Representations of Uq(sl2S1) at roots of unity

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Universiteit van Amsterdam

MSc Mathematical Physics

Master Thesis

Representations of U

q

(sl

2∣1

)

at roots of unity

Tim Weelinck

June 26, 2015

Supervisor:

Prof. Dr. N.R.

Reshetikhin

Second Reader:

Dr. H.B. Posthuma

Korteweg-de Vries Instituut voor Wiskunde

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at roots of unity

Tim Weelinck

Abstract

In this thesis representations of Uq(sl2∣1) at roots of unity are studied. The representation theory is studied by applying techniques and methods developed

by C. De Concini, V. Kac and C. Procesi for quantum groups at roots of unity [DCK90, DCKP92].

The study of simple representations of Uq(sl2∣1) is reduced to the study of a family, Ax, of finite dimensional quotients of Uε(sl_2∣1)parametrized by an affine variety Ω = Spec(Z0). Here Z0 is some central sub-Hopf algebra of Uε(sl2∣1). It is

shown that Z0 has a canonical Poisson bracket, inducing an isomorphism between

Poisson manifolds Ω and (SL∗

2∣1)∅, the base space of the dual Poisson-Lie group.

An action of an infinite group on Ω is defined, the so-called quantum coadjoint action. It is shown that orbits of the quantum coadjoint action are exactly the symplectic leaves in Ω. It is deduced that Ax is generically semimsimple over Ω

by concretely describing the generic simple representations. Moreover, it is shown that the family of algebras Ax can be seen as a trivial vector bundle over

Ω, and that for two points x, y in the same symplectic leaf we have that Ax is

isomorphic to Ay as algebras.

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Acknowledgements

First and foremost, I would like to thank my supervisor Nicolai Reshetikhin. I am thankful for the time taken supervising this project and the wealth of ideas -and new approaches- he shared with -and suggested to- me. I am also thankful for pleasant conversations on visions on mathematics, and life in and outside of math. Secondly, I wish to thank Hessel Posthuma for being second reader, and for helpful discussions on Poisson geometry.

Thirdly, I want to give my thanks to Sacha, Iordan and David for fruitful conversations while in Berkeley.

Fourthly, I want to thank all the students at KdV for enjoyable coffee breaks, lunches, and especially Reinier for assisting me geometrically.

Lastly, I want to express my sincerest gratitude towards Juultje, for everything.

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Introduction

In this thesis we study the representation theory of the quantum supergroup Uq(sl2∣1)at roots of unity. We should add that U_q(sl_2∣1)is not really quantum, nor is it a supergroup. What then, is Uq(sl2∣1)? To answer this question we will jump

back in time to the early 80s.

Theoretical physicsts were studying the quantum inverse scattering method (QISM), a method to solve quantum integrable systems developed by Ludvig Faddeev and his group at the Leningrad School. In 1981 Petr Kulish and Nicolai Reshetikhin [KN83] described what would later be recognized as Uq(sl2), the first example of

a quantized universal enveloping algebra.

We refer to chapter one to learn what is meant by ‘universal enveloping algebra’. We will take some time to explain the term ‘quantized’. Nowadays, the word has its own meaning in mathematics, but this is of course a term borrowed from physics. Let us start with the physics and then distill from it what we shall mean with quantization.

Intermezzo: quantization

In the early 20th century physics was split into two seemingly separate worlds: the classical (macroscopic) world and the quantum (smallest scale) world. To describe a physical system in classical mechanics, one specifies certain data: a smooth Poisson manifold M playing the role of the space of possible states, and a set of observables, which are functions on the manifold M . The evolution of the system over time is represented by a smooth path m(t) in M . Given a starting point m(t0), the following set of equations

d

dtf (m(t)) = {Hcl, f }(m(t))

determines the evolution of m(t) over time. Here Hcl denotes some special

function called the classical Hamiltonian, and {⋅, ⋅} denotes the Poisson bracket.

In quantum mechanics, one specifies similar data, but the mathematical structures are different. The space of states is now a complex Hilbert space, and the observables are non-commutative operators on the Hilbert space. A special operator, called the quantum Hamiltonian Hqu, now encodes the

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time evolution of an operator A as follows dA

dt = [Hqu, A]

Can one pass from the classical situation to a the quantum situation? This is exactly the problem of quantization. One approach, due to Jos´e Moyal, is to keep the space of functions on the manifold as space of observables, but to replace the product between functions by some non-commutative product. Let us make this more precise: denote F (M ) the set of functions

on M , and denote ⋆0 the normal commutative product on F (M ). We

define a family of products ⋆h depending on some parameter h (h can be

a complex number for example), such that ⋆0 is the old product, and ⋆h

satisfies some smoothness condition with respect to h. One should think of h as Planck’s constant. As Planck’s constant tends to 0, h → 0, quantum mechanics becomes classical mechanics, and from Physical motivation we ask that

lim

h→0

f ⋆hg − g ⋆hf

h = {f, g} (*)

Of course a mathematician does not need to restrict its attention to func-tion algebras on spaces of states, one can consider any commutative algebra A with a Poisson bracket and wonder whether we can deform the product to some product ⋆h satisfying (∗). This is what we will mean by a

‘quan-tizations’.1

Returning to Kulish and Reshetikhin, it was later realized they had defined a family of algebras Uq(sl2), dependent on some complex parameter q, that should

also be considered as a quantization. Mid 1980’s Vladimir Drinfel’d and Mi-chio Jimbo independently generalized the construction of Uq(sl₂), by associat-ing a quantum group Uq(g) to any simple Lie group g. In his famous address

[Dri87] in 1986 in Berkeley, Drinfel’d popularized the name ‘quantum group’ for these objects. Drinfel’d developed the basis for the theory of quantum groups, introducing concepts such as Poisson-Lie groups and Lie bialgebras. Although quantum groups were initially conceived to produce non-trivial solutions to the Yang-Baxter equation, nowadays they are connected to many diverse mathemati-cal fields.2 _{For example, quantum groups have become connected to knot theory}

and low-dimensional topology, most famously perhaps in the construction of the renowned Reshetikin-Turaev invariants [RT91].

In the early 90s Victor Kac, Corrado De Concini and Claudio Procesi published a series of papers studying the properties of quantum groups at roots of unity,

1_{This of course does not constitute a rigorous definition, but it is the right idea. See definition}

1.2.3 for an improved definition.

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3

amongst others [DCK90, DCKP92], and an enlarged summary in [DCP93]. By utilizing methods from smooth and algebraic geometry they were able to lay bare many interesting properties of the representation theory of the quantum group at roots of unity. This thesis should be seen as a direct succesor to the ideas in these papers. We consider a specific quantum supergroup and utilize the techniques developed by De Concini, Kac and Procesi to study its representation theory. The purpose of this project is twofold. Firstly, we show that many of the techniques and methods developed in [DCK90, DCP93, DCKP92] are almost directly appli-cable to Uq(sl_2∣1). This gives strong evidence that the same should hold for more general basic Lie superalgebras. Secondly, although the papers [DCK90, DCKP92] have a certain status in the field, the author has found that many PhD students working in areas closely related to quantum groups have not actually read these papers themselves.

The author conjectures that the style of writing of these papers has a lot to do with that. To illustrate this point, an important result in [DCP93, prop. 11.8] which has but four lines as proof, has grown in size to a proof of more than a page in this thesis as proposition 3.4.1. We do not believe this difference in size is due to a misunderstanding of the theorem or its proof.

The author has modest hopes that by providing a detailed ‘worked example’ in this thesis, the original work of De Concini, Kac and Procesi becomes accesible to a wider audience.

Organisation and main results

The thesis is written having a first year master student, or advanced undergraduate student, in mind as audience. We hope the style does not come across as pedantic, but we have tried to consistently choose clarity over brevity. We hasten to add that this holds for the ‘main part’ of the thesis, being chapter two and three. We have added chapter one to provide a fuller picture to the reader, but the chapter is in no way meant as a comprehensive introduction to quantum supergroups.

Chapter One: Preliminaries on Quantum Supergroups

This chapter is written with two goals in mind. On the one hand, we try to get the reader up to speed with quantum supergroups. We introduce the mathemat-ical definitions underlying the object Uq(sl2∣1) such as super Hopf algerbras, Lie superbialgebras and Poisson-Lie supergroups. The treatment is rather brief, but references are provided on all important topics.

On the other hand, the chapter is written to make precise in what way Uq(sl2∣1) quantizes what Poisson manifold. If the reader is to take anything from the chapter it should be the following statement.

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• Uq(sl_2∣1) endows O(SL∗

2∣1) with the structure of a Poisson-Hopf algebra.

Hence O((SL∗

2∣1)∅) also has the structure of a Poisson-Hopf algebra.

Chapter Two: Representation Theory of Uε(sl2∣1)

We study Uq(sl2∣1) at a root of unity. We prove the following structure theorems in section one.

• Let ε be an `th root of unity. Uε(sl_2∣1) has a central sub-Hopf algebra Z₀ generated by the `th powers of the even generators.

• Uε(sl2∣1) is a free Z₀-module of dimension 16`4.

In the second and third section we define a family of Uε(sl2∣1)-modules dependent

on a set of four parameters living in C2_×

C∗2. Excluding some Zariski closed set in C2

×_C∗2 this family consists of `2 non-isomorphic simple modules of dimension 4`.

