lecture 10 - stanford universitysporadic.stanford.edu/quantum/lecture10.pdf · lecture 10 daniel...

Borel subgroups New Hopf Algebras from Old A duality property of QTHA Quantized enveloping algebras

Lecture 10

Daniel Bump

May 29, 2019

1−aij∑k=0

(−1)k[

1− aij

k

]qE1−aij−k

i EjEki ,

1−aij∑k=0

(−1)k[

1− aij

k

]qF1−aij−k

i FjFki ,


Borel subgroups

A special role in the theory of Lie groups is played by the Borelsubgroups. For GL(r) these are the subgroups conjugate to thegroup B of upper triangular matrices.

If T is a maximal torus, then the Lie algebra g decomposes intoT-eigenspaces, called root spaces. The roots are decomposedinto positive and negative roots. The Lie algebra of B consists ofjust the positive root spaces, together with the Lie algebra of T.

There is an opposite Borel subgroup B− that includes T and thenegative roots. Although B has many conjugates, these two (Band B−) work together and we will take a closer look at them,using GL(3) as an example.


The Borel subgroup and its relatives GL(3)

Any complex reductive group G will have Borel subgroups, ofwhich we will pick a particular “positive” one B, and its negativeB−. The intersection of these will be a maximal torus T.Moreover B is a semidirect product TU where U is a unipotentsubgroups, and similarly B− = TU−. For GL(3):

B =

∗ ∗ ∗∗ ∗∗

, T =

∗ ∗∗

, U =

1 ∗ ∗1 ∗

1

,

B− =

∗∗ ∗∗ ∗ ∗

, U− =

1∗ 1∗ ∗ 1

.

Note that U is normal in B and T ∼= B/U.


The Bruhat decomposition

Let N(T) be the normalizer of T. Then W = N(T)/T is the Weylgroup, famous for acting on everything in sight. We have theBorel decomposition:

G =⋃

w∈W

BwB (disjoint).

In theis decomposition, each “Bruhat cell” BwB or BwB− hasdimension `(w), the length of w in the Coxeter group W. This isthe smallest number k such that w can be written as a productof k simple reflections.

Let w0 be the long Weyl group element, having maximal length.Then Bw0B is open and dense in G.


The Big Cell

We have w0Bw0 = B−. We have another decomposition

G =⋃

w∈W

BwB− (disjoint)

which may be deduced from the first. In this decomposition it isBB− that is the open cell. We may write B = UT and B− = TU−.

So UTU− = BB− is dense in G. In fact the multiplication map:

U × T × U− −→ G

is a homeomorphism of U × T × U− onto its image, which is adense open subset of G.


The triangular decomposition (continued)

This fact has an analog for the Lie algebras. Let

u = Lie(U), h = Lie(T), u− = Lie(U−).

Theng = u⊕ h⊕ u−.

This direct sum decomposition is fundamental. Together withthe Poincaré-Birkhoff-Witt theorem, it implies

U(g) = U(u)⊗ U(h)⊗ U(u−).

This decomposition is also fundamental.


In which Uq(u) is not a Hopf algebra

Let B = TU be the standard Borel subgroup of a complex Liegroup G. Here T is the standard maximal torus of B and U is themaximal unipotent subgroup. Then U is normal in B but T is not.

Let g, h, u and b be the Lie algebras of G, T, U and B. Theneach of the enveloping algebras U(h), U(t) and U(b) are allHopf subalgebras of U(g), which are closed under bothmultiplication and comultiplication.

For example, if G = SL(2) then u is spanned by E and since∆(E) = 1⊗ E + E ⊗ 1 it is closed under comultiplication. Thisfails for Uq(u) since

∆(E) = E ⊗ K + I ⊗ E.


Coideal subalgebras of a Hopf algebra

Dualizing the notion of an ideal of an algebra, a left coideal I ina coalgebra A is a subspace such that ∆(I) ⊂ A⊗ I.

