vertex algebras, hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfvertex...
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![Page 1: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/1.jpg)
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
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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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 2 / 30
![Page 3: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/3.jpg)
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 2 / 30
![Page 4: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/4.jpg)
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 2 / 30
![Page 5: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/5.jpg)
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 3 / 30
![Page 6: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/6.jpg)
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 3 / 30
![Page 7: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/7.jpg)
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 3 / 30
![Page 8: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/8.jpg)
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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 4 / 30
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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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 5 / 30
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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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 5 / 30
![Page 11: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/11.jpg)
Vertex algebras Definitions
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).
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 6 / 30
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Vertex algebras Definitions
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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 7 / 30
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Vertex algebras Definitions
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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 7 / 30
![Page 14: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/14.jpg)
Vertex algebras Definitions
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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 7 / 30
![Page 15: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/15.jpg)
Vertex algebras Definitions
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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 7 / 30
![Page 16: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/16.jpg)
Vertex algebras Definitions
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 8 / 30
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Vertex algebras Definitions
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 8 / 30
![Page 18: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/18.jpg)
Vertex algebras Definitions
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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 9 / 30
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Vertex algebras Definitions
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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 9 / 30
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Vertex algebras Examples
Example 1: Heisenberg vertex algebra
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)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 10 / 30
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Vertex algebras Examples
Example 1: Heisenberg vertex algebra
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)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 10 / 30
![Page 22: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/22.jpg)
Vertex algebras Examples
Example 1: Heisenberg vertex algebra
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)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 10 / 30
![Page 23: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/23.jpg)
Vertex algebras Examples
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
Mλ
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”.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 11 / 30
![Page 24: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/24.jpg)
Vertex algebras Examples
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
Mλ
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”.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 11 / 30
![Page 25: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/25.jpg)
Vertex algebras Examples
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 12 / 30
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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)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 13 / 30
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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)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 13 / 30
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Hopf algebras Definitions
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 14 / 30
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Hopf algebras Definitions
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 14 / 30
![Page 30: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/30.jpg)
Hopf algebras Definitions
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 14 / 30
![Page 31: Vertex algebras, Hopf algebras and latticestherisingsea.org/notes/albert-thesis-talk.pdfVertex algebras De nitions Alternative de nition The data (V;1;F) is called a vertex algebra,](https://reader036.vdocuments.us/reader036/viewer/2022071105/5fdf06e6480dc85b4a2e9eb9/html5/thumbnails/31.jpg)
Hopf algebras Definitions
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◦)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 15 / 30
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Hopf algebras Definitions
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◦)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 15 / 30
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Hopf algebras Definitions
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 16 / 30
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Hopf algebras Definitions
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 16 / 30
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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
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 17 / 30
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Hopf algebras Examples
Lemma
There is a bijection:
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))
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 18 / 30
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Hopf algebras Examples
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 19 / 30
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Universal measuring algebra
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))?
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 20 / 30
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Universal measuring algebra
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))?
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 20 / 30
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Universal measuring algebra
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))?
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 20 / 30
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Universal measuring algebra
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:
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 21 / 30
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Universal measuring algebra
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 22 / 30
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Universal measuring algebra
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 23 / 30
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Universal measuring algebra
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 23 / 30
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Universal measuring algebra
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 23 / 30
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Universal measuring algebra
Proof
Construct β(C ,A) as a quotient of Sym(C ⊗ A) making the followingdiagrams commute:
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 24 / 30
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Universal measuring algebra
Proof
Construct β(C ,A) as a quotient of Sym(C ⊗ A) making the followingdiagrams commute:
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 24 / 30
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Universal measuring algebra
Universal property
Algk(β(C ,A), k) ∼= Algk(A,C ∗)
Lemma
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 ∗)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 25 / 30
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Universal measuring algebra
Universal property
Algk(β(C ,A), k) ∼= Algk(A,C ∗)
Lemma
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 ∗)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 25 / 30
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Universal measuring algebra
Universal property
Algk(β(C ,A), k) ∼= Algk(A,C ∗)
Lemma
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 ∗)
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 25 / 30
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Universal measuring algebra
Jets
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))
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 26 / 30
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Universal measuring algebra
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 .
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 27 / 30
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Universal measuring algebra
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 .
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 27 / 30
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Universal measuring algebra
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 .
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 27 / 30
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Universal measuring algebra
Identify Hn = C[x ]≤n ∼= (C[t]/tn+1)∗ then H ∼= limn→∞
Hn.
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 28 / 30
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Universal measuring algebra
Identify Hn = C[x ]≤n ∼= (C[t]/tn+1)∗ then H ∼= limn→∞
Hn.
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 28 / 30
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Universal measuring algebra
Identify Hn = C[x ]≤n ∼= (C[t]/tn+1)∗ then H ∼= limn→∞
Hn.
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.
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 28 / 30
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Universal measuring algebra
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).
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 29 / 30
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Universal measuring algebra
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).
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 29 / 30
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Universal measuring algebra
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).
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 29 / 30
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Universal measuring algebra
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).
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 29 / 30
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Universal measuring algebra
References
http://therisingsea.org/
Borcherds, Quantum vertex algebras
Tuite, Vertex algebras according to Isaac Newton
Albert Zhang (University of Melbourne ) Vertex algebras, Hopf algebras and lattices October 2018 30 / 30