In the fourth section we prove that

Z0 ≅ O((SL2∣1)∅) (*)

as Hopf algebras. This allows us to study representations of Uε(sl2∣1)over the affine variety Ω ∶= maxSpec(Z0) ≅ (SL∗_2∣1)∅. In particular we identify the parameters

that defined a family of representations, as points in Ω, and prove the following theorem.

• There exists a Zariski open subset of Ω such that Ax∶=U_ε/m_xU_εis a semisim-ple algebra of dimension 16`4 _{over those points.}3 _{The semisimple algebra}

splits up as a direct sum of `2 _{simple modules of dimension 4`.}

Chapter Three: The Quantum Coadjoint Action

We begin this chapter by substantially improving the isomorphism *, with the following result.

• Z0 has a canonical Poisson bracket defined by

{a, b} = lim

q→ε

ab − ba `(q`₋_q−`₎

this endows Z0 with the structure of a Hopf algebra and as

Poisson-Hopf algebras Z0 ≅ O((SL∗_2∣1)∅).

Now that Ω is endowed with a Poisson structure, we wish to study the interaction between Poisson geometry and representation theory. In section two we prove the following results.

3_{Here m}

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5

• We can naturally associate a Lie algebra ˜g of complete vector fields on Ω to the generators of Z0.

• The global flows corresponding to the complete vector fields in ˜g define an infinite group ˜G acting on Ω (the quantum coadjoint action). The orbits of

˜

G are are exactly the symplectic leaves of Ω.

In section three we elucidate the structure of the symplectic leaves, by proving the following results

• There exists a natural 4-1 covering map π from Ω to the big cell G0 _⊂

(SL_2∣1)∅.

• The Lie algebra generated by the infinitesimal generators of the conjugation action of (SL2∣1)∅ lift to a Lie algebra of vector fields g on Ω.

• At every point of Ω we have that g and ˜g span the same space in the tangent space to Ω.

• Let O denote a conjugacy class in SL2∣1. Then O0 = G0 ∩ O is a smooth

connected variety and the connected components of π−1_(O0₎_{are orbits of the}

group ˜G.

Hence, we can use the structure of conjugacy classes in (SL2∣1)∅ to study the

symplectic leaves in Ω. For example, we immediately obtain the structure of the fixed points of ˜G.

• Denote F the fixed points of ˜G, then F = {(z1, z2, b, c) ∈ Ω ∶ z1 =1, b = c = 0}. The last section is devoted to proving the following important theorem

• The family of algebras Ax ∶= U_ε(sl_2∣1)/m_xU_ε(sl_2∣1) defines a trivial vector bundle of rank 16`4 _{over Ω. For two points x, y in the same sympletic leaf}

we have that Ax≅Ay as algebras.

Appendices

In the appendices we have placed the theory that we could not give a satisfying place within the thesis.

Appendix A

The first three appendices give a short introduction to super mathematics, and are intended to provide the necessary background knowledge to read chapter one. Similar to the approach in chapter one, many results are stated without proof, but rather are references provided.

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Appendix B

The appendix on the PBW theorem, provides a brute force proof of the PBW theorem for Uq(sl2∣1).

Appendix C

The appendix on representation theory is a compact introduction to Jacobson the-ory of semisimple algebra, culminating in the proof of the following statement Let A be a finite dimensional algebra of dimension n, let Mi be a set of

non-isomorphic simple modules of A such that dim(A) = ∑idim(Mi)2. Then 1. A is a semimsimple algebra.

2. A ≅ ∏iEnd(Mi) as algebra.

3. Mi constitute a complete set of simple modules.

The material in this appendix is standard and can be found in many books, for example [Lan02]. However, the proof of theorem C.0.35 is original, in the sense that the author is unaware of a similar proof in the literature.

Appendix D

The introduction on Spec(R) was written with the student in mind who has never attended a course on algebraic geometry. The appendix is rather brief, as we do not need much algebraic geometry.

Appendix E

The final appendix deals with quantum calculus, whose identities we will need at several points of the thesis. We have chosen to give a elaborate introduction there, more than necessary at least, mainly because we feel q-calculus is a lot of fun to play around with.

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Chapter 1 Preliminaries on Quantum

Supergroups

In this first chapter we will introduce the concepts that are preliminary to the object that we wish to study: the quantum supergroup Uq(sl_2∣1). We have tried to introduce those concepts that a ‘generic first year graduate student’, will not be familiar with. For basics on super mathematics we refer to the appendix. In contrast with the rest of the thesis the emphasis will be on stating results without proof, instead providing references. The idea being that the this section creates context and motivation for the further thesis, but is not vital to appreciating the representation theoretic results in the rest of the thesis. The main goal of this chapter can be summarized as: explaing what it means that the super Hopf algebra Uq(sl2∣1) quantizes O(SL∗_2∣1).

1.1 Hopf superalgebras and enveloping

superal-gebras

A Hopf algebra is an algebra that has a compatible structure as a coalgebra. Moreover, a Hopf algebra is endowed with a special antihomomorphism. They naturally appear as the function algebras on (smooth, algebraic) groups.

Historical Remark. In [Dri87] Drinfel’d writes that quantum groups are more or less the same as Hopf algebras. Unfortunately, there is still no satisfactory definition of quantum group today. Drinfel’d reasons as follows. If one considers a group (Lie group resp. algebraic group) the (smooth resp. regular) functions on that group naturally have the structure of a commutative Hopf algebra. In fact this gives an anti-equivalence of categories, if one matches the right type of group to the right

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type of Hopf algebra. The non-commutative Hopf algebras are to be considered as non-commutative functions on a ‘quantum group’.

A thorough treatment of Drinfel’ds philosophy would require too much space. Instead, we will define the central concepts of super Hopf algebra, Lie superbialge-bra and Poisson-Lie supergroup and work out the example of how the super Hopf algebra Uq(sl2∣1) quantizes the function algebra O(SL∗_2∣1).

Remark. Super Hopf algebras are simple generalizations of Hopf algebras. Readers familiar with Hopf algebras and universal enveloping algebras are advised to skip to the end of this section.

The part ‘co’ in coalgebra refers to a dual structure. In this case a dualized algebra structure, similarly supercoalgebras are dual superalgebras.1 _{We give a}

definition of superalgebra different from the one in the appendix, because it is easier to dualize.

Definition 1.1.1. An associative superalgebra with unit is a triple (A, µ, η) where A is a super vector space and two even linear maps, the product µ ∶ A ⊗ A → A and the unit η ∶ C → A, satisfying the conditions (Un) and (Ass).

(Un) The diagram

A ⊗ A ⊗ A A ⊗ A A ⊗ A A id ⊗ µ µ ⊗ id µ µ commutes.

(Ass) The diagram

C ⊗ A A ⊗ A A ⊗ C

A

η ⊗ id id ⊗ η

≅ ≅

commutes.

A superalgebra A is called (super)commutative if, in addition, the condition (Comm) holds

(Comm) The diagram

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Hopf superalgebras and enveloping superalgebras 9 A ⊗ A A ⊗ A A τ η η commutes.

Here τ ∶ A ⊗ A → A ⊗ A is the graded flip, defined by τ (a ⊗ b) = (−1)∣a∣∣b∣_{b ⊗ a.}

A morphism of superalgebras f ∶ (A, µ, η) → (A′_{, µ}′_{, η}′

) is an even linear map f ∶ A → A′ _{such that}

η′_{○ (f ⊗ f ) = f ○ η} _and _{f ○ η = η}′_.

Example 1.1.2. Let V be a super vector space. We define super vector space T (V ) = ⊕n>0Tn(V ), where T0(V ) = C, Tn(V ) = V⊗n for n > 0. The canonical isomorphisms Tn_{(V ) ⊗ T}m_{(V ) ≅ T}n+m_{(V ) induce an associative product on T (V )}

explicitly given by

(x1⊗. . . ⊗ x_n) ⋅ (x_n+1⊗. . . x_n+m) =x₁⊗. . . ⊗ x_n+m. We call this the tensor algebra of V .

The definition of superalgebra is readily dualized by reversing all arrows. Definition 1.1.3. A coassociative supercoalgebra with counit is a triple (C, ∆, ε) where C is a super vector space and two even linear maps, the coproduct ∆ ∶ C → C ⊗ C and the counit ε ∶ C → C, such that the conditions (Coun) and (Coass) hold. (Coun) The diagram

C ⊗ C C ⊗ C C ⊗ C C ε ⊗ id id ⊗ ε ∆ ≅ ≅ commutes.

(Coass) The diagram

C C ⊗ C

C ⊗ C C ⊗ C ⊗ C

∆

∆ id ⊗ ∆

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commutes.

A supercoalgebra C is called co(super)commutative if the condition (Cocomm) holds.2

(Cocomm) The diagram

C ⊗ C C ⊗ C C τ ∆ ∆ commutes. A morphism of supercoalgebras f ∶ (C, ∆, ε) → (C′_{, ∆}′_{, ε}′

) is an even linear map f ∶ C → C′ _{such that}

(f ⊗ f ) ○ ∆ = ∆′

○f and ε′○f = ε.

Remark. (i) Note that in the definitions of supercoalgebra and superalgebra we are interpreting C as C1∣0_{. Here C}1∣0_{denotes the super vector space with even part}

C1∣00 =C and odd part C 1∣0

1 = {0}. In particular we ask that η(C) ⊂ A0 and that

ε(C1) = {0}.