Thus Uq(u), if we define it to be the subalgebra of Uq(g)generated by E is not a Hopf algebra, but it is a right coideal inthe Hopf algebra Uq(g).

Thus closed Lie groups of G correspond to Lie subalgebras ofg, or to Hopf subalgebras of U(g), but when we pass to thequantum group, the corresponding subalgebra of Uq(g) is notalways a Hopf subalgebra. But it may be a coideal subalgebra,and coideal subalgebras are the key to quantum analogs ofsymmetric spaces.

Letzter: Coideal subalgebras and quantum symmetric pairs

https://arxiv.org/abs/math/0103228


Borel subgroups and quasitriangularity

It is easy enough to construct Uq(sl2) from scratch. Similarconstructions “by hand” for more general quantum groups canbe given. But such constructions do not produce the R-matrix.

Drinfeld realized that it is possible to create quasitriangular thequantized enveloping algebra of G from the enveloping algebraof G by a doubling method.

One starts with the quantized enveloping algebra of B, doublesit, notices that there are two copies of U(h) and removes one ofthem. This procedure has the advantage of provingquasitriangularity.


Axioms of a Hopf Algebra are self-dual

The axioms of a Hopf algebra are self-dual. That is, each axiombecomes another axiom on the reversing the direction of thearrows. For example the associativity of the multiplication µ andthe coassociativity of the comultiplication ∆ are dual:

H ⊗ H ⊗ H H ⊗ H

H ⊗ H H

µ⊗1

1⊗µ µ

µ

H H ⊗ H

H ⊗ H H ⊗ H ⊗ H

∆

∆ ∆⊗1

1⊗∆

The Hopf axiom itself is self-dual:

H ⊗ H H ⊗ H ⊗ H ⊗ H H ⊗ H ⊗ H ⊗ H

H H ⊗ H

∆⊗∆

µ

1H⊗τ⊗1H

µ⊗µ

∆


The dual of a finite-dimensional Hopf algebra

An immediate consequence is that if H is a finite-dimensionalHopf algebra then the dual vector space H∗ is also a Hopfalgebra. If µ : H ⊗ H → H is the multiplication, the dual mapµ∗ : H∗ → H∗ ⊗ H∗ becomes the comultiplication in H∗ andsimilarly ∆∗ is the multiplication in H∗.

To avoid confusion, let us avoid using the notations µ∗ and ∆∗

for comultiplication and multipliction in H∗. Instead we will writem = ∆∗ : H∗ ⊗ H∗ → H∗ and D = µ∗ : H∗ → H∗ ⊗ H∗. We willdenote the dual pairing H∗ ⊗ H → K by 〈 , 〉.

Implicit in this is the identification (H ⊗ H)∗ = H∗ ⊗ H∗. So ifλ, µ ∈ H∗ then λ⊗ µ ∈ H∗ ⊗ H∗ is the functional

〈λ⊗ µ, x⊗ y〉 = 〈λ, x〉〈µ, y〉.


Duality in Sweedler notation

Now remembering that D : H∗ → H∗ ⊗ H∗ is dual to µ : H ⊗ Hgives the identity

〈Dλ, x⊗ y〉 = 〈λ, µ(x⊗ y)〉 = 〈λ, xy〉.

Now we employ Sweedler notation and write D(λ) = λ(1) ⊗ λ(2).(As usual, there is an implied summation.) Then

〈λ, xy〉 = 〈λ(1), x〉〈λ(2), y〉.

Dually, let us remember that the multiplication m in H∗ isdefined to be dual to the comultiplication in H. This gives us thecompanion formula

〈λµ, x〉 = 〈λ, x(1)〉〈µ, x(2)〉.


More general dual Hopf algebras

More generally suppose we have two Hopf algebras H and Aover a field K with a bilinear pairing A× H → K. Changingnotation slightly we will write the multiplication andcomultiplication in both Hopf algebras as µ and ∆. We assume

〈λ, xy〉 = 〈λ(1), x〉〈λ(2), y〉, 〈λµ, x〉 = 〈λ, x(1)〉〈µ, x(2)〉.

