πT (y) + U (m+s−1) (g)
= πT (x ⊗ y) + U (m+s−1) (g)
(28)
= π T (x ⊗ y)
so that π T is a homomorphism. Further, if x, y ∈ T 1 g ≈ g, then
x̄ȳ − ȳx̄ = πT (x)πT (y) − πT (y)πT (x) + U (1) (g)
= πT ([x, y]) + U (1) (g)
(29)
= 0 + U (1) (g)
Therefore IS ⊆ Ig so the diagram of algebras (27) commutes.
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Theorem 2.2 (Poincaré-Birkhoff-Witt, highbrow version.)
π S : Sg
≈
(30)
U (g).
Following Humphreys, we give a few fundamental corollaries along with their proofs.
Corollary 2.3 Assume W ⊆ T m g is a vector subspace, and that the canonical projection
T g → Sg sends W isomorphically onto S m g. Then πT (W ) is a compliment to U (m−1) (g)
in U (m) (g).
Pf. The following diagram of vector spaces commutes:
U (m) (g)
πT
π
T mg
U
can.
m
(31)
πS
m
S g
Using can.(W ) = S m g and the theorem, the composition of the maps on the bottom line
is an isomorphism from W onto its image, so therefore too the top line. Since π factors
U (m−1) (g) out of U (m) (g), the image of W must be complimentary to it.
Corollary 2.4 The canonical map g → U (g) is injective.
Pf. This is the case m = 1 in the previous lemma.
Corollary 2.5 (Poincare-Birkhoff-Witt, blue collar version.) If {x1 , . . . , xn } is an ordered basis of g, the monomials of the form
xi1 xi2 . . . xik ,
i1 ≤ i2 ≤ · · · ≤ ik
where
(32)
constitute a vector space basis of U (g).
Pf. Such monomials form a vector space basis of Sg.
Corollary 2.6 (Functorial properties) If π T : T g → U (g) is the canonical map and if
l ⊆ g is a subalgebra, then we have a natural embedding U (l) ⊆ U (g), and U (l) is generated
by π T (l). Further, U (g) is naturally a U (l)-module in the algebra sense.
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3
Proof of Poincare-Birkhoff-Witt
We begin by constructing an action of g on Sg, which we then prove gives Sg a module
structure.
Let {x1 , . . . , xn } be a basis of g, and let zi be their images in Sg under g → T g → Sg.
If Λ = (λ1 , . . . , λk ) is any multi-index, define
zΣ = zλ1 . . . zλk ∈ Sg
(33)
We write |Σ| = k, and if λ < λi (or λ ≤ λi ) for all λi ∈ Σ, we write λ < Σ (of λ ≤ Σ). The
following lemma defines the desired action.
Lemma 3.1 The symmetric space Sg has a unique g-module structure under the conditions
that if Σ is a multi-index with |Σ| = m then
Am ) xi .zσ = zi zΣ when i ≤ Σ
0
Bm ) xi .zΣ0 − zi zΣ0 ∈ S (|Σ |−1) whenever |Σ0 | ≤ |Σ|
Cm ) xi .xj .zT − xj .xi .zT = [xi , xj ].zT whenever |T | = |Σ| − 1.
Pf. By induction we can assume the action has been defined on S (m−1) ⊂ Sg satisfying
(Am−1 ), (Bm−1 ), and (Cm−1 ). Assuming that |T | = |Σ| − 1 we have that [xi , xj ].zT is
already defined. By (Am ) we have that
xi .zΣ
(34)
is defined when i < Σ. Let λ ∈ Σ be a member of Σ with λ < i and λ ≤ Σ, and let Σ0 be Σ
with λ missing. Since xλ .zΣ0 = zλ zΣ0 = zΣ is defined, we have by (Bm−1 ) that
xi .zΣ = xi .zλ zΣ0 = xi .xλ .zΣ0
= xλ .xi .zΣ0 + [xi , xλ ].zΣ0
(35)
Since xi .zΣ0 is defined and since λ ≤ (i, Σ0 ), the left side is now defined as well, and so the
action is entirely defined. It remains to check that (Cm ) holds. Going down one further
step, let µ ∈ Σ0 be so that µ ≤ Σ0 and let Σ00 be Σ0 without µ. Then by (Am−2 ) we have
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zΣ0 = xµ .zΣ00 , so
xi .xλ .zΣ0 − xλ .xi .zΣ0 = xi .xλ .xµ .zΣ00 − xλ .xi .xµ .zΣ00
= xi .xµ .xλ .zΣ00 − xλ .xµ .xi .zΣ00 + xi .[xλ , xµ ].zΣ00 − xλ .[xi , xµ ].zΣ00
= xµ .xi .xλ .zΣ00 − xµ .xλ .xi .zΣ00 + [xi , xµ ].xλ .zΣ00 − [xλ , xµ ].xi .zΣ00
+ xi .[xλ , xµ ].zΣ00 − xλ .[xi , xµ ].zΣ00
= xµ .[xi , xλ ].zΣ00 + [xi , xµ ].xλ .zΣ00 − [xλ , xµ ].xi .zΣ00
(36)
+ xi .[xλ , xµ ].zΣ00 − xλ .[xi , xµ ].zΣ00
= xµ .[xi , xλ ].zΣ00 + [[xi , xµ ].xλ ].zΣ00 − [[xλ , xµ ], xi ].zΣ00
= [xi , xλ ].xµ .zΣ00 + [xµ , [xi , xλ ]].zΣ00 + [xλ , [xµ , xi ]].zΣ00 + [xi , [xλ , xµ ]].zΣ00
= [xi , xλ ].zΣ0
Now we conclude the proof of the Poincare-Birkhoff-Witt theorem. The various canonical maps give us the following commutative diagram:
g
iU
iT
U (g)
(37)
Tg
πT
The previous lemma provided a Lie algebra homomorphism g → gl(Sg), which extends via
the various universal properties to the maps
g
iU
U (g)
θ
iT
gl(Sg)
ψ
(38)
Tg
πT
which, by uniqueness, implies
ψ = θ ◦ πT .
(39)
Now, lets consider an element t ∈ T g that maps under π T to 0 in U (g). By diagram (27)
we will have proven π S is an isomorphism if and only if
ker can = ker π T
8
(40)
Clearly ker can ⊆ ker π T , so we must prove the opposite inclusion. Assume t ∈ ker π T .
Because π T is a graded homomorphism, we can assume t ∈ T m g is homogeneous.
What π T (t) = 0 means is that πT (t) ∈ U (m−1) = πT (T m−1 g), so that there is some
t ∈ T m−1 g with πT (t) = πT (t0 ). But then
0
ψ(t − t0 ) = θ(πT (t − t0 )) = 0.
(41)
However we know how ψ works: it is symmetrization followed by multiplication in Sg. Thus
the symmetrization of t−t0 is zero, so t−t0 is purely antisymmetric, meaning t−t0 ∈ ker can.
However ker can is graded, and since t is the m-homogeneous component of t − t0 , we have
t ∈ ker can.
(42)
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