Alexandre Bouayad
Research Fellow


Department of Pure Mathematics
and Mathematical Statistics
University of Cambridge
>> Contact details

I am also a Fellow of
St John's College, where I am Director of Studies in Pure Mathematics.
Research interests:
representation theory, Lie theory, quantum groups, deformation theory, geometric Langlands correspondence.
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Publications & Preprints
Coloured KacMoody algebras, Part II
[PDF]
To appear soon
Coloured KacMoody algebras, Part I
[PDF]
arXiv:1412.8606 (2014)
We introduce a parametrization of formal deformations of Verma modules of $\mathfrak{sl}_2$. A point in the moduli space is called a colouring.
We prove that for each colouring $\psi$ satisfying a regularity condition, there is a formal deformation $U_h(\psi)$ of $U(\mathfrak{sl}_2)$ acting on the deformed Verma modules.
We retrieve in particular the quantum algebra $U_h(\mathfrak{sl}_2)$ from a colouring by $q$numbers.
More generally, we establish that regular colourings parametrize a broad family of formal deformations of the ChevalleySerre presentation of $U(\mathfrak{sl}_2)$.
The present paper is the first of a series aimed to lay the foundations of a new approach to deformations of KacMoody algebras and of their representations.
We will employ in a forthcoming paper coloured KacMoody algebras to give a positive answer to E. Frenkel and D. Hernandez’s conjectures on Langlands duality in quantum group theory.
Groupes quantiques d’interpolation de Langlands de rang 1
[PDF]
Int. Math. Res. Not. IRMN 2013, No. 6, 12681323 (2013)
We study a certain family, parameterized by an positive integer $g$, of double deformations of the envelopping algebra $U(\mathfrak{sl}_2)$, in the spirit of
E. Frenkel and D. Hernandez [1]. We prove that each of these double deformations simultaneously deforms two rank 1 quantum groups. We show this interpolating
property explains the Langlands duality for the representations of the quantum groups in rank 1. Hence we prove a conjecture of E. Frenkel and D. Hernandez in this case:
we prove for all g the existence of representations which simultaneously deform two Langlands dual representations. We also study more generaly the finite rank representation
theory of this family of double deformations.
[1] 
Frenkel, E.; Hernandez, D., 'Langlands duality for representations of quantum groups', Math. Ann. 349, No. 3, 705–746 (2011). 
Generalized quantum enveloping algebras, coloured KacMoody algebras and Langlands interpolation
PhD thesis, Université Paris VII (2013)
[Abstract]
We propose in this thesis a new deformation process of KacMoody algebras and their representations.
The direction of deformation is given by a collection of numbers, called a colouring.
The natural numbers lead for example to the classical algebras, while the quantum numbers lead to the associated quantum algebras.
We first establish sufficient and necessary conditions on colourings to allow the process depend polynomially on a formal parameter and to provide the
generalised quantum enveloping (GQE) algebras. We then lift the restrictions and show that the process still exists via the coloured KacMoody algebras.
We formulate the GQE conjecture which predicts that every representation in the category $\mathcal O$ integrable of a KacMoody algebra can be deformed into a
representation of an associated GQE algebra. We give various evidences for this conjecture and make a first step towards its resolution by proving that KacMoody algebras
without Serre relations can be deformed into GQE algebras without Serre relations.
In case the conjecture holds, we establish an analog result for coloured KacMoody algebras, we prove that the deformed representation theories are parallel to the classical one,
we explicit a deformed Serre presentation for GQE algebras, we prove that the latter are the representatives of a natural class of formal deformations of KacMoody algebras
and are trivial in finite type. As an application, we explain in terms of interpolation both classical and quantum Langlands dualities between representations of Lie algebras,
and we propose a new approach which aims at proving a conjecture of FrenkelHernandez.
In general, we prove that representations of two isogenic coloured KacMoody algebras can be interpolated by representations of a third one.
Observing that standard quantum algebras satisfy the GQE conjecture, we give a new proof of the previously mentioned classical Langlands duality
(the first proofs are due to Littelmann and McGerty).
Curriculum Vitae
2013: Research Fellow, University of Cambridge
2013: Fellow, St John's College, Cambridge
Fall 2012: Post Doctoral Fellow, Harvard University
200913: PhD Student, Université Paris VII
200810: École Polytechnique
200509: École Normale Supérieure de Lyon
200305: Lycée Hoche, classes préparatoires
Teaching
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2KacMoody Algebras (Part III Essay, Lent 2016)