We also require

〈1A, x〉 = ε(x), 〈λ, 1H〉 = ε(λ), 〈S(λ), S(x)〉 = 〈λ, x〉.

Then we say H and A are in duality.

For example let G be an affine algebraic group over C. Let g bethe Lie algebra of the complex Lie group G(C). We may takeH = U(g) and A = O(G) to be the affine ring of rationalfunctions on G. Then H and A are Hopf algebras in duality.


The opposite Hopf algebra

Let (H, µ, η,∆, ε) denote the Hopf algebra H with multiplicationµ, unit η, comultiplication ∆ and unit ε.

Now we may reverse the multiplication, and define µop = µ ◦ τ .So µop(x⊗ y) = µ(y⊗ x) = yx. We do not need to reverse thecomultiplication. It is easy to check that (H, µop, η,∆, ε) is aHopf algebra. For example consider the Hopf axiom:

H ⊗ H H ⊗ H ⊗ H ⊗ H H ⊗ H ⊗ H ⊗ H

H H ⊗ H

∆⊗∆

µop

1H⊗τ⊗1H

µop⊗µop

∆

This diagram commutes since, both ways to x⊗ y

(yx)(1) ⊗ (yx)(2) = y(1)x(1) ⊗ y(1)x(2).


The opposite Hopf algebra (continued)

Alternatively, we may reverse the comultiplication and denote∆op = τ ◦∆. Assume that the antipode of H is invertible. Then(H, µ, η,∆op, ε) is also a Hopf algebra. There are thus four Hopfalgebras altogether:

H = (H, µ, η,∆, ε), Hop = (H, µop, η,∆, ε),Hcop = (H, µ, η,∆op, ε), H = (H, µop, η,∆op, ε).

The algebra (H∗)cop is important in the quasitriangularity storyand we will denote it H◦. (Again S must be invertible.)


Tensor product of Hopf algebras

If A and B are algebras, then A⊗ B is an algebra withmultiplication (a⊗ b)(a′ ⊗ b′) = aa′ ⊗ bb′. This multiplicationµA⊗B is

(A⊗ B)⊗ (A⊗ B) A⊗ A⊗ B⊗ B A⊗ B1A⊗τ⊗1B µ⊗µ .

Similarly if A, B are coalgebras, so is A⊗ B with comultiplication

A⊗ B A⊗ A⊗ B⊗ B (A⊗ B)⊗ (A⊗ B)∆⊗∆ 1A⊗τ⊗1B

And if A, B are Hopf algebras, so is A⊗ B.


Drinfeld Twisting

Twisting modifies the comultiplication of a Hopf algebra.

PropositionLet H be a Hopf algebra and let F be an invertible element ofH ⊗ H. Assume that

F12(∆⊗ 1)(F) = F23(1⊗∆)(F)

and that (1⊗ ε)(F) = (ε⊗ 1)(F) = 1. Define

∆F(x) = F∆(x)F−1.

Then we may replace the comultiplication in H by ∆F to obtainanother Hopf algebra with the same algebra structure.


Proof (I): coassociativity

The assumption F12(∆⊗ 1)(F) = F23(1⊗∆)(F) implies

(∆F ⊗ 1)(F)F12 = (1⊗∆F)(F)F23.

For coassociativity we want

(∆F ⊗ 1)(F∆(x)F−1) = (1⊗∆F)(F∆(x)F−1).

Since ∆F : H → H ⊗H is a homomorphism, this is equivalent to

(∆F ⊗ 1)(F) · F12(∆⊗ 1)(∆(x))F−112 · (∆F ⊗ 1)(F−1) =

(1⊗∆F)(F) · F23(1⊗∆)(∆(x))F−123 · (1⊗∆F)(F−1).

We see that these are equal.