(ii) To any (co)superalgebra we can associate two natural (co)algebras. Firstly, since all defining maps are even, (A0, µ, η) resp. (C0, ∆, ε) define an algebra resp.

coalgebra. However, using again that all maps are even, we can also define the algebra (A/A1, µ, η) resp. coalgebra (C/C1, ∆, ε). These do not define the same

(co)algebras in general.3 _{This observation is closely related to the fact that for a}

supermanifold (M, A) the algebra C∞_{(M ) is not a subalgebra of A(M ) in general,}

but rather a quotient.

Example 1.1.4. [Kas95, p. 41]

(i) (Dual Coalgebra) If (C, ∆, ε) is a supercoalgebra, then C∗

=Hom_C(C, C) natu-rally carries a superalgebra structure. Denote λ ∶ C∗

⊗C∗→ (C ⊗ C)∗, the natural map defined by λ(f ⊗ g)(v ⊗ u) ∶= f (u) ⊗ g(v), τ the graded flip.

Then (C∗_{, ∆}∗

○λ ○ τ, ε∗) is a superalgebra.

(ii) (Dual Algebra) Let (A, µ, η) be a finite dimensional superalgebra. Then λ ∶ A∗

⊗A∗→ (A ⊗ A)∗ is an isomorphism. A∗ has a natural supercoalgebra structure defined by (A∗_{, τ ○ λ}−1

○µ∗, η∗).

Notation. (Sweedler’s sigma notation) Let (C, ∆, ε) be a supercoalgebra. For x ∈ C the element ∆(x) ∈ C ⊗ C is of the form

∆(x) = ∑ i x′ i⊗x ′′ i.

2_{We will always drop the super in cosupercommutative. For example, we will speak of}

co-commutative supercoalgerbas.

3_{In fact we can improve these associations to functors from the category of (co)superalgebras}

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Hopf superalgebras and enveloping superalgebras 11

Henceforth, we shall agree to drop the subscripts and write ∆(x) = ∑

(x)

x′

⊗x′′

instead.

Coassociativity allows us to unamibiguously write ∆ ⊗ id ○ ∆(x) = id ⊗ ∆ ○ ∆(x) = ∑

(x)

x′

⊗x′′⊗x′′′,

and in fact using coassociativity repeatedly allows us to use the same notation for higher powers of the coproduct. Using Sweedler’s sigma notation we can rewrite the conditions (Coun) and (Cocomm) in the following short form:

∑ (x) ε(x′ )x′′ = ∑ (x) ε(x′′ )x′ =x ∀x ∈ C, (Coun) ∑ (x) x′ ⊗x′′= ∑ (x) (−1)∣x′∣∣x′′∣_x′′ ⊗x′ ∀x ∈ C. (Cocomm)

If a super vector space has both superalgebra and supercoalgebra structures we can ask for two natural compatility conditions.

Definition 1.1.5. A superbialgebra is a quintuple (B, µ, η, ∆, ε) such that (A, µ, η) is a superalgebra and (B, ∆, ε) is a supercoalgebra and one of the following equiv-alent conditions holds

• ∆ and ε are superalgebra morphisms. • µ and η are supercoalgebra morphisms.

Proof. The proof can be made entirely visual, by drawing the four commuting diagrams that correspond to each condition and observing they are the same four diagrams[Kas95, th. 3.2.1].

Using Sweedler’s notation we can reformulate the first condition into the fol-lowing short form:

∑ (xy) (xy)′ ⊗ (xy)′′ = ∑ (x)(y) (−1)∣y′∣∣x′′∣ x′ y′ ⊗x′′y′′, ∆(1) = 1 ⊗ 1, ε(xy) = ε(x)ε(y), ε(1) = 1. Example 1.1.6. [Kas95, p. 46-47]

(i)Let V be a super vector space, there exists a unique superbialgebra structure on the tensor algebra T (V ) such that ∆(v) = v ⊗ 1 + 1 ⊗ v and ε(v) = 0 for any element v ∈ V . It is cocommutative.

(ii) (Dual Bialgebra) Let B be a finite dimensional superbialgebra. Then the dual space B∗ _{naturally has the structure of a superbialgebra.}

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Definition 1.1.7. A Hopf superalgebra is a sextuple (H, µ, η, ∆, ε, S) such that (H, µ, η, ∆, ε) is a superbialgebra and S ∶ H → H is an even linear anti-homomorphism of superalgebras satisfying the condition (An-Po)

(An-Po) The diagram H ⊗ H H ⊗ H H _C H H ⊗ H H ⊗ H S ⊗ id id ⊗ S ∆ ∆ µ µ ε η commutes.

We can reformulate the condition (An-Po) in compact Sweedler’s notation: ∑ (x) S(x′ )x′′ = ∑ (x) x′ S(x′′ ) =ε(x)η(1), ∀x ∈ H. (An-Po)

Remark. Let (B, µ, η, ∆, ε) is a superbialgebra. The bialgebra structure has im-portant consequences for the representation theory of B. The algebra B has a trivial module C on which B acts through ε. Moreover, let M, N be A-modules. Then ∆ makes M ⊗ N into an B-module, by b ⋅ m ⊗ n = b′_{m ⊗ b}′′_{n. Rephrased}

categorically, M od(B), the category of B-modules, can be given the structure of a monoidal category [KRT97,§2.1].

Example 1.1.8. (i) (Group algebra) This group algebra is not a super Hopf alge-bra, but nevertheless a very important Hopf algebra. Let G be a finite group. The group algebra C[G] is defined as the C vector space with basis {δg}_g∈G. The unit η ∶ C ↪ C[G] is given by η(c) = cδe. The product is defined by bilinearly extending

δg⋅δ_h =δ_gh, associativity of G makes this into an associative product. We can en-dow C[G] with a cocommutative Hopf algebra structure by letting ∆(δg) =δ_g⊗δ_g, ε(δg) =1 and S(δ_g) =δ_g−1.

(ii) As a dual vector space k[G]∗

=F un(G) is naturally a commutative bialgebra. We can give it antipode S∗_{. By interpreting F un(G) ⊗ F un(G) as functions on}

G × G we see that coproduct, counit and antipode are given by

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(iii) Let H be a finite dimensional super Hopf algebra. The dual superbialgebra H∗ _{is naturally a super Hopf algebra with antipode S}∗_.

As indicated in the following proposition, example (ii) has a very important ‘extension’ to Lie- and algebraic groups.

Proposition 1.1.9. Let G be a smooth or complex Lie group resp. algebraic group. The commutative algebra F un(G) (C∞_{(G) or H(G) resp. O(G)) can be given the}

structure of a Hopf algebra by letting

∆(f )(g1⊗g₂) =f (g₁g₂), ε(f ) = f (e), S(f )(g) = f (g−1).

Proof. First of all we need to make sense of ∆(f ) acting on g1⊗g2. We consider

the following map

F un(G) ⊗ F un(G) → F un(G × G), f ⊗ f′

(g ⊗ g′

) ∶=f (g)f′(g′).

We will show that this map is an injection. Let ∑ifi⊗f_i′ ∈ F un(G) ⊗ F un(G), we may assume the fi are linearly independent over C. If not, we could rewrite

f1⊗f₁′+ (αf₁) ⊗f₂′ =f₁⊗ (f₁′+αf₂′). Suppose ∑ f_i⊗f_i′ ↦0 then ∀g ∈ G we have that f ∑ifi(g)fi′ is the 0-function. By linear independence of the f

′

i we have that

fi(g) = 0 for all g ∈ G. Hence ∑_if_i⊗f_i′=0.

Note that ∆ lands in the image of F un(G)⊗2 _{inside F un(G × G), therefore we can}

pullback the map ∆ ∶ F un(G) → F un(G×G) to ∆ ∶ F un(G) → F un(G)⊗F un(G). It remains to check that we have defined a bialgebra structure. We will show this by checking whether the defining diagrams commute:

id ⊗ ε ○ ∆(f )(g) = f (ge) = f (g) = f (eg) = ε ⊗ ε ○ ∆(f )(g), ∆ ⊗ id ○ ∆(f )((g1⊗g₂) ⊗g₃) =f (g₁g₂g₃) =id ⊗ ∆ ○ ∆(f )(g₁⊗ (g₂⊗g₃)), ε(f g) = f g(e) = f (e)g(e) = ε(f )ε(g), ∆(f f′ )(g ⊗ g′) = (f f′)(gg′) =f (gg′)f′(gg′), µ ○ id ⊗ S ○ ∆f (g) = f (gg−1) =f (e) = ε(f ), µ ○ S ⊗ id ○ ∆f (g) = f (g−1g) = f (e) = ε(f ).

F un(G) is commutative, therefore S is both an anti-homomorphism and homo-morphism.

The last example of a Hopf algebra that we will treat is central to the whole thesis. It is the object that we will deform to obtain our quantum group.

Definition 1.1.10. A universal enveloping algebra of g is a pair (U, i) where U is an associative algebra with 1 over C and i ∶ g → U is a linear map satisfying

i([X, Y ]) = i(X)i(Y ) − (−1)∣X∣∣Y ∣i(Y )i(X)

such that if (A, j) is another such pair, there exists a unique homomorphism φ ∶ U → A such that φ ○ i = j.

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We summarize some of the properties of the universal enveloping algebra in the following theorem.

