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Academic literature on the topic 'LIE ALGEBRAS, REPRESENTATION THEORY'

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Published: 10 March 2023

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Journal articles on the topic "LIE ALGEBRAS, REPRESENTATION THEORY"

1

Cheng, Yongsheng, and Huange Qi. "Representations of Bihom-Lie Algebras." Algebra Colloquium 29, no. 01 (January 13, 2022): 125–42. http://dx.doi.org/10.1142/s1005386722000104.

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A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.

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Rouquier, Raphaël. "Quiver Hecke Algebras and 2-Lie Algebras." Algebra Colloquium 19, no. 02 (May 3, 2012): 359–410. http://dx.doi.org/10.1142/s1005386712000247.

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We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

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3

Friedlander, Eric M., and Brian J. Parshall. "Modular Representation Theory of Lie Algebras." American Journal of Mathematics 110, no. 6 (December 1988): 1055. http://dx.doi.org/10.2307/2374686.

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4

BARANOV, A. A., and A. E. ZALESSKII. "PLAIN REPRESENTATIONS OF LIE ALGEBRAS." Journal of the London Mathematical Society 63, no. 3 (June 2001): 571–91. http://dx.doi.org/10.1017/s0024610701002101.

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In this paper we study representations of finite dimensional Lie algebras. In this case representations are not necessarily completely reducible. As the general problem is known to be of enormous complexity, we restrict ourselves to representations that behave particularly well on Levi subalgebras. We call such representations plain (Definition 1.1). Informally, we show that the theory of plain representations of a given Lie algebra L is equivalent to representation theory of finitely many finite dimensional associative algebras, also non-semisimple. The sense of this is to distinguish representations of Lie algebras that are of complexity comparable with that of representations of associative algebras. Non-plain representations are intrinsically much more complex than plain ones. We view our work as a step toward understanding this complexity phenomenon.We restrict ourselves also to perfect Lie algebras L, that is, such that L = [L, L]. In our main results we assume that L is perfect and [sfr ][lfr ]2-free (which means that L has no quotient isomorphic to [sfr ][lfr ]2). The ground field [ ] is always assumed to be algebraically closed and of characteristic 0.

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5

Liu, Shanshan, Lina Song, and Rong Tang. "Representations and cohomologies of regular Hom-pre-Lie algebras." Journal of Algebra and Its Applications 19, no. 08 (August 8, 2019): 2050149. http://dx.doi.org/10.1142/s0219498820501492.

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In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of regular Hom-pre-Lie algebras in terms of the cohomology theory of regular Hom-Lie algebras. As applications, we study linear deformations of regular Hom-pre-Lie algebras, which are characterized by the second cohomology groups of regular Hom-pre-Lie algebras with the coefficients in the regular representations. The notion of a Nijenhuis operator on a regular Hom-pre-Lie algebra is introduced which can generate a trivial linear deformation of a regular Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a regular Hom-pre-Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an [Formula: see text]-operator on a regular Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure.

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6

Mirković, I., and D. Rumynin. "Geometric representation theory of restricted Lie algebras." Transformation Groups 6, no. 2 (June 2001): 175–91. http://dx.doi.org/10.1007/bf01597136.

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Goodwin, Simon M., Gerhard Röhrle, and Glenn Ubly. "On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type." LMS Journal of Computation and Mathematics 13 (September 2, 2010): 357–69. http://dx.doi.org/10.1112/s1461157009000205.

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AbstractWe consider the finiteW-algebraU(𝔤,e) associated to a nilpotent elemente∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem forU(𝔤,e), we verify a conjecture of Premet, thatU(𝔤,e) always has a 1-dimensional representation when 𝔤 is of typeG2,F4,