Proof (concluded)

Note that ∆F(x) = F∆(x)F−1 is an algebra homomorphismH → H ⊗ H because ∆ is. The other assumption(1⊗ ε)F = (ε⊗ 1)(F) implies the counit axioms

H ⊗ H

H ⊗ I H

IH⊗ε∼=

∆

H ⊗ H

I ⊗ H H

ε⊗IH

∼=

∆

Indeed, apply ε⊗ 1 to ∆F(x) = F∆(x)F−1. Since the counitε : H → K is a homomorphism we get

(ε× 1)(F) · (ε× 1)(∆(x)) · (ε× 1)(F)−1 = 1 · x · 1−1 = x,

and similarly (1⊗ ε)∆F(x) = x.


Reminder of A◦

Let A be a finite-dimensional Hopf algebra. Its dual A∗ is a Hopfalgebra, and A◦ = (A∗)cop is obtained from A∗ by reversing thecomultiplication. Denote the comultiplication in A◦ by ∆◦.

Since A∗ and A are in duality if λ, µ ∈ A∗ and x, y ∈ A we have

〈λµ, x〉 = 〈λ⊗ µ,∆(x)〉, 〈∆λ, x⊗ y〉 = 〈λ, xy〉 .

These formulas may be taken to be the definitions of themultiplication and comultiplication in A∗. For A◦, themultiplication is the came as A∗, but the comultiplication isreversed.

〈∆◦λ, x⊗ y〉 = 〈λ, yx〉 .In Sweedler notation, if ∆(λ) = λ(1) ⊗ λ(2) then

∆◦(λ) = τ∆(λ) = λ(2) ⊗ λ(1).


A duality theorem

The following result may be due to Drinfeld but we are followingReshetikhin and Semenov-Tian-Shansky, Quantum R-matricesand factorization problems. We will prove:

TheoremLet A be a finite-dimensional quasi-triangular Hopf algebra withR-matrix R = R(1) ⊗ R(2). Define a map ρ : A◦ → A by

ρ(λ) = 〈λ,R(1)〉R(2).

Then ρ is a Hopf algebra homomorphism.

The homomorphism ρ may not be an isomorphism. Forexample if A is cocommutative and R = 1 then ρ = ε.


Proof (I)

Reminder:ρ(λ) = 〈λ,R(1)〉R(2).

First we prove that ρ(λµ) = ρ(λ)ρ(µ). Write the axiom(∆⊗ 1)R = R13R23 in the form

∆(R(1))⊗ R(2) = R(1) ⊗ R(1) ⊗ R(2)R(2)

where we’ve used two copies R on the right-hand side. Now

ρ(λµ) = 〈λµ,R(1)〉R(2)

= 〈λ⊗ µ,∆R(1)〉R(2) = 〈λ⊗ µ,R(1) ⊗ R(1)〉R(2)R(2)

= 〈λ,R(1)〉R(2)〈µ, R(1)〉R(2) = ρ(λ)ρ(µ).

Thus ρ : A◦ → A is an algebra homomorphism.


Proof (II)

Now let us show that (ρ⊗ ρ)∆◦(λ) = ∆ρ(λ), so that ρ is acoalgebra homomorphism. This time we will use(1⊗∆)R = R13R12, that is

R(1) ⊗∆R(2) = R(1)R(1) ⊗ R(2) ⊗ R(2).

Apply a⊗ b⊗ c 7→ 〈λ, a〉(b⊗ c) to both sides:

∆ρ(λ) = 〈λ,R(1)〉∆R(2) = 〈λ,R(1)R(1)〉(R(2) ⊗ R(2)) =

〈λ(1),R(1)〉〈λ(2), R

(1)〉(R(2) ⊗ R(2))

= 〈λ(2), R(1)〉R(2) ⊗ 〈λ(1),R

(1)〉R(2)

= ρ(λ(2))⊗ ρ(λ(1)) = (ρ⊗ ρ)∆◦(λ)

since ∆◦(λ) = λ(2) ⊗ λ(1).