Theorem 1.1.11. [Mus12] Let g be a Lie superalgebra.

1. U = U (g) exists and equals the associative superalgebra T (g)/I where I is the ideal in the tensor algebra T (g) generated by elements of the form X ⊗ Y − (−1)∣Y ∣∣X∣_{Y ⊗ X − [X, Y ] where X, Y ∈ g.}

2. Any g-representation has a unique structure as a U (g)-module.

3. Let X1, . . . , Xn be a basis of g0 over C, and Xn+1, . . . , Xn+ma basis of g1 over

C. The set X1j1⋯X jn+m

n+m with j1, . . . , jn ∈ _{N, j}_n+1, . . . , j_n+m ∈ {0, 1} defines a basis of U g over C.

4. U (g) has a canonical structure as a Hopf algebra where ∆ and ε are defined on g ↪ U (g) by

∆(X) = X ⊗ 1 + 1 ⊗ X, ε(X) = 0, S(X) = −X.

Remark. It is not hard to check that the algebra we define in (1) has the right properties. The basis in (3) is called the PBW-basis, see appendix B, and shows g injects into U (g). To prove (2) and (4) one uses the universal property of U . Example 1.1.12. U (sl_2∣1)is the unital associative superalgebra with even gener-ators H1, H2, E1, F1 and odd generators E2, F2, subject to the relations

HiHj =H_jH_i, (E0) HiEj−E_jH_i=a_ij, E_j (E1) HiFj−F_jH_i= −a_ij, F_j (E2) E₂2 =0 = F₂2, (E3) EiFj− (−1)ijFjEi=δ_ijH_i, (E4) E2 1E2−2E₂E₁E₂+E₁2E₂ =0, (S1) F₁2F2−2F₂F₁F₂+F₁2F₂ =0, (S2) where A = (2 −1 −1 0).

Remark. Note that universal enveloping algebras of Lie superalgebras are not do-mains in general, another important difference between super Lie theory and Lie theory.

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Definition 1.1.13. (The Quantum Supergroup Uq(sl_2∣1)) Let U_q denote the quan-tum supergroup Uq(sl2∣1), defined to be the C-superalgebra, dependent on param-eter q ∈ C∗_{∖ {±1}, and having even generators K}±

1, K ±

2, E1, F1 and odd generators

E2, F2, subject to the following relations

KiKi−1 =1, KiKj =K_jK_i, (E0) KiEj =qaijE_jK_i, K_iF_j =q−aijF_jK_i, where A = (2 −1 −1 0), (E1) EiFj− (−1)¯i¯jF_jE_i=δ_ij Ki−K_i−1 q − q−1 , (E2) E₂2=0 = F₂2, (E3) E₁2E2− (q + q−1)E1E2E1+E2E12=0, (S1) F₁2F2− (q + q−1)F1F2F1+F₂F₁2 =0. (S2)

In the following proposition, we show that we can endow Uq with a super Hopf

algebra structure.

Proposition 1.1.14. There exist unique even algebra morphisms ∶ U → C, ∆ ∶ U → U ⊗ U , and a unique even algebra anti-morphism S ∶ U → U that are defined on the generators of Uq as follows:

∆(Ei) =E_i⊗K_i+1 ⊗ E_i, (E_i) =0, ∆(Fi) =F_i⊗1 + K_i−1⊗F_i, (F_i) =0, ∆(Ki) =K_i⊗K_i (K_i±1), = 1, S(Ei) = −EiKi−1, S(Fi) = −KiFi, S(Ki) =K_i−1.

Denote µ and η the product and unit in Uq. The sextuple (Uq, µ, η, ∆, ε, S) is a

super Hopf algebra.

Proof. The proof of a statement such as ‘algebra A with these generators and relations can be given a Hopf algebra structure’ are standard. We will give an elaborate description. First we will show that existence implies uniqueness. We know that a general element in Uqcan written as the sum of monomials a = ai1⋯ain with aij generators. Suppose there exist two algebra morphisms ∆, ∆

′ _{defined on}

generators as above. The computation ∆(a) = ∆(ai1)⋯∆(ain) =∆

′ (ai1)⋯∆ ′ (ain) =∆(a ′ )

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shows that ∆ and ∆′ _{are equal on all monomials in A, and hence equal on all of}

A. The proofs that existence of ε and S imply uniqueness are analogous.

To check that ∆ extends to a morphism of algebras one reasons as follows. De-note F the free superalgebra with the same generators as Uq (but no relations).

Then Uq is a quotient of F by some set of relations. One defines ∆(A1⋯An) ∶= ∆(A1)⋯∆(An)for all generators A_i and extends linearly. Note that this is a well-defined (even) algebra morphism F → F ⊗F . We wish to check whether this algebra morphism descends to the quotient Uq of F i.e. are all relations in Uq respected by

∆. Concretely one checks the following. Let u = 0 be a relation in Uq. We wish to

check whether ∆(u) is in the kernel of the quotient map F ⊗ F → Uq⊗U_q. These are straightforward computations, we check two of such computations:

∆(E2)∆(E₂) = (E₂⊗K₂+1 ⊗ E₂)(E₂⊗K₂+1 ⊗ E₂),

= (−1)1E2⊗K₂E₂+ (−1)0E₂⊗E₂K₂=0 = ∆(E₂2), ε(E1)ε(F1) −ε(F₁)ε(E₁) =0 =

ε(K1) −ε(K₁−1) q − q−1 .

Checking ∆ descends to Uq establishes existence, and hence uniqueness. Similarly

one shows existence for ε. It remains to check whether they define a cosuperalgebra structure on Uq. The computation

id ⊗ ε ○ ∆(xy) = ∑ (xy) id ⊗ ε((xy)′ ⊗ (xy)′′) = ∑ (x)(y) id ⊗ ε(x′ y′ ⊗x′′y′′) = ∑ (x)(y) id(x′ y′ )ε(x′′ y′′ ) = ∑ (x)(y) x′ y′ ε(x′′ )ε(y′′ ) = ∑ (x)(y) id ⊗ ε(x′ ⊗x′′)id ⊗ ε(y′⊗y′′) =id ⊗ ε ○ ∆(x)id ⊗ ε ○ ∆(y)

shows us it sufficies to check the conditions (Coun) and (Coass), as rewritten using Sweedler’s notation, on the generators. These are straightforward calculations that we leave to the reader. In conclusion, Uq has a unique bialgebra structure with

coproduct and counit extending our definitions above.

It remains to check the bialgebra Uq has a antipode extending the definition above.

For the same reasons existence will imply uniqueness, and existence is shown in the same way as before. We show the computation

S(K₁−1)S(K₁) =K₁K₁−1 =1 = S(1)

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Superbialgebras and Poisson-Lie supergroups 17

It remains to check that S is an antipode for the bialgebra U . The computation µ ○ id ⊗ S ○ ∆(ab) = (ab)′

S((ab)′′

) =a′b′S(a′′b′′) =a′b′S(b′′)S(a′′) =a′ε(b)η(1)S(a′′) =ε(a)ε(b)η(1) = ε(ab)η(1)

shows that it suffices checking the condition (An-Po) on generators.4 _{These are}

straightforward computations, such as

µ ○ id ⊗ S ○ ∆(Ei) =E_iK_i−1+ −E_iK_i−1 =0 = 0 ⋅ 1 = ε(E_i)η(1), that we leave to the reader.

Historical Remark. Following Drinfel’d we will call Uq(sl2∣1)a quantized univer-sal enveloping algebra. The first example of a quantized univeruniver-sal enveloping algebra, Uq(sl2), is due to Kulish and Reshetikhin [KN83]. Drinfel’d [Dri87] and

Jimbo [Jim85] independently generalized this construction to any simple Lie alge-bra. QUEAs are non-commutative, non-cocommutative Hopf algebras. Before the ‘quantum-group era’ only a handful of examples were known.

Remark. Uq(sl2∣1) is not strictly a deformation of U (sl_2∣1), although the two are intimitely related. In fact one could think of the new elements Ki as qHi , but we

will not pursue this train of thought here. We refer to the book of Christian Kassel [Kas95] to see this connection elucidated for Uq(sl2).

References and further reading

Although not treating super mathematics, we highly recommend any reader un-familiar with Hopf algebras or quantized enveloping algebras the excellent book of Christian Kassel [Kas95]. The Rule of Signs heuristic is enough to adapt all definitions to the super case. For enveloping superalgebras we recommend [Mus12].

1.2 Superbialgebras and Poisson-Lie supergroups

Let us recall what a Poisson structure is on (Hopf) (super)algebras.

Definition 1.2.1. 1. Let A be a commutative superalgebra, a Poisson struc-ture on A is a Lie superalgebra bracket {, } ∶ A × A → A that satisfies the (super) Leibniz rule {a, bc} = {a, b}c + (−1)∣a∣∣b∣_{b{a, c} for any a, b, c ∈ A. We}

call the pair (A, {, }) a Poisson superalgebra.

2. A morphism between Poisson algebras (A, {, }A)and (B, {, }_B)is a morphism of superalgebras f ∶ A → B such that f ({a, a′

}_A) = {f (a), f (a′)}_B.

4_{To be precise, a similar calculation shows the same reasoning holds for the condition with S}

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3. Let (A, {, }A} and (B, {, }_B) be Poisson superalgebras. Then A ⊗ B has a canonical Poisson bracket defined by

{a ⊗ b, a′

⊗b′}_A⊗B∶= (−1)∣a ′_∣∣b∣

({a, a′

}_A⊗bb′+aa′⊗ {b, b′}_B).

