Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex...

Vertex algebras, Hopf algebras and lattices

Albert Zhang

University of Melbourne

October 2018

Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 1 / 30

Introduction

Overview

Borcherds defines a special class of vertex algebras as commutativealgebras in a category with singular multilinear maps.

Goal 1: Understand this construction.

Goal 2: Construct lattice vertex algebras via the universal measuringalgebra associated to the formal additive group and group algebra ofthe lattice. Then by a universal construction, lift the pairing on thelattice to a bicharacter inducing a singular multiplication map.

Introduction

Overview

Introduction

Overview

Vertex algebras

Introduction

Vertex algebras axiomise:

correlation functions of 2-dimensional conformal field theories,

an infinite dimensional Z-graded representation of the Monster groupwith modular functions j as the generating function of the dimensionsof its homogeneous subspaces,

singular commutative algebras in a certain category.

Vertex algebras

Introduction

Vertex algebras

Introduction

Vertex algebras

Introduction

”... in vertex operator algebra theory, there are essentially no examples,[...] that are easy to construct and for which the axioms can be easilyproved.” - Lepowsky and Huang

Vertex algebras Definitions

Definition

The space of A-valued formal distributions is defined by:

A[[z±1]] = {∑n∈Z

anz−n−1 | an ∈ A}

The commutation of two formal distributions is defined by:

[a(z), b(w)] =∑

m,n∈Z

[am, bn]z−m−1w−n−1

Definition

The space of A-valued formal distributions is defined by:

A[[z±1]] = {∑n∈Z

anz−n−1 | an ∈ A}

The commutation of two formal distributions is defined by:

[a(z), b(w)] =∑

m,n∈Z

[am, bn]z−m−1w−n−1

Definition: Mutually local

Two formal distributions α(z), β(z) ∈ End(V )[[z±1]] are said to bemutually local if there exists some integer n ∈ Z such that:

(z − w)n[a(z), b(w)] = 0(z)

This is denoted by a(z) ∼ b(z).

Definition

The data (V ,1,T ,Y ) is called a vertex algebra, where V is a vectorspace, 1 ∈ V is a distinguished vacuum vector, T ∈ End(V ) is a linearendomorphism, and Y is a linear map Y ( , z) : V → End(V )[[z±1]],satisfying the following axioms:

1 locality: Y (v , z) ∼ Y (w , z), for all v ,w ∈ V

2 creativity: Y (v , z)(1) = v + O(z), for all v ∈ V

3 translation covariance: [T ,Y (v , z)] = ∂Y (v , z),T (1) = 0

Definition

Alternative definition

The data (V ,1,F) is called a vertex algebra, where V is a vector space,1 ∈ V is a distinguished vacuum vector, and F is a collection of fields,satisfying the following axioms:

1 creativity: for all v ∈ V , there exists precisely one field Cv (z) ∈ Fsuch that Cv (z)(1) = v + O(z),

2 locality: Cv (z) ∼ Cw (z) for all v ,w ∈ V

Theorem: Uniqueness theorem

If (V ,1,F) is a vertex algebra, then there exists at most one translationoperator T ∈ End(V ) compatible with these fields.

Alternative definition

The data (V ,1,F) is called a vertex algebra, where V is a vector space,1 ∈ V is a distinguished vacuum vector, and F is a collection of fields,satisfying the following axioms:

1 creativity: for all v ∈ V , there exists precisely one field Cv (z) ∈ Fsuch that Cv (z)(1) = v + O(z),

2 locality: Cv (z) ∼ Cw (z) for all v ,w ∈ V

Theorem: Uniqueness theorem

If (V ,1,F) is a vertex algebra, then there exists at most one translationoperator T ∈ End(V ) compatible with these fields.

Example 1: Heisenberg vertex algebra

Definition

The Heisenberg Lie algebra is the vector space:

H =⊕n∈Z

Chn ⊕ C1

with Lie bracket defined by:

[hm, hn] = mδm,−n1, [1, hn] = 0

Definition

The Heisenberg Lie algebra is the vector space:

H =⊕n∈Z

Chn ⊕ C1

with Lie bracket defined by:

[hm, hn] = mδm,−n1, [1, hn] = 0

Vertex algebras Examples

The Heisenberg vertex algebra is the vector space:

M0 := U(H)/U(H+) = span{h−n1 ...h−nk (1) | n1, ..., nk ≥ 1}

with vacuum vector h−1(1), translation operator T =∑

n∈Z h−n−1hn, andthe field for h−1(1) defined by:

h(z) =∑n∈Z

hnz−n−1

By the reconstruction theorem, the other fields are determined by:

Y (h−n1 ...h−nk (1)) = (h ∗n1 (h ∗n2 ...(h ∗nk I )))(z)

h(z) =∑n∈Z

hnz−n−1

h(z) =∑n∈Z

hnz−n−1

Example 2: Lattice vertex algebras

Let N be even and L =√NZ the one-dimensional lattice. The lattice

vertex algebra VL is vector space:

VL =⊕λ∈L

where Mλ = M0 for all λ ∈ L.