A look ahead to the Drinfeld Double

Let A be a finite-dimensional Hopf algebra. Drinfeld defined aHopf algebra D(A) which contains copies of A and A◦. As acoalgebra it is A⊗ A◦. However the multiplication is modified.

Since A is finite-dimensional the antipode of A is known to beinvertible, which is required for the construction of the double.

The Drinfeld double is quasitriangular. The quasitriangularity ofUq(g) with q a root of unity can be deduced from this.


Weights

Let g be a complex reductive Lie algebra, and h a Cartansubalgebra. If g is the Lie algebra of a complex reductive Liegroup G, then h is the Lie algebra of a maximal torus T. It isabelian, and its elements are semisimple (diagonalizable) inany representation of G.

The group T is abelian, so its irreducible representations areone-dimensional. The group X∗(T) of characters is called theweight lattice. Each such character (weight) λ induces a linearfunctional on h so sometimes the weight lattice is identified witha subgroup of h∗.


Root space decomposition

Let Λ be the weight lattice. It may be identified with the group ofrational characters of T, or as a group of linear functional on h.Since T and h act on g via the adjoint representation, itdecomposes into weight eigenspaces. The nonzero charactersof T that occur are called roots; the set of roots is denoted Φ. Ifα ∈ Φ let Xα ⊂ g be the corresponding eigenspace. It isone-dimensional.

We haveg = h⊕

⊕α∈Φ

Xα.

We may decompose Φ = Φ+ ∪ Φ− with Borel subalgebras

b = h⊕⊕α∈Φ+

Xα, b− = h⊕⊕α∈Φ−

Xα.


Simple roots

If α is a positive root that cannot be decomposed into otherpositive roots, then α is called simple. If g is semisimple, thenthe number of simple roots equals the dimension of h;otherwise of h can be slightly larger. Let αi be the simple roots.Every positive root can be decomposed as a sum of αi.

For example let G = GL(r). Then we may identify Λ = Zr. Thusif λ = (λ1, · · · , λr) we identify λ with the weight (character of T)

λ

t1. . .

tr

=∏

i

tλii .

If ei are the standard basis vectors in Zr the roots are ei − ej

where i 6= j. The root is positive if i < j and simple if j = i + 1.


Cartan matrix

We choose a W-invariant inner product on the weight lattice Λ.With αi be simple roots, let

aij =〈2αi, βj〉〈αi, αi〉

.

The matrix A = (aij) is called the Cartan matrix.

For example, if g = sl4, the Cartan matrix is:

A =

2 −1 0 0−1 2 −1 00 −1 2 −10 0 −1 2

.

If g is semisimple the Cartan matrix contains enoughinformation to reconstruct g.


Cartan matrix (continued)

The Cartan matrices that produce finite dimensional Liealgebras were classified by Cartan. In the 1970’s Kac showedmuch theory generalizes to infinite dimensional Lie algebrasobtained from more general Cartan matrices. For example

A =

2 −1 0 0 −1−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −1−1 0 0 −1 2

produces the affine Lie algebra sl4 (See Lecture 6).

Kac-Moody Lie algebras occur naturally in mathematicalphysics, e.g. as current algebras. Kac-Moody Lie algebras alsohave nice quantized enveloping algebras.


Chevalley basis

Before Chevalley, finite groups of Lie type such as SL(n,Fq) andthe finite orthogonal and symplectic groups were constructedon a case-by-case basis. There was interest in these becausemany finite simple groups were obtained this way; for examplethe quotient of SL(n,Fq) by its center is usually simple.

Chevalley showed that every complex semisimple Lie algebra ghad a basis with structure constants in Z. Then the adjointgroup becomes a group scheme over Z. We can take its pointsover a finite field and obtain the groups of finite type over finitefields. This procedure had to be supplemented by Galoistwisting (Steinberg, Suzuki, Tits, Ree). Together with thealternating groups, finite groups of Lie type account for all but26 of the finite simple groups.