4. Let A be a Hopf algebra, and let {, } be a Poisson structure on the algebra A. We call A a Poisson-Hopf algebra if the coproduct ∆ ∶ A → A ⊗ A is a morphism of Poisson algebras i.e. ∆{a, a′_{} = {∆(a), ∆(a}′_)}.

For us, the most important examples of Poisson structures will come from so-called deformations.

Definition 1.2.2. Let A be a commutative superalgebra. An analytical (torsion free) deformation of A is a family of superalgebras (Ah, µh, ηh) and linear isomor-phisms φh∶A_h→A₀ smoothly dependent on parameter h ∈ C such that m₀ ∶A₀→ A is an isomorphism of algebras. By smoothly dependent we will mean following. Define a multiplication ⋆h on A as follows a ⋆hb = φh⊗φ_h○µ_h○φ−1_h ⊗φ−1_h (a ⊗ b). Let {Xi}_i∈I be a basis in A and ask that

a ⋆hb = ∑

i∈I′,∣I′∣<∞

ma,b_i (h)Xi

for some analytic functions ma,b_i (h) ∶ C → C.

Two deformations (Ah, φh), (A′_h, φ′_h) are called equivalent if µ′_h =φ′−1_h ○φ_h○µ_h ○ φ−1

h ○φ ′

h i.e. if they define the same multiplication ⋆h on A.

Remark. There are many other ways of defining deformations, and the smooth-ness condition in the above is a very specific choice. It will suffice for our purposes however, and is a natural condition in the case that the algebra has a filtration, and you require the deformation to respect the filtration. Besides analytical de-formations, there are many other dede-formations, see for example [Dri87, §2] and [DCP93, §11].

Proposition 1.2.3. Let Ah be a deformation of A. Then A has a canonical

Poisson structure, defined by {a, b} ∶= lim

h→0

a ⋆hb − (−1)∣a∣∣b∣b ⋆ha

h .

Proof. First we should explain how to interpret the limit. Recall what it means that ⋆h depends smoothly, and note that limh→0a ⋆hb − b ⋆ha = 0, we define the

limits as follows {a, b} ∶= ∑ i∈I′ lim h→0 ma,b_i (h) − mb,a_i (h) h Xi = ∑ i∈I′ (ma,b_i −mb,a_i )′(0)X_i.

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Therefore, the bracket is well-defined. Obviously this bracket is supersymmetric. For the Jacobi identity we have

{a, {b, c}}+ ⟲ = lim h′→0limh→0 a ⋆h(b ⋆h′c − c ⋆h′b) − (b ⋆h′c − c ⋆h′b) ⋆ha) hh′ + ⟲ =lim h→0 a ⋆h(b ⋆hc − c ⋆hb) − (b ⋆hc − c ⋆hb) ⋆ha h2 + ⟲ =lim h→0 a ⋆hb ⋆hc − a ⋆hc ⋆hb − b ⋆hc ⋆ha + c ⋆hb ⋆ha+ ⟲ h2 =lim h→0 0 h2 =0

Where we are allowed to replace the seperate limits (h, h′_{) → (0, 0) by the diagonal}

limit h → 0 since all seperate limits exists and hence must coincide with the diagonal one. Using the rule of signs one easily uses the above computation to prove the super Jacobi identity.

For the Leibniz identity we use a similar technique {a, bc} = lim h→0 a ⋆h(bc) − (bc) ⋆ha h =lim h→0 a ⋆hb ⋆hc − b ⋆hc ⋆ha h =lim h→0 a ⋆hb ⋆hc − b ⋆ha ⋆hc + b ⋆ha ⋆hc − b ⋆hc ⋆ha h =lim h→0( (a ⋆hb)c − (b ⋆ha)c h + b(a ⋆hc) − b(c ⋆ha) h ) = {a, b}c + b{a, c}

Where we can replace h′ _{by h and vice versa, by looking at the analytic functions}

corresponding to a⋆h(b⋆h′c)−(b⋆h′c)⋆ha, call them fi(h, h′). Note that fi(0, 0) = 0

for all i, such that we are calculating their derivatives in line two. Also note that fi(0, h) = 0 for all i, such that the product rule yields

d dh∣h=0 fi(h, h) = d dh∣h=0 (fi(0, h) + f (h, 0)) = d dh∣h=0 fi(h, 0)

Which is exactly the expression of line one. Similarly one moves from line three to four. Using the rule of signs yields the super version.

The non-commutativity of Ah ‘encodes’ the Poisson bracket of A0. We will

say that Ah is a quantization of the Poisson algebra A0. Conversely, if we have

such a family of non-commutative algebras Ah we can dequantize it and call the

Poisson algebra A0 the quasi-classical limit of Ah. This language is borrowed

from physics where one would interpret the parameter h as Plankc’s constant, and sending h → 0 corresponds to going from a quantum mechanical sysmtem to a

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classical system. Our Poisson bracket comes from the first order in h, hence the quasi in quasi-classical.

If the function algebra of a Lie supergroup is endowed with a ‘compatible Poisson structure’ (to be explained later) we will call it a Poisson-Lie supergroup. We will introduce the linearized version first.

Definition 1.2.4. A Lie superbialgebra is a pair (g, δ) where g is a Lie superalgebra and δ ∶ g → Λ2_{(g), the cobracket is an even morphism of super vector spaces such}

that δ is a cocycle, i.e.

δ[X, Y ] = [δ(X), Y ⊗ 1 + 1 ⊗ Y ] + [1 ⊗ X + X ⊗ 1, δ(Y )], and δ satisfies the super co Jacobi identity, i.e.

Alt(δ ⊗ id) ○ δ(X) = 0;

where Alt(a ⊗ b ⊗ c) = a ⊗ b ⊗ c + (−1)∣a∣∣b∣_{c ⊗ a ⊗ b + (−1)}∣a∣∣c∣_{b ⊗ c ⊗ a.}

Remark. The language is consistent with our earlier treatment of co-, bialgebras: we have endowed g with a bialgebra structure in the category of Lie superalgebras. Definition 1.2.5. A finite dimensional Manin supertriple is a triple (g, g1, g2) of finite dimensional Lie superalgebras, where g is endowed with a non-degenerate bilinear form ⟨, ⟩ such that

1. The bilinear form is invariant i.e. ⟨[X, Y ], Z⟩ = ⟨X, [Y, Z]⟩ for all X, Y, Z ∈ g. 2. g = g1⊕g₂ as vector spaces, and g_i↪g as Lie algebras.

3. g1 and g2 are isotropic subspaces w.r.t ⟨, ⟩.

A linear subspace V ⊂ g is called isotropic if ⟨, ⟩∣_{V ×V} =0.

Proposition 1.2.6. [And93, prop. 1] Let (g, δ) be a finite dimensional Lie su-perbialgebra, then g∗ _{inherits a Lie superalgebra bracket from δ}∗_{. The triple (g ⊕}

g∗_{, g, g}∗

) naturally has the structure of a finite dimensional Manin supertriple. Conversely, let (g, g1, g2) be a Manin supertriple, with g finite dimensional, the Lie superalgebra structure on g2 induces a superbialgebra structure on g1.

Proof. Let x, y ∈ g, α, β ∈ g∗_{. The bilinear form on g ⊕ g}∗ _{is given by}

(x + α, y + β) = ⟨α, y⟩ + −(−1)∣x∣∣β∣⟨β, x⟩

where ⟨, ⟩ is the natural pairing between g and g∗_{. The Lie superalgebra bracket}

is given by [x, α] = [x, α]1+ [x, α]2 where [x, α]i∈p_i are determined by ([x, α]1, β) = (x, [α, β]), (y, [x, α]2) = ([y, x], α)

The rest of the proof consists of checking that identities are satisfied, and can be found in [And93].

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Corollary 1.2.7. (Dual Lie Superbialgebra) Let g be a finite dimensional Lie superbialgebra, then g∗ _{naturally has the structure of a Lie superbialgebra.}

Proof. This is immediate from the previous proposition by noting that the Manin supertriple is symmetric with respect to the gi↪g.

Proposition 1.2.8. [And93] ( Drinfel’ds Double) Let g be a Lie superbialgebra and (g ⊕ g∗_{, g, g}∗

) be the associated Manin supertriple. Then δ = δ_g−δ_g∗ defines a Lie superbialgebra structure on g ⊕ g∗_{. It is called the Drinfel’d double of g and}

denoted D(g).

Example 1.2.9. (Standard superbialgebra structure on sl2∣1) Let g = sl2∣1. Recall

that g has a non-degenerate invariant supersymmetric bilinear form (, ) ∶ g × g → C given by the supertrace. We can use it to identify the positive and negative borel as dual super vector spaces.

b+=h ⊕ n+ b−=h ⊕ n−

=h ⊕_α∈−Φ+g_α, =h ⊕_α∈Φ−g_α.

See appendix A.1 for the definition of the positive root system. Concretely b+

(resp. b−) is spanned by the Hi and Ei (resp. Hi and Fi).

Claim: (b+⊕b−, b+, b−) naturally has the structure of a finite dimensional Manin

supertriple.