The field for |λ〉 ∈ Mλ is given by:

Vλ(z) = Sλzλh0exp(−λ

∑n<0

hnnzn)exp(−λ

∑n>0

hnnzn)

called the ”bosonic vertex operator”.

Example 2: Lattice vertex algebras

Let N be even and L =√NZ the one-dimensional lattice. The lattice

vertex algebra VL is vector space:

VL =⊕λ∈L

where Mλ = M0 for all λ ∈ L. The field for |λ〉 ∈ Mλ is given by:

Vλ(z) = Sλzλh0exp(−λ

∑n<0

hnnzn)exp(−λ

∑n>0

hnnzn)

called the ”bosonic vertex operator”.

Lattice vertex algebras

Lemma (Borcherds, Quantum vertex algebras)

The lattice vertex algebra VL has the universal measuring algebraβ(H,C[L]) as the underlying vector space, where H = C[x ] is the Hopfalgebra associated to the algebraic group (C,+) and C[L] is the group ringof L.

Hopf algebras Definitions

Definition

A coalgebra is the triple (C ,∆, ε) with C a vector space, a map∆ : C → C ⊗ C called comultiplication, and a map ε : C → k called thecounit, making the following diagrams commute:

Notation

For x ∈ C , we denote the image of the comultiplication by:

∆(x) =∑

x(1) ⊗ x(2)

Definition

A coalgebra is the triple (C ,∆, ε) with C a vector space, a map∆ : C → C ⊗ C called comultiplication, and a map ε : C → k called thecounit, making the following diagrams commute:

Notation

For x ∈ C , we denote the image of the comultiplication by:

∆(x) =∑

x(1) ⊗ x(2)

Definition

The group-like elements of a coalgebra are the elements x ∈ C satisfying:

∆(x) = x ⊗ x , ε(x) = 1

Denote G (C ) the set of group-like elements of C .

Example

Let S be a set and k a field. Denote kS the vector space with S as abasis. The comultiplication and counit are defined by:

∆(s) = s ⊗ s, ε(s) = 1

There is a bijection:

(kS)∗ = Homk(kS , k) ∼= Maps(S , k)

that identifies (kS)∗ as an algebra.

Definition

∆(x) = x ⊗ x , ε(x) = 1

Example

∆(s) = s ⊗ s, ε(s) = 1

Definition

∆(x) = x ⊗ x , ε(x) = 1

Example

∆(s) = s ⊗ s, ε(s) = 1

Definition

The Sweedler dual space is defined for an algebra A by:

A◦ = {f ∈ Homk(A, k) | ker(f ) contains a cofinite two-sided ideal I}

Theorem

The Sweedler dual ( )◦ is a functor from the category of algebras to thecategory of coalgebras that is adjoint to the dual ( )∗. In other words, wehave the adjuction:

Algk(A,C ∗) ∼= CoAlgk(C ,A◦)

Definition

The Sweedler dual space is defined for an algebra A by:

A◦ = {f ∈ Homk(A, k) | ker(f ) contains a cofinite two-sided ideal I}

Theorem

The Sweedler dual ( )◦ is a functor from the category of algebras to thecategory of coalgebras that is adjoint to the dual ( )∗. In other words, wehave the adjuction:

Algk(A,C ∗) ∼= CoAlgk(C ,A◦)

Definition

A bialgebra is the data (H,∇, η,∆, ε) where (H,∇, η) is an algebra and(H,∆, ε) is a coalgebra such that ∆, ε are algebra maps.

Definition

A Hopf algebra is a bialgebra with an antipode S : H → H.

Definition

A bialgebra is the data (H,∇, η,∆, ε) where (H,∇, η) is an algebra and(H,∆, ε) is a coalgebra such that ∆, ε are algebra maps.

Definition

A Hopf algebra is a bialgebra with an antipode S : H → H.

