Abstract
Lie groups and Lie algebras occupy a prominent and central place in mathematics, connecting differential geometry, representation theory, algebraic geometry, number theory, and theoretical physics. In some sense, the heart of (classical) representation theory is in the study of the semisimple Lie groups. Their study is simultaneously simple in its beauty, as well as complex in its richness. From Killing, Cartan, and Weyl, to Dynkin, Harish-Chandra, Bruhat, Kostant, and Serre, many mathematicians in the twentieth century have worked on building up the theory of semisimple Lie algebras and their universal enveloping algebras. Books by Borel, Bourbaki, Bump, Chevalley, Humphreys, Jacobson, Varadarajan, Vogan, and others form the texts for (introductory) graduate courses on the subject.
This work was supported in part by DARPA Grant # YFA N66001-11-1-4131.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
A.A. Beilinson and J.N. Bernstein, Localisation de 𝔤-modules, C. R. Acad. Sci. Paris, Ser. 1 292 (1981), 15–18.
A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Inventiones Mathematicae 143 (2001), 77–128.
J. Bernstein, I.M. Gelfand, and S.I. Gelfand, A category of 𝔤 modules, Functional Analysis and Applications 10 (1976), 87–92.
R. Brauer, Sur la multiplication des caractéristiques des groups continus et semi-simples, C. R. Acad. Sci. Paris 204 (1937), 1784–1786.
F. Bruhat, Sur les réprésentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97–205.
V. Chari and J. Greenstein, Current algebras, highest weight categories and quivers, Advances in Mathematics 216 (2007), no. 2, 811–840.
—, Minimal affinizations as projective objects, Journal of Geometry and Physics 61 (2011), no. 3, 594–609.
V. Chari, A. Khare, and T.B. Ridenour, Faces of polytopes and Koszul algebras, Journal of Pure and Applied Algebra 216 (2012), no. 7, 1611–1625.
V. Chari and A. Pressley, A new family of irreducible, integrable modules for affine Lie algebras, Mathematische Annalen 277 (1987), no. 3, 543–562.
I. Dimitrov and M. Roth, Geometric realization of PRV components and the Littlewood-Richardson cone, Contemporary Mathematics 490: Symmetry in Mathematics and Physics, D. Babbitt, V. Chari, and R. Fioresi, Eds. (2009), 83–95.
M. Duflo, Construction of primitive ideals in an enveloping algebra, Lie groups and their representations (1971 János Bolyai Math. Soc. Summer School in Mathematics, Budapest), I.M. Gelfand, Ed. (1975), 77–93.
—, Représentations irréductibles des groupes semi-simples complexes, Springer Lecture Notes in Mathematics 497 (1975), 26–88.
—, Sur la classification des idéaux primitifs dans l’algèbre enveloppante d’une algèbre de Lie semi-semisimple, Ann. Math. 105 (1977), 107–120.
D.R. Farkas and G. Letzter, Quantized representation theory following Joseph, Progress in Mathematics 243, Part I: Studies in Lie Theory (2006), 9–17.
I.M. Gelfand and M.A. Naimark, Unitary representations of the classical groups, Trudy Mat. Inst. Steklov 36, Moscow-Leningrad, 1950.
M. Gorelik and E. Lanzmann, The annihilation theorem for Lie superalgebra 𝔬𝔰𝔭 (1, 2l), Inventiones Mathematicae 137 (1999), 651–680.
B.C. Hall, Lie groups, Lie algebras, and representations: an elementary introduction, Graduate Texts in Mathematics, no. 222, Springer-Verlag, Berlin-New York, 2004.
Harish-Chandra, On some applications of the universal enveloping algebra of a semi-simple Lie algebra, Trans. Amer. Math. Soc. 70 (1951), 28–96.
—, Representations of a semi-simple Lie group on a Banach space: I, Trans. Amer. Math. Soc. 75 (1953), 185–243.
—, Representations of semi-simple Lie groups: II, Trans. Amer. Math. Soc. 76 (1954), 26–65.
—, The Plancherel formula for complex semi-simple Lie groups, Trans. Amer. Math. Soc. 76 (1954), 485–528.
M. Hayashi, The moduli space of SU (3)-flat connections and the fusion rules, Proc. Amer. Math. Soc. 127 (1999), 1545–1555.
J.E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, no. 9, Springer-Verlag, Berlin-New York, 1972.
—, Representations of semisimple Lie algebras in the BGG Category O, Graduate Studies in Mathematics 94, American Mathematical Society, Providence, RI, 2008.
A. Joseph, Quantum groups and their primitive ideals, Ergeb. Math. Grenzgeb. (3) 29, Springer, Berlin, 1995.
—, A completion of the quantized enveloping algebra of a Kac-Moody algebra, Journal of Algebra 214 (1999), no. 1, 235–275.
—, On the Kostant-Parthasarathy-Ranga Rao-Varadarajan determinants, I. Injectivity and multiplicities, Journal of Algebra 241 (2001), 27–45.
A. Joseph and G. Letzter, Separation of variables for quantized enveloping algebras, American Journal of Mathematics 116 (1994), 127–177.
—, Verma modules annihilators for quantized enveloping algebras, Ann. Ecole Norm. Sup. 28 (1995), 493–526.
—, On the Kostant-Parthasarathy-Ranga Rao-Varadarajan determinants, II. Construction of the KPRV determinants, Journal of Algebra 241 (2001), 46–66.
A. Joseph, G. Letzter, and D. Todoric, On the Kostant-Parthasarathy-Ranga Rao-Varadarajan determinants, III. Computation of the KPRV determinants, Journal of Algebra 241 (2001), 67–88.
A. Joseph and D. Todoric, On the quantum KPRV determinants for semisimple and affine Lie algebras, Algebras and Representation Theory 5 (2002), 57–99.