Proof claim. We write ˜g ∶= b+⊕b− = n+⊕h+⊕n−⊕h− and define the following

commutation relations [H+ i , H − j] =0, [H ± i , Ej] =a_ijE_j, [H_i±, F_j] = −a_ijF_j [Ei, Fj] =δ_ij 1 2(H + i +H − i ) where H±

i ∈h±. It is easy to see this defines a Lie superalgebra structure on ˜g. We

endow ˜g with ⟨x+ 1+x − 1+h + 1+h − 2, x + 2+x − 2+h + 2+h − 2⟩ =str(x + 1x − 2) +str(x − 1x + 2) +2(str(h + 1h − 2) +str(h − 1h + 2))

as bilinear form. Clearly b± ↪ ˜g, and b± are isotropic with respect to ⟨, ⟩. It

remains to check whether the pairing is non-degenerate and invariant, but this is immediate since the supertrace has those properties for sl2∣1.

Proposition 1.2.6 now immediately yields that b+ and b− can be given dual

su-perbialgebras structures by dualizing their Lie superbrackets via the bilinear form. Let us compute the cobracket on b+. The dual basis for b+ in terms of the basis

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of b− is given as follows: E∨ 1 =F1 E2∨=F2, E∨ 12=F12 H1∨= − 1 2H2, H∨ 2 = − 1 2(H1+2H2). Using the identification (b+)

∗ ≅b−, the brackets [H∨ 1, E ∨ 1] = − 1 2E ∨ 1 [H ∨ 2, E ∨ 1] =0, [H∨ 1, E ∨ 2] =0 [H ∨ 2, E ∨ 2] = − 1 2E ∨ 2, [H∨ 1, E ∨ 12] = − 1 2E ∨ 12 [H ∨ 2, E ∨ 12] = − 1 2E ∨ 12, [E∨ 1, E ∨ 2] =E ∨ 12 [E ∨ i, E ∨ 12] =0, [H∨ i , H ∨ j] =0,

are the Lie bracket transported from b− to b ∗

+. We will dualize the bracket on b ∗ +

to define a cobracket δ+∶b+→b+∧b+, by setting

⟨δ₊(X), Y ∧ Z⟩ = ⟨X, [Y, Z]⟩.

We will compute δ+(E1) as example. E1, only pairs non-zero with E ∨

1 i.e. only

when [Y, Z] = E∨

1. By looking at the relations in b∗+ we see that the only non-zero

pairing is given by ⟨δ+(E1), 1 2[E ∨ 1, H ∨ 1]⟩ =1. We conclude that δ+(E1) = 1 2E1∧H1.

The other cobrackets can be computed similarly, and are given as follows δ+(Hi) =0, δ+(E2) = 1 2E2∧H2, δ+(E12) = 1 2E12∧ (H1+H2) +E1∧E2.

Proposition 1.2.6, and actually its proof, now ensure us that we have defined a Lie superbialgebra structure on b+. Analogously we can dualize the Lie bracket on b+

to find the following cobracket on b−

δ−(Hi) =0, δ−(Fi) = − 1 2Fi∧Hi, δ−(F12) = − 1 2F12∧ (H1+H2) +F2∧F1,

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Moreover, ˜g can be given a Lie superbialgebra as the Drinfel’d double of b+ by

letting δ˜g=δ_b₊−δ_b₋.

Observe that we have chosen the Lie superalgebra on ˜g in such a way that ˜

g Ð→ sl2∣1

Ei ↦E_i, F_i↦, F_i, H_i±↦H_i

is a well-defined Lie superalgebra morphism. Equivalently we could say that sl2∣1is

a Lie superalgebra quotient of ˜g. We claim that this is actually a Lie superbialgebra quotient. Indeed, as δ±(H

±

i) = 0 the cobracket δ˜g descends to sl2∣1 defining a Lie

superbialgebra structure. This is called the standard bialgebra structure on sl2∣1

[ES02, §4.4]. The cobracket on sl2∣1 is given by

δ(H1) =0, δ(H₂) =0, δ(Ei) = 1 2Ei∧Hi, δ(Fi) = 1 2Fi∧Hi, δ(E12) = 1 2E12∧ (H1+H2) +E1∧E2, δ(F12) = 1 2F12∧ 1 2(H1+H2) +F1∧F2, note that we have extra minusses for the F s coming from the minus in δ˜g=δb+−δb−. Example 1.2.10. (Standard superbialgebra structure on sl∗

2∣1) Corrolary 1.2.7 now

immediately gives that we can induce a dual Lie superbialgebra structure on sl∗ 2∣1

from the standard structure on sl2∣1. Let us begin by describing the Lie

super-algebra structure on sl∗

2∣1. We define a Lie superbracket on sl ∗

2∣1 by solving the

conditions

⟨[X, Y ], Z⟩ = ⟨X ∧ Y, δ(Z) for all pairs X, Y ∈ sl∗

2∣1. For example, there is no Z ∈ sl2∣1such that δ(Z) = Ei∧F_i, hence [E∨

i, F ∨

i ] =0. We obtain the following conditions

[E_i∨, F_j∨] =0, [H_i∨, H_j∨] =0, [E∨ 1, E ∨ 2] =E ∨ 12, [F ∨ 1, F ∨ 2] =F ∨ 12, [H_i∨, E_j∨] = −δ_ij1 2E ∨ j, [H ∨ i , F ∨ j] = −δij 1 2F ∨ j , [H_i∨, E₁₂∨] = −1 2E ∨ 12, [H ∨ i , F ∨ 12] = − 1 2F ∨ 12,

defining a Lie superbracket on sl∗

2∣1. It is easy to check that map

sl∗ 2∣1Ð→sl2∣1⊕sl_2∣1, E∨ i ↦ (0, Fi), E₁₂∨ ↦ (0, F₁₂), F∨ i ↦ (Ei, 0), F12∨ ↦ (−E12, 0), H∨ 1 ↦ ( 1 2H2, − 1 2H2), H ∨ 2 ↦ ( 1 2H1+H2, − 1 2H1−H2),

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defines a morphism of Lie superalgebras. We give the calculation [H₂∨, F₁₂∨] = [(1 2H1+H2, − 1 2H1−H2), (−E12, 0)] = (1 2E12, 0) = − 1 2F ∨ 12

as example. Henceforth we will identify the Lie superalgebra sl∗

2∣1 with its image

inside sl2∣1⊕sl2∣1. It remains to compute the cobracket on sl ∗

2∣1. As before we solve

the equations

⟨δsl∗_2∣1(X), Y ∧ Z⟩ = ⟨X, [Y, Z]sl2∣1⟩ to find the cobracket. On the dual basis we find

δ(H∨ 1) =E ∨ 1 ∧F ∨ 1 +E ∨ 12∧F ∨ 12, δ(H ∨ 2) =E ∨ 2 ∧F ∨ 2 +E ∨ 12∧F ∨ 12, δ(E∨ 1) =2H ∨ 1 ∧E ∨ 1 −H ∨ 2 ∧E ∨ 1, δ(F ∨ 1) = −2H ∨ 1 ∧F ∨ 1 +H ∨ 2 ∧F ∨ 1 δ(E∨ 2) = −H ∨ 1 ∧E ∨ 2, δ(F ∨ 2) =H ∨ 1 ∧F ∨ 2 δ(E∨ 12) = (H ∨ 1 −H ∨ 2) ∧E ∨ 12+E ∨ 2 ∧E ∨ 1 δ(F ∨ 12) = −(H ∨ 1 −H ∨ 2) ∧F ∨ 12+F ∨ 1 ∧F ∨ 2, as cobracket on sl∗

2∣1. We can translate this to the following cobracket

δ(H1, −H1) =4(E₁, 0) ∧ (0, F₁) +4(E₁₂, 0) ∧ (0, F₁₂) +2(E₂, 0) ∧ (0, F₂), (1.2) δ(H2, −H2) = −2(E1, 0) ∧ (0, F1) −2(E₁₂, 0) ∧ (0, F₁₂), (1.3) δ(E1, 0) = 1 2(E1, 0) ∧ (H1, −H1), (1.4) δ(0, F1) = − 1 2(0, F1) ∧ (H1, −H1), (1.5) δ(E2, 0) = 1 2(H1, −H1) ∧ (E2, 0), (1.6) δ(0, F2) = − 1 2(H1, −H1) ∧ (0, F2), (1.7) δ(E12, 0) = 1 2(H1+H2, −H1−H2) ∧ (E12, 0) + (E2, 0) ∧ (E1, 0), (1.8) δ(0, F12) = −1 2(H1+H2, −H1−H2) ∧ (0, F2) − (0, F1) ∧ (0, F2), (1.9) on sl∗

2∣1as sup Lie superalgebra of sl2∣1⊕sl_2∣1. We will give two sample calculations, to show how to translate the cobracket

δ(H2, −H2) =2δ(H₁∨) =2E₁∨∧F₁∨+2E₁₂∨ ∧F₁₂∨ =2(0, F₁) ∧ (E₁, 0) + 2(0, F₁₂) ∧ (−E₁₂, 0) = −2(E1, 0) ∧ (0, F1) −2(E₁₂, 0) ∧ (0, F₁₂), δ(H1, −H1) =2δ(H₂∨) −2δ(H₂) =E₂∨∧F₂∨+E₁₂∨ ∧F₁₂∨) −2(−2(E₁, 0) ∧ (0, F₁) −2(E₁₂, 0) ∧ (0, F₁₂)) =4(E₁, 0) ∧ (0, F₁) + (E₁₂, 0) ∧ (0, F₁₂) + (E₂, 0) ∧ (0, F₂).