Hopf algebras Examples

Example

The polynomial algebra C[x ] forms a Hopf algebra with:

∆(x) = 1⊗ x + x ⊗ 1, ∆(1) = 1⊗ 1

ε(x) = 0, ε(1) = 1

S(1) = 1, S(x) = −x

G (H◦) ∼= CoAlgk(k ,H◦) ∼= Algk(H, k)

and the set Algk(H, k) forms an algebraic group with the convolutionproduct defined by:

(f ∗ g)(x) =∑

f (x(1))g(x(2))

Proposition

There is a functor G ( ◦) from the category of Hopf algebras to thecategory of algebraic groups:

G ( ◦) : Hopfk → AlgGrp.

Example

The associated Hopf algebra to the algebraic additive group (C,+) isgiven by (C[x ],∆, ε) where ∆(x) = 1⊗ x + x ⊗ 1 and ε(x) = 0.

Universal measuring algebra

Suppose A is an algebra, B is a commutative algebra and C is a coalgebra:

Φ : Homk(A⊗ C ,B)∼=−→ Homk(A,Homk(C ,B))

Consider the subset Algk(A,Homk(C ,B)) ⊂ Homk(A,Homk(C ,B)).

Question

What maps in Homk(A⊗ C ,B) correspond to algebra maps inHomk(A,Homk(C ,B))?

Question

Proposition

Suppose ψ ∈ Homk(A⊗ C ,B). Then ρ := Φ(ψ) ∈ Homk(A,Homk(C ,B))is an algebra morphism if and only if ψ makes the following diagramscommute:

Definition

The pair (C , ψ), where C is a coalgebra and ψ : A⊗ C → B is a mapsatisfying the previous diagrams, is said to measure A to B.

Theorem

Suppose A is a commutative algebra and C is a cocommutative coalgebra.Then there exists a commutative algebra β(C ,A) with a measuring:

θ : A⊗ C → β(C ,A)

which is universal, in the sense that for any ψ ∈ Meas(A,C ;B) there is aunique algebra morphism F : β(C ,A)→ B that makes the followingdiagram commute:

This algebra with the morphism θ is defined to be the universalmeasuring algebra.

Theorem

θ : A⊗ C → β(C ,A)

Theorem

θ : A⊗ C → β(C ,A)

Construct β(C ,A) as a quotient of Sym(C ⊗ A) making the followingdiagrams commute:

Universal property

Algk(β(C ,A), k) ∼= Algk(A,C ∗)

This induces the bijection:

HomSchk (Spec(k),Spec(β(C ,A)) ∼= HomSchk (Spec(C ∗), Spec(A))

Example

Consider the scheme X = Spec(A) and C = (k[t]/t2)∗. Then the schemeTX = Spec(β(C ,A)) has the k-points given by:

(TX )k = HomSchk (Spec(k[t]/t2),X ) = Algk(A,C ∗)

Universal property

HomSchk (Spec(k), Spec(β(C ,A)) ∼= HomSchk (Spec(C ∗), Spec(A))

Example

Universal property

HomSchk (Spec(k), Spec(β(C ,A)) ∼= HomSchk (Spec(C ∗), Spec(A))

Example

Definition

The n-jet algebra is defined JnA = β(C ,A) for C = (k[t]/(tn+1))∗, andobserve that the scheme JnX = Spec(JnA) has the k-points:

(JnX )k = Algk(A, (k[t]/(tn+1))

Example

Consider an one-dimensional lattice L, its group ring A = C[L] andthe scheme X = Spec(A).

Recall the Hopf algebra H = C[x ] viewed as a coalgebra.

The universal measuring algebra β(H,A) can be considered thecoordinate ring of the infinite jet space of the scheme X .

Example

Identify Hn = C[x ]≤n ∼= (C[t]/tn+1)∗ then H ∼= limn→∞

Consider the scheme JnX = Spec(JnA) = Spec(β(Hn,A))

The inverse limit J∞X = lim←− JnX identifies the schemeSpec(β(H,C )) as the infinite jet scheme.

Summary

Take an even integral one-dimensional lattice L.

Take the algebraic additive group (C,+) and identify the associatedHopf algebra H = C[x ].

Take the group ring A = C[L] and construct the universal measuringalgebra β(H,A).

The algebra β(H,A) then forms the underlying vector space of thelattice vertex algebra which can be described as the coordinate ring ofthe infinite jet space of the scheme Spec(A).

Summary

References

http://therisingsea.org/

Borcherds, Quantum vertex algebras

Tuite, Vertex algebras according to Isaac Newton

Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex...

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