M. Kashiwara, On crystal bases, Canadian Math. Soc. Conf. Proc. 16 (1995), 155–197.
D. Kazhdan, M. Larsen, and Y. Varshavsky, The Tannakian Formalism and the Langlands Conjectures, preprint, http://arxiv.org/abs/1006.3864.
A. Khare, Axiomatic framework for the BGG Category O, preprint, http://arxiv.org/abs/0811.2080.
A.U. Klimyk, On the multiplicities of weights of representations and the multiplicities of representations of semisimple Lie algebras, Dokl. Acad. Nauk SSSR 177 (1967), 1001–1004.
A.W. Knapp and G. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. Math. 116 (1982), 389–455.
B. Kostant, A formula for the multiplicity of a weight, Trans. Amer. Math. Soc. 93 (1959), 53–73.
—, Lie group representations on polynomial rings, American Journal of Mathematics 85 (1963), no. 3, 327–404.
—, On the existence and irreducibility of certain series of representations, Lie groups and their representations (1971 János Bolyai Math. Soc. Summer School in Mathematics, Budapest), I.M. Gelfand, Ed. (1975), 231–331.
—, Clifford algebra analogue of the Hopf-Koszul-Samelson Theorem, the ρ-decomposition C(𝔤) = End V ρ ⊗ C(P), and the 𝔤-module structure of ⋀𝔤, Advances in Mathematics 125 (1997), 275–350.
S. Kumar, Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, Inventiones Mathematicae 93 (1988), 117–130.
—, Existence of certain components in the tensor product of two integrable highest weight modules for Kac-Moody algebras, Advanced series in Mathematical Physics 7: Infinite dimensional Lie algebras and groups, V.G. Kac, Ed. (1989), 25–38.
—, A refinement of the PRV conjecture, Inventiones Mathematicae 97 (1989), 305–311.
—, Proof of Wahl’s conjecture on surjectivity of the Gaussian map for flag varieties, American Journal of Mathematics 114 (1992), 1201–1220.
—, Tensor Product Decomposition, Proceedings of the International Congress of Mathematicians (2010).
R.P. Langlands, On the classification of irreducible representations of real algebraic groups, Mathematical Surveys and Monographs 31: Representation theory and harmonic analysis on semisimple Lie groups, P.J. Sally and DA. Vogan, Eds. (1989), 101–170.
J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973), 1–44.
J. Lepowsky and G.W. McCollum, On the determination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc. 176 (1973), 45–57.
P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Inventiones Mathematicae 116 (1994), 329–346.
G. Lusztig, Canonical bases arising from quantized enveloping algebras II, Prog. Theor. Phys. 102 (1990), 175–201.
O. Mathieu, Construction d’un groupe de Kac-Moody et applications, Compositio Mathematica 69 (1989), no. 1, 37–60.
—, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Inventiones Mathematicae 107 (1992), 225–234.
—, Classification of irreducible weight modules, Annales de l’institut Fourier 50 (2000), no. 2, 537–592.
P.L. Montagard, B. Pasquier, and N. Ressayre, Two generalizations of the PRV conjecture, Compositio Mathematica 147 (2011), no. 4, 1321–1336.
—, Generalizations of the PRV conjecture, II, preprint, http://arxiv.org/abs/1110.4621.
R.V. Moody and A. Pianzola, Lie algebras with triangular decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley Interscience, New York-Toronto, 1995.
D.I. Panyushev and O.S. Yakimova, The PRV-formula for tensor product decompositions and its applications, Functional Analysis and Applications 42 (2008), no. 1, 45–52.
K.R. Parthasarathy, R. Ranga Rao, and V.S. Varadarajan, Representations of complex semisimple Lie groups and Lie algebras, Bull. Amer. Math. Soc. 72 (1966), 522–525.
—, Representations of complex semisimple Lie groups and Lie algebras, Ann. Math. 85 (1967), 383–429.
P. Polo, Variétés de Schubert et excellentes filtrations, Astérisque (Orbites unipotentes et représentations) 173–174 (1989), 281–311.
K.N. Rajeswari, Standard monomial theoretic proof of PRV conjecture, Communications in Algebra 19 (1991), 347–425.
N.N. Shapovalov, On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Functional Analysis and Its Applications 6 (1972), 307–312.
R. Steinberg, A general Clebsch-Gordan Theorem, Bull. Amer. Math. Soc. 67 (1961), 406–407.
V.S. Varadarajan, Some mathematical reminiscences, Methods and Applications of Analysis 9 (2002), no. 3, v–xviii.
D.N. Verma, Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968), no. 1, 160–166.
D.A. Vogan Jr., The algebraic structure of the representation of semisimple Lie groups. I, Ann. Math. 109 (1979), no. 1, 1–60.
C.A.S. Young and R. Zegers, Dorey’s rule and the q-characters of simplylaced quantum affine algebras, Commun. Math. Phys. 302 (2011), 789–813.
D.P. Zhelobenko, The analysis of irreducibility in the class of elementary representations of a complex semisimple Lie group, Math. USSR Izv. 2 (1968), no. 1, 105–128.
—, Harmonic analysis on complex semisimple Lie groups (Russian), Mir, Moscow, 1974.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2013 Hindustan Book Agency
About this chapter
Cite this chapter
Khare, A. (2013). Representations of Complex Semi-simple Lie Groups and Lie Algebras. In: Bhatia, R., Rajan, C.S., Singh, A.I. (eds) Connected at Infinity II. Texts and Readings in Mathematics, vol 67. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-56-9_5
Download citation
DOI: https://doi.org/10.1007/978-93-86279-56-9_5
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-93-80250-51-9
Online ISBN: 978-93-86279-56-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)