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Drinfel’d introduce Lie bialgebras in [Dri87] as the tangent spaces of Poisson-Lie groups. Andruskiewitsch generalized this to Poisson-Poisson-Lie supergroups.

Definition 1.2.11. 1. A Poisson supermanifold is a triple (M, A, ,) where (M, A) is a supermanifold and {, } ∶ A(M )×A(M ) → A(M ) turns A(M ) into a Pois-son superalgebra.

2. A Poisson-Lie supergroup is a quadruple (G, A, i, ∆, {, }) where (G, A, i, ∆) is a Lie supergroup i.e. (G, A) is a supermanifold and i ∶ A(G) → A(G), ∆ ∶ A(G) → A(G) ⊗ A(G) induce a Hopf algebra structure on A(G)⋆_.

Addi-tionally (G, A, {, }) is a Poisson manifold and ∆ ∶ A(G) → A(G) ⊗ A(G) is a map of Poisson superalgebras.

Theorem 1.2.12. [And93, prop. 5] Let (G, A, {, }) be a Poisson-Lie supergroup. Then g has a natural superbialgebra structure induced by {, }. Conversely, let (g, δ) be a finite dimensional Lie superbialgebra. Then the simply connected Lie supergroup with Lie superalgebra g is a Poisson-Lie supergroup with respect to some bracket {, } that induces δ.

Proof. See [And93] for details. If we have {, } a superbracket on A(G) then we can dualize to a map δ ∶ A(G)⋆

→ A(G)⋆⊗ A(G)⋆, defined by

⟨δ(x), f ⊗ g⟩ = ⟨x, {f, g}⟩

This map can be restricted to g ⊂ A(G)⋆ _{and yields a cobracket δ ∶ g → g ⊗ g which}

endows g with a Lie superbialgebra structure.

Conversely, for f, g ∈ A(G) we can define {f, g} by requiring ⟨{f, g}, x⟩ = ⟨f ⊗ g, δ(x)⟩.

Remark. Although the proof of Andruskiewitsch is short, it does not provide a tangeable way to actually compute one from the other. As we will see in the last section, we will compute δ as the tangent map to the Poisson 2-tensor.

Definition 1.2.13. Let (G, A) be a Poisson-Lie supergroup. We call (G∗_{, A}′

) a Poisson dual group to (G, A) if their Lie algebras are dual as Lie superbialgebras.

References and further reading

For basics on Lie bialgebras and Poisson-Lie groups we refer the original article [Dri87] of Drinfeld, the book by Etingof and Schiffmann [ES02] and chapter one in [CP94]. For general theory on Poisson structures, including Poisson-Lie groups, we refer to the book [LGPV13]. For Poisson-Lie supergroups, the reader should consult the article [And93] of Andruskiewitsch.

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1.3 U

q

(sl

2∣1

)

as quantization of O(SL

∗_2∣1

)

We introduce an auxiliary algebra U′

q that is isomorphic to Uq(sl2∣1)for all q ∈ C∗∖ {±1} but is also well-defined at q = 1. This will turn out to be a supercommutative Hopf algebra of dimension (4, 4) isomorphic to O(SL∗

2∣1). U ′

1 is canonically a

Poisson algebra by proposition 1.2.3. In fact it is a Poisson-Hopf algebra. In this sense Uq quantizes O(SL∗_2∣1), the Poisson structure induced on O(SL∗_2∣1) from U_q is exactly the one corresponding to the standard bialgebra structure on sl∗

2∣1.

Proposition 1.3.1. Let U′

q be defined as the superalgebra with even generators

¯

E1, ¯F1, K1, K2 and odd generators ¯E2, ¯E12, ¯F2, ¯F12 subject to the following relations

KiKi−1 =1, KiKj =K_jK_i, K_iE¯_j=qaijE¯_jK_i, (1.10) KiE¯12=qai1+ai2E¯12Ki, KiF¯12=q−ai1−ai2F¯12Ki, (1.11) where A = (2 −1 −1 0), ¯ EiF¯j− (−1)∣i∣∣j∣F¯jE¯i=δ_ij(q − q−1)(K_i−K_i−1), (1.12) ¯ E12F¯12+ ¯F₁₂E¯₁₂= (q − q−1)(K₁K₂−K₁−1K₂−1), (1.13) ¯ E2 2 = ¯E122 =0, F¯22= ¯F122 =0, (1.14) ¯ E2E¯1−q−1E¯₁E¯₂= (q − q)−1E¯₁₂, F¯₁F¯₂−q ¯F₂F¯₁= (q − q−1) ¯F₁₂, (1.15) ¯ E12E¯1 =q ¯E1E¯12, F¯1F¯12=q−1F¯12F¯1, (1.16) ¯ E2E¯12= −q−1E¯12E¯2, F¯12F¯2 = −q ¯F2F¯12, (1.17) ¯ F1E¯12− ¯E₁₂F¯₁= (q − q−1)2K₁E¯₂, E¯₁F¯₁₂− ¯F₁₂E¯₁ = (q − q−1)2F¯₂K₁−1, (1.18) ¯ F2E¯12+ ¯E₁₂F¯₂= −(q − q−1)2E¯₁K₂−1, E¯₂F¯₁₂+ ¯F₁₂E¯₂ = (q − q−1)2K₂F¯₁. (1.19) For all q ∈ C∗ _{we have that U}

q ≅U_q′ as superalgebras. Hence, the the super Hopf algebra structure of Uq induces a super Hopf algebra structure on Uq′.

Proof. The isomorphism is given on generators by U′ qÐ→Uq, ¯ Ei↦ (q − q−1)Ei, ¯ Fi ↦ (q − q−1)Ei, Ki↦K_i, ¯ E12↦ (q − q−1)(E2E1−q−1E1E2), ¯ F12↦ (q − q−1)(F1F2−qF₂F₁).

It is an easy, but tedious proof to check that all the map respects all the conditions. We will not write this out. Similarly, one can define the obvious inverse map on

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Uq(sl2∣1) as quantization of O(SL∗_2∣1) 27

generators of Uq and check that it defines an algebra homomorphism. In effect

we have rewritten Uq in terms of new generators, and all the conditions between

them.

The reason we introduced U′

q is that the superalgebra is well defined in the

limit q → 1, and it is easy to see U′

q becomes supercommutative in that limit.

Proposition 1.3.2. U′

1 is endowed with a canonical Poisson structure by

propo-sition 1.2.3. U′

1 has the structure of a Poisson-Hopf algebra with this bracket.

Proof. We need to check whether the canonical bracket in 1.2.3 is well-defined. Although we do not want to use the canonical bracket, but rescale the bracket by a factor 1₂ by defining {a, b} ∶= lim q→1 ab − (−1)∣b∣∣a∣_ba q − q−1 =lim_q→1 ab − (−1)∣a∣∣b∣_ba q − 1 q 1 + q.

The proof will consist in computing the bracket. A typical calculation will use L’Hˆopital’s rule, for example:

{ ¯E₂, ¯E₁} =lim q→1 ¯ E2E¯1− ¯E₁E¯₂ q − q−1 =lim q→1 (q−1−1) ¯E₁E¯₂+ (q − q−1E¯₁₂ q − q−1 = − 1 2 ¯ E1E¯2+ ¯E₁₂.

The brackets on the generators are all well-defined and given as follows

{Ki, Kj} =0, { ¯E_i, ¯F_j} =δ_ij(K_i−K_i−1), (1.20) {K± i, ¯Ej} = ± aij 2 ¯ EjKi±, {K ± i, ¯Fj} = ∓ aij 2 K ± iF¯j, (1.21) {K± i, ¯E12} = ± ai1+a_i2 2 ¯ E12Ki±, {K ± i, ¯F12} = ∓ ai1+a_i2 2 K ± i F¯12, (1.22) { ¯F_i, ¯E₁₂} =0, { ¯E_i, ¯F₁₂} =0, (1.23) { ¯E₁₂, ¯F₁₂} =K₁K₂−K₁−1K₂−1, (1.24) { ¯E₂, ¯E₁} = − 1 2 ¯ E1E¯2+ ¯E₁₂, { ¯F₂, ¯F₁} = − 1 2 ¯ F1F¯2+ ¯F₁₂, (1.25) { ¯E₁₂, ¯E₁} = 1 2 ¯ E1E¯12, { ¯F₁₂, ¯F₁} = 1 2 ¯ F1F¯12, (1.26) { ¯E₂, ¯E₁₂} = − 1 2 ¯ E2E¯12, { ¯F₂, ¯F₁₂} = − 1 2 ¯ F2F¯12. (1.27)

As the bracket is well-defined on generators, proposition 1.2.3 now immediately yields we have defined a Poisson bracket.