The aim of this essay is to give an account of 2KacMoody algebras, and of their representation theory.
A 2KacMoody algebra is a 2category, obtained by "doubling" a categorification of one half of a KacMoody algebra $\mathfrak g$.
The categorication in question consists in a monoidal category $\mathcal B$, where quiver Hecke algebras associated to a same quiver $\Gamma$ are put together.
Quiver Hecke algebras have been introduced by Khovanov, Lauda and Rouquier, and are sometimes also called KLR algebras. Geometric methods are available in the
case of symmetrizable KacMoody algebras. Namely, the monoidal category $\mathcal B$ is equivalent to Lusztig's category of perverse sheaves on the moduli space
of representations of the quiver $\Gamma$, and projective modules for the quiver Hecke algebras in characteristic zero relate to the canonical basis of the
quantised KacMoody algebra $U_q(\mathfrak g)$.
Relevant courses
'Lie Algebras and their Representations', 'Introduction to Category Theory' are recommended.
'Representation Theory', 'Topics on Category Theory', 'Algebraic Geometry' can be helpful.
References
[1] 
Beilinson, A.; Bernstein, J.; Deligne, P., 'Faisceaux pervers', Analysis and topology on singular spaces I (Luminy, 1981), Astérisque 100, 5171
(Soc. Math. France, Paris, 1982).

[2] 
Lusztig, G., 'Canonical bases arising from quantized enveloping algebras', J. Amer. Math. Soc. 3, No. 2, 447498 (1990). 
[3] 
Khovanov, M.; Lauda, A., 'A diagrammatic approach to categorification of quantum groups I', Represent. Theory 13, 309347 (2009). 
[4] 
Khovanov, M.; Lauda, A., 'A diagrammatic approach to categorification of quantum groups II', Trans. Amer. Math. Soc. 363, No. 5, 26852700 (2011). 
[5] 
Rouquier, R., '2KacMoody algebras', arXiv:0812.5023 (2008). 
[6] 
Rouquier, R., 'Quiver Hecke algebras and 2Lie algebras', Algebra Colloq. 19, No. 2, 359410 (2012). 
[7] 
Varagnolo, M.; Vasserot, E., 'Canonical bases and KLR algebras', J. Reine Angew. Math. 659, 67100 (2011). 
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FeiginFrenkel Theorem (Part III Essay, Lent 2016)

The local Langlands correspondence is a profound conjecture in the representation theory of a reductive group $G(F)$ over $p$adic numbers and their extensions.
It predicts a parametrisation for smooth representations in terms of Langlands parameters, which are roughly representations of the absolute Galois group $\mathrm{Gal}(\bar F/F)$
into the Langlands dual group $G^\vee$. In the local geometric Langlands correspondence, $p$adics are replaced with the field $\mathbb C((t))$ of Laurent series in one complex
variable. This geometric analogue involves replacing the $p$adic group $G(F)$ with the loop algebra $\mathfrak g \otimes \mathbb C((t))$ (for $\mathfrak g$ a complex semisimple
Lie algebra), and with its central extension $\hat{\mathfrak g}$ which is an affine KacMoody algebra. The Langlands parameters in the geometric setting become
$G^\vee$connections over the punctured disc $\mathrm{Spec} \, \mathbb C((t))$. The aim of this essay is to give an account of the fundamental discovery of Feigin and Frenkel
(resolving a conjecture of Drinfel'd), namely that the enveloping algebra of $\hat{\mathfrak g}$ acting at the critical level has a huge center, analogous to the HarishChandra
center for $U(\mathfrak g)$, and that this center is canonically identified with functions on the space of $G^\vee$opers on the punctured disc.
Relevant courses
'Algebraic Geometry' is recommended.
'Lie Algebras and their Representations', 'Introduction to Category Theory' can be helpful.
References
[1] 
Feigin, B.; Frenkel, E., 'Affine KacMoody algebras at the critical level and GelfandDikii algebras', Infinite analysis, Part A, B (Kyoto, 1991)}, Adv. Ser. Math. Phys. 16,
197215 (World Sci. Publ., River Edge, NJ, 1992).

[2] 
Frenkel, E., 'Langlands correspondence for loop groups', Cambridge Studies in Advanced Mathematics 103 (Cambridge University Press, Cambridge, 2007). 
[3] 
Frenkel, E., 'Wakimoto modules, opers and the center at the critical level', Adv. Math. 195, No. 2, 297404 (2005). 
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Geometric Satake Equivalence (Part III Essay, Lent 2016)

This aim of this essay is to give an account of the geometric Satake equivalence, which is one of the cornerstones of the geometric Langlands correspondence.
The connected complex reductive groups have a combinatorial classification by their root data. In a root datum, the roots and the coroots appear in a symmetric manner,
and so the connected reductive algebraic groups come in pairs. The companion of a reductive group $G$ is the socalled Langlands dual group $G^\vee$. The geometric Satake
equivalence is an equivalence between the category of representations of $G$ and the category of spherical perverse sheaves on the complex affine Grassmannian associated
to the dual group $G^\vee$. This constitutes a geometric version of the classical Satake isomorphism. One can view this result also as a remarkable construction of the
Langlands dual group $G^\vee$ in terms of the group $G$ that does not refer to the underlying root data.
Relevant courses
'Algebraic Geometry', 'Introduction to Category Theory' are recommended.
'Topics on Category Theory' can be helpful.
References
[1] 
Beilinson, A.; Bernstein, J.; Deligne, P., 'Faisceaux pervers', Analysis and topology on singular spaces I (Luminy, 1981), Astérisque 100, 5171
(Soc. Math. France, Paris, 1982).

[2] 
Beilinson, A.; Drinfeld, V., 'Quantization of Hitchin's integrable system and Hecke eigensheaves',
pdf (1991).

[3] 
Deligne, P.; Milne, J., 'Tannakian Categories', Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics 900, 101228
(SpringerVerlag, BerlinNew York, 1982).

[4] 
Gross, B., 'On the Satake isomorphism', Galois representations in arithmetic algebraic geometry (Durham, 1996)}, London Math. Soc. Lecture Note Ser. 254, 223237
(Cambridge Univ. Press, Cambridge, 1998).

[5] 
Humphreys, J., 'Linear algebraic groups', Graduate Texts in Mathematics, No. 21 (SpringerVerlag, New YorkHeidelberg, 1975). 
[6] 
Mirkovic, I; Vilonen, K., 'Geometric Langlands duality and representations of algebraic groups over commutative rings', Ann. of Math. (2) 166, No. 1, 95143 (2007). 
[7] 
Richarz, T., 'A new approach to the geometric Satake equivalence', Doc. Math 19, 209246 (2014). 
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Infinite dimensional Lie Algebras (Graduate Course, Lent 2015)

The aim of this course is to give an introduction to KacMoody algebras and their representations, to present some applications of the theory, and to have a look at the
quantum counterpart. KacMoody algebras are infinitedimensional analogs of semisimple Lie algebras, and we will first study the structure of these algebras in general.
The first significant examples will be the affine Lie algebras. These algebras have another realisation; namely they are central extensions of loop algebras, and therefore
have important applications in theoretical physics. We will precise the relations with the Virasoro and the Heisenberg algebras, and we will study the category of
finitedimensional representations of an affine Lie algebra. By construction, an affine algebra has a central element, which then acts as a scalar (called level) on every
simple representation. We will look at a certain category formed by representations with fixed level, describe the fusion product within this category, and present
applications to the KnizhnikZamolodchikov equations. Quantum groups (or to be more precise, quantized enveloping algebras in this course) are quantum analogs of semisimple
Lie algebras, and more generally of KacMoody algebras. If time permits, we will define these objects, study their representation theory (with an emphasis again on the affine
case, which should lead us to the socalled $q$characters), and discusss the relations between representations of classical and quantum affine algebras.
Prerequisites
Essential: 'Lie algebras and their representations'.
Useful: 'Representation theory'.
References
[1] 
Chari, V.; Pressley, A., 'A guide to quantum groups' (Cambridge University Press, Cambridge, 1995). 
[2] 
Etingof, P.; Frenkel, I.; Kirillov, A., 'Lectures on representation theory and KnizhnikZamolodchikov equations',
Mathematical Surveys and Monographs 58 (American Mathematical Society, Providence, RI, 1998).

[3] 
Frenkel, E.; BenZvi, D., 'Vertex algebras and algebraic curves', Mathematical Surveys and Monographs 88 (American Mathematical Society, Providence, RI, 2004).

[4] 
Frenkel, E., 'Langlands correspondence for loop groups', Cambridge Studies in Advanced Mathematics 103 (Cambridge University Press, Cambridge, 2007). 
[5] 
J. Humphreys, 'Introduction to Lie algebras and representation theory', Graduate Texts in Mathematics 9 (SpringerVerlag, New YorkBerlin, 1978). 
[6] 
Jantzen, J., 'Lectures on Quantum Groups', Graduate Studies in Mathematics 6 (American Mathematical Society, Providence, RI, 1996). 
[7] 
Kac, V., 'Infinitedimensional Lie algebras' (Cambridge University Press, Cambridge, 1990). 
[8] 
Serre, JP., 'Lie algebras and Lie groups', 1964 lectures given at Harvard University, Lecture Notes in Mathematics 1500 (SpringerVerlag, Berlin, 2006). 
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Infinite dimensional Lie Algebras (Part III Essay, Lent 2014)

The aim of this essay is to give an introduction to KacMoody algebras and their representations, present some applications of the theory, and have a look at the quantum side.
You will start from the Serre presentation of complex semisimple Lie algebras, together with the celebrated CartanKilling classification by Cartan matrices (this material has
been covered in the course 'Lie algebras and their representations'). KacMoody algebras are infinitedimensional analogs of semisimple Lie algebras; they also admit a Serre
presentation, associated to generalised Cartan matrices. You will first investigate the structure of KacMoody algebras, and their representation theory (the category $\mathcal O$).
The "first" infinitedimensional KacMoody algebras are said affine. An affine algebra $\hat{\mathfrak g}$ can be realised as a central extension of the loop algebra of a semisimple
Lie algebra $\mathfrak g$. Thanks to this description, you will learn more about $\hat{\mathfrak g}$, precise the relations with the Virasoro algebras and the Heisenberg algebras,
and study the category of finitedimensional representations of $\hat{\mathfrak g}$. By construction, the affine algebra $\hat{\mathfrak g}$ has a central element, which then acts as
a scalar (called level) on every simple representation. Going back to the category $\mathcal O$ of $\hat{\mathfrak g}$, you will focus on the subcategory $\mathcal O_k$ formed by the
representations with fixed level $k$, and experience once again the richness of the representation theory of affine algebras. Affine KacMoody algebras provide a perfect example
where abstract mathematical motivations lead to objects which are closely related to other fields as Mathematical Physics. You will eventually study more advanced topics, as the
fusion product within the category $\mathcal O_k$, and applications to KnizhnikZamolodchikov equations. Quantum groups are quantum analogs of semisimple Lie algebras. If time
permits, you will learn about the quantized enveloping algebra $U_q(\mathfrak g)$, investigate its representation theory, and look at the relations between between representations
of (classical) affine algebras and representation of quantum groups. Affine algebras also admit a quantization. Again, if time permits, you may be interested in the
finitedimensional representation theory of quantum affine algebras: classification of simple representations by the Drinfeld polynomials, the Grothendieck ring, and the
$q$characters.
Relevant courses
Essential: 'Lie algebras and their representations'.
Useful: 'Representation theory'.
References
[1] 
Chari, V.; Pressley, A., 'A guide to quantum groups' (Cambridge University Press, Cambridge, 1995). 
[2] 
Etingof, P.; Frenkel, I.; Kirillov, A., 'Lectures on representation theory and KnizhnikZamolodchikov equations',
Mathematical Surveys and Monographs 58 (American Mathematical Society, Providence, RI, 1998).

[3] 
Frenkel, E.; BenZvi, D., 'Vertex algebras and algebraic curves', Mathematical Surveys and Monographs 88 (American Mathematical Society, Providence, RI, 2004).

[4] 
Frenkel, E., 'Langlands correspondence for loop groups', Cambridge Studies in Advanced Mathematics 103 (Cambridge University Press, Cambridge, 2007). 
[5] 
J. Humphreys, 'Introduction to Lie algebras and representation theory', Graduate Texts in Mathematics 9 (SpringerVerlag, New YorkBerlin, 1978). 
[6] 
Jantzen, J., 'Lectures on Quantum Groups', Graduate Studies in Mathematics 6 (American Mathematical Society, Providence, RI, 1996). 
[7] 
Kac, V., 'Infinitedimensional Lie algebras' (Cambridge University Press, Cambridge, 1990). 
[8] 
Serre, JP., 'Lie algebras and Lie groups', 1964 lectures given at Harvard University, Lecture Notes in Mathematics 1500 (SpringerVerlag, Berlin, 2006). 
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Classes and supervisions in Pure Mathematics (Parts IA, IB & II)

Next IA class: beginning of Lent term.