Serre presentation

If g is a Lie algebra, the adjoint representation ad : g→ End(g)is the map ad(X)(Y) = [X,Y]. It is a Lie algebra homomorphism.

Let A be a Cartan matrix of finite type. Let r be the rank, that is,the dimension of h. Then A is r × r. The finite-dimensionalsemisimple Lie algebra g has 3r generators Hi, Ei and Fi withrelations

[Hi,Hj] = 0, [Ei,Fj] = δijHi,

[Hi,Ej] = aijEi, [Hi,Fj] = −aijFj,

ad(Ei)1−aijEj = ad(Fi)

1−aijFj = 0, (i 6= j).

The last relations are called the Serre relations. Note that theexponent 1− aij is positive since aij 6 0 when i 6= j.


Serre relations

For example, if g = sl4,

A =

2 −1 0 0−1 2 −1 00 −1 2 −10 0 −1 2

.

the Serre relation tells us [Ei,Ej] = 0 if |i− j| > 1 but

[Ei, [Ei,Ej]] = 0, j = i± 1.

To translate the Serre presentation to generators and relationsfor U(g) we replace [A,B] by AB− BA. So this relation becomes

E2i Ej − 2EiEjEi + EjE2

i = 0.


Generators and relations for U(g)

To translate the Serre presentation to generators and relationsfor U(g) we replace [A,B] by AB− BA. We obtain:

HiHj − HjHi = 0, EiFj − FjEi = δijHi,

HiEj − EjHi = aijEi, HiFj − FjHi = −aijFi,

1−aij∑k=0

(−1)k(

1− aij

k

)E1−aij−k

i EjEki ,

1−aij∑k=0

(−1)k(

1− aij

k

)F1−aij−k

i FjFki .

The Serre relation has been written using binomial coefficients.


The quantized enveloping algebra

To obtain the quantized enveloping algebra, we replace thegenerators Hi by group-like invertible generators Ki. These areassumed to commute.

The root system is called simply-laced if all roots have thesame length. For simplicity we will assume this to be the case.We normalize the inner product on Λ so that 〈α, α〉 = 2 if α is aroot. If

KiEjK−1i = q〈αi,αj〉Ej, KiFjK−1

i = q−〈αi,αj〉Fj,

EiFj − FjEi = δijKi − K−1

iq− q−1 .

We still have to discuss the Serre relations.


The quantum Serre relations

For the Serre relations we replace the binomial coefficients bythe corresponding Gaussian binomial coefficients.

1−aij∑k=0

(−1)k[

1− aij

k

]qE1−aij−k

i EjEki ,

1−aij∑k=0

(−1)k[

1− aij

k

]qF1−aij−k

i FjFki ,

Here i 6= j and[mn

]q

=[m]q!

[n]q![m− n]q!, [m](q) =

qm − q−m

q− q−1 ,

[m]q! =

m∏k=1

[k]q.


The quantum Serre relations (continued)

Actually we are assuming that the root system is simply-laced,so if i 6= j then aij = 0 or −1. The quantum Serre relationrelation says that if αi and αj are orthogonal then Ei and Ej

commute, otherwise aij = −1 and

E2i Ej − (q + q−1)EiEjEi + EjE2

i = 0

because [21

]q

= [2]q = q + q−1.


comultiplication

The quantized enveloping algebra is a Hopf algebra. Thecomultiplication is given on generators by:

∆(Ki) = Ki ⊗ Ki,

∆(Ei) = Ei ⊗ Ki + 1⊗ Ei, ∆(Fi) = Fi ⊗ 1 + K−1i ⊗ Fi.

If q is a root of unity, then Uq(g) has a finite-dimensionalquotient that is quasitriangular. If q is not a root of unity, thenUq(g) is quasitriangular in a generalized sense.

In order to see why these Hopf algebras are quasitriangular wewill take a different approach based on the Drinfeld double.

lecture 10 - stanford universitysporadic.stanford.edu/quantum/lecture10.pdf · lecture 10 daniel...

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