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It remains to show whether U′

q is a Poisson-Hopf algebra, it suffices to check the

equality

∆{a, b} = {∆(a), ∆(b)} (1.28)

on generators by the following argument. Suppose 1.28 holds for the pairs a, b and a, c then we also have

{∆(a), ∆(bc)} = {∆(a)∆(b), ∆(c)} = (−1)∣a∣∣b∣∆(b){∆(a), ∆(c)} + {∆(a), ∆(b)}∆(c) = (−1)∣a∣∣b∣_{∆(b)∆({a, c}) + ∆({a, b})∆(c)}

=∆((−1)∣a∣∣b∣b{a, c} + {a, b}c) = ∆({a, bc})

We will give one of these computations as example. Note that since we are trans-porting the coproduct of Uq some of the coefficients change. For example,

∆(F12) =F12⊗1 + K1−1K2−1⊗F12− (q − q−1)F2K1−1⊗F1,

∆( ¯F12) = ¯F₁₂⊗1 + K₁−1K₂−1⊗ ¯F₁₂− ¯F₂K₁−1⊗ ¯F₁.

We will now check that ∆({ ¯F1, ¯F2} = {∆( ¯F1), ∆( ¯F2)}. The reader should take

care to closely observe the rule of signs, adding minusses for every exchange of odd elements in a formula. We compute:

{∆( ¯F2), ∆( ¯F1)} = { ¯F₂⊗1 + K₂−1⊗ ¯F₂, ¯F₁₂⊗1 + K₁−1K₂−1⊗ ¯F₁₂− ¯F₂K₁−1⊗ ¯F₁} = { ¯F₂, ¯F₁₂} ⊗1 + { ¯F₂, K₁−1K₂−1} ⊗ ¯F₁₂+ {K₂, ¯F₂K₁−1} ⊗F₁ − {K₂−1, ¯F₁₂} ⊗ ¯F₂+K₁−1K₂−1⊗ { ¯F₂, ¯F₁₂} + {K2, ¯F2K1−1} ⊗ ¯F2F¯1+K₂−1F¯₂K₁−1⊗ { ¯F₂, ¯F₁} =1 2 ¯ F2F¯12⊗1 + 1 2 ¯ F2K1−1K2−1⊗ ¯F12+ 1 2K −1 2 F¯12⊗ ¯F2 +1 2K −1 1 K2−2⊗ ¯F2F¯12−1 2K2 ¯ F2K1−1⊗ ¯F1F¯2 =1 2( ¯F2⊗1 + K −1 2 ⊗ ¯F2)( ¯F₁₂⊗1 + K₁−1K₂−1⊗ ¯F₁₂− ¯F₂K₁−1⊗ ¯F₁) = 1 2∆(F2)∆( ¯F12) =∆({ ¯F₂, ¯F₁}). We leave out the other computations.

Before defining the Lie supergroup, we would like to warn the reader that the following argument is not completely rigorous. We will be defining a complex Lie supergroup, for which the theory is more subtle than the real theory we treated in appendix A.3. For example, the global holomorphic functions do not contain all the information of the complex-analytic supermanifold. We feel that without

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Uq(sl2∣1) as quantization of O(SL∗_2∣1) 29

discussing the theory of complex supermanifolds and giving all the necessary de-tails, the arguments below do not hold up to the standard of rigour we hope to have achieved in the other parts of the thesis. We do note that the impact on the thesis is minimal, so that the reader should feel free to read the rest of the chapter as heuristics. The important thing to note is that corrolory 1.3.6 remains valid regardless of the rigour in defining this supergroup, which is the only result we will actually use in the rest of the thesis.

Definition 1.3.3. (Defining the Lie supergroup SL∗

2∣1) In defining the Lie

supergroup, we will move in three steps. First we will identify the base space (SL∗

2∣1)∅, which we will denote G0. Then we construct the sheaf of functions on

the supermanifold, denoted A, and define the Hopf algebra structure on A(G)⋆

by defining a Hopf algebra structure on A(SL∗

2∣1). Finally we check the tangent

space to the supermanifold is indeed g = sl∗ 2∣1.

Defining the base space G0= (SL∗_2∣1)∅

We will define the base space by integrating the complex Lie algebra g0= (sl ∗ 2∣1)0

to a complex Lie group G0:

g0= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ h1 e1 0 0 −h1+h₂ 0 0 0 h2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −h1 0 0 f1 −h1−h₂ 0 0 0 −h₂ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟ ⎠ ∶h₁, h₂, e₁, f₁∈_C ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ , G0 = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z2 c1 0 0 z1z2 0 0 0 z2 2z1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z−1 2 0 0 b1 z1−1z−12 0 0 0 z−2 2 z1−1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟ ⎠ ∶z_i ∈_C∗, a, b ∈ C ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ .

Note that the zi, b, c uniquely parametrize elements of the G0. We will use this

parametrization of G0 to define a Hopf algebra structure on O(G).

Defining the super Hopf algebra of global functions A(G0)

We declare G0 to be a globally split supermanifold i.e. A(U ) ≅ H(U ) ⊗ Λ(C4),

where H(U ) denote the holomorphic functions on U . This is actually natural to ask, as all complex Lie supergroups all split supermanifolds [Vis11, cor 2.2.2]. Note that we are using here that as dim(g1) = 4 we must have four odd coordinates. We will generalize the formulas defining the Hopf algebra structure of coordinate functions on a Lie group to define a super Hopf algebra structure on A(G0). Let

us denote the odd coordinates c2, c12, b2, b12. We interpret them as ‘coordinates in

a zero part of G0’ ⎛ ⎜ ⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z2 c1 c12 0 z1z2 c2 0 0 z2 2z1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z−1 2 0 0 b1 z1−1z2−1 0 b12 b2 z₂−2z−1₁ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟ ⎠

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and use the formulas of 1.1.9 to define the super Hopf algebra structure. For example, for the upper triangular part we have

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z2 c1 c12 0 z1z2 c2 0 0 z2 2z1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z′ 2 c ′ 1 c ′ 12 0 z′ 1z ′ 2 c ′ 2 0 0 z′2 2 z′1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z2z2′ z2c′1+c1z1′z ′ 2 z2c′12+c1c′2+c12z1′z ′₂ 2 0 z1z2z1′z ′ 2 z1z2c′2+c2z1′z ′₂ 2 0 0 z2 2z1z ′₂ 2 z ′ 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

We find the following coproduct candidate

∆(z1) =z₁⊗z₁ ∆(z−1₁ ) =z₁−1⊗z₁−1 ∆(z2) =z₂⊗z₂ ∆(z−1₂ ) =z₂−1⊗z₂−1

∆(c1) =c₁⊗z₁z₂+z₂⊗c₁ ∆(b₁) =b₁⊗z₂−1+z₁−1z₂−1⊗b₁ ∆(c2) =c₂⊗z₁z₂2+z₁z₂⊗c₂ ∆(b₂) =b₂⊗z₁−1z−1₂ +z₂−2z₁−1⊗b₂

∆(c12) =c₁₂⊗z₁z₂2+c₁⊗c₂+z₂⊗c₁₂ ∆(b₁₂) =b₁₂⊗z₂−1+b₂⊗b₁+z−2₂ z₁−1⊗b₁₂ where the second row is obtain from a similar calculation on the lower triangular part. The candidate counit is evaluating the coordinates on the identity element, i.e. (zi) =1, (c_i) =0 = (b_i). Since O(G₀) is just a supersymmetric algebra, we can define ∆(ab) = ∆(a)∆(b) and it is easy to see from the formulas that we have defined a superbialgebra structure on O(G0).

For the antipode we will follow the same strategy, and compute the normal an-tipode, as if we were dealing with normal coordinate functions:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z2 c1 c12 0 z1z2 c2 0 0 z2 2z1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ z−1 2 −c1z1−1z2−2 −z1z2−3(c12+z₂−1c₁c₂) 0 z−1 1 z2−1 −c2z−21 z2−3 0 0 z−2 2 z−11 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

We read off the candidate antipode:

S(z1) =z₁−1, S(z₁−1) =z₁, S(z2) =z₂−1, S(z₂−1) =z₂, S(c1) = −c1z1−1z2−2, S(b1) = −b1z1z22, S(c2) = −z1−2z−32 c2, S(b2) = −b2z21z32, S(c12) = −z1−1z −3 2 (c12−z₁−1z−1₂ c₁c₂), S(b₁₂) = −z₁z3₂(b₁₂−z₁z₂b₂b₁). The first thing to note is, that the Hopf algebra maps restricted to the even coordinates are the same as they would have been coming from 1.1.9 on the Lie group G0. This is coincidental, because of the triangular form of our matrices.5

Normally one would have to take the Hopf quotient O(G0)/O(G₀)₁ to obtain the ‘usual’ Hopf algebra. Nevertheless, this is a nice bonus as it ensures we have

5_{We expect that one could have predicted this behaviour by noting that SL}∗

2∣1has the structure

of a split Lie supergroup (a group object in the category of split supermanifolds), which is true because [(sl∗_2∣1)1, (sl∗2∣1)1] =0 [Vis11, prop 4.4].

Representations of Uq(sl2S1) at roots of unity

Universiteit van Amsterdam

MSc Mathematical Physics

Master Thesis

Representations of U

q

(sl

2∣1

)

at roots of unity

Tim Weelinck

June 26, 2015

Supervisor:

Prof. Dr. N.R.

Reshetikhin

Second Reader:

Dr. H.B. Posthuma

at roots of unity

Tim Weelinck

Abstract

Acknowledgements

Contents

Introduction

Organisation and main results

Chapter 1

Preliminaries on Quantum

Supergroups

1.1

Hopf superalgebras and enveloping

superal-gebras

1.2

Superbialgebras and Poisson-Lie supergroups

1.3

U

(sl

)

as quantization of O(SL

)