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MIT International Research Program

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The aim of this essay is to give an account of 2-Kac-Moody algebras, and of their representation theory. A 2-Kac-Moody algebra is a 2-category, obtained by "doubling" a categorification of one half of a Kac-Moody algebra $\mathfrak g$. The categorification 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 Kac-Moody 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 Kac-Moody algebra $U_q(\mathfrak g)$. 'Lie Algebras and their Representations', 'Introduction to Category Theory' are recommended.
'Representation Theory', 'Topics on Category Theory', 'Algebraic Geometry' can be helpful.

[1]	Beilinson, A.; Bernstein, J.; Deligne, P., 'Faisceaux pervers', Analysis and topology on singular spaces I (Luminy, 1981), Astérisque 100, 5-171 (Soc. Math. France, Paris, 1982).
[2]	Lusztig, G., 'Canonical bases arising from quantized enveloping algebras', J. Amer. Math. Soc. 3, No. 2, 447-498 (1990).
[3]	Khovanov, M.; Lauda, A., 'A diagrammatic approach to categorification of quantum groups I', Represent. Theory 13, 309-347 (2009).
[4]	Khovanov, M.; Lauda, A., 'A diagrammatic approach to categorification of quantum groups II', Trans. Amer. Math. Soc. 363, No. 5, 2685-2700 (2011).
[5]	Rouquier, R., '2-Kac-Moody algebras', arXiv:0812.5023 (2008).
[6]	Rouquier, R., 'Quiver Hecke algebras and 2-Lie algebras', Algebra Colloq. 19, No. 2, 359-410 (2012).
[7]	Varagnolo, M.; Vasserot, E., 'Canonical bases and KLR algebras', J. Reine Angew. Math. 659, 67-100 (2011).

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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 Kac-Moody 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 Harish-Chandra 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. 'Algebraic Geometry' is recommended.
'Lie Algebras and their Representations', 'Introduction to Category Theory' can be helpful.

[1]	Feigin, B.; Frenkel, E., 'Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras', Infinite analysis, Part A, B (Kyoto, 1991)}, Adv. Ser. Math. Phys. 16, 197-215 (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, 297-404 (2005).

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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 so-called 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. 'Algebraic Geometry', 'Introduction to Category Theory' are recommended.
'Topics on Category Theory' can be helpful.

[1]	Beilinson, A.; Bernstein, J.; Deligne, P., 'Faisceaux pervers', Analysis and topology on singular spaces I (Luminy, 1981), Astérisque 100, 5-171 (Soc. Math. France, Paris, 1982).
[2]	Beilinson, A.; Drinfel'd, 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, 101-228 (Springer-Verlag, Berlin-New York, 1982).
[4]	Gross, B., 'On the Satake isomorphism', Galois representations in arithmetic algebraic geometry (Durham, 1996)}, London Math. Soc. Lecture Note Ser. 254, 223-237 (Cambridge Univ. Press, Cambridge, 1998).
[5]	Humphreys, J., 'Linear algebraic groups', Graduate Texts in Mathematics, No. 21 (Springer-Verlag, New York-Heidelberg, 1975).
[6]	Mirkovic, I; Vilonen, K., 'Geometric Langlands duality and representations of algebraic groups over commutative rings', Ann. of Math. (2) 166, No. 1, 95-143 (2007).
[7]	Richarz, T., 'A new approach to the geometric Satake equivalence', Doc. Math 19, 209-246 (2014).

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The aim of this course is to give an introduction to Kac-Moody algebras and their representations, to present some applications of the theory, and to have a look at the quantum counterpart. Kac-Moody algebras are infinite-dimensional analog 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 finite-dimensional 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 Knizhnik-Zamolodchikov equations. Quantum groups (or to be more precise, quantized enveloping algebras in this course) are quantum analog of semisimple Lie algebras, and more generally of Kac-Moody 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 so-called $q$-characters), and discuss the relations between representations of classical and quantum affine algebras. Essential: 'Lie algebras and their representations'. Useful: 'Representation theory'.

[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 Knizhnik-Zamolodchikov equations', Mathematical Surveys and Monographs 58 (American Mathematical Society, Providence, RI, 1998).
[3]	Frenkel, E.; Ben-Zvi, 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 (Springer-Verlag, New York-Berlin, 1978).
[6]	Jantzen, J., 'Lectures on Quantum Groups', Graduate Studies in Mathematics 6 (American Mathematical Society, Providence, RI, 1996).
[7]	Kac, V., 'Infinite-dimensional Lie algebras' (Cambridge University Press, Cambridge, 1990).
[8]	Serre, J-P., 'Lie algebras and Lie groups', 1964 lectures given at Harvard University, Lecture Notes in Mathematics 1500 (Springer-Verlag, Berlin, 2006).

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The aim of this essay is to give an introduction to Kac-Moody 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 Cartan-Killing classification by Cartan matrices (this material has been covered in the course 'Lie algebras and their representations'). Kac-Moody algebras are infinite-dimensional analog of semisimple Lie algebras; they also admit a Serre presentation, associated to generalised Cartan matrices. You will first investigate the structure of Kac-Moody algebras, and their representation theory (the category $\mathcal O$). The "first" infinite-dimensional Kac-Moody 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 finite-dimensional 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 Kac-Moody 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 Knizhnik-Zamolodchikov equations. Quantum groups are quantum analog 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 finite-dimensional representation theory of quantum affine algebras: classification of simple representations by the Drinfel'd polynomials, the Grothendieck ring, and the $q$-characters. Essential: 'Lie algebras and their representations'. Useful: 'Representation theory'.

[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 Knizhnik-Zamolodchikov equations', Mathematical Surveys and Monographs 58 (American Mathematical Society, Providence, RI, 1998).
[3]	Frenkel, E.; Ben-Zvi, 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 (Springer-Verlag, New York-Berlin, 1978).
[6]	Jantzen, J., 'Lectures on Quantum Groups', Graduate Studies in Mathematics 6 (American Mathematical Society, Providence, RI, 1996).
[7]	Kac, V., 'Infinite-dimensional Lie algebras' (Cambridge University Press, Cambridge, 1990).
[8]	Serre, J-P., 'Lie algebras and Lie groups', 1964 lectures given at Harvard University, Lecture Notes in Mathematics 1500 (Springer-Verlag, Berlin, 2006).

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St John's College

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both pure and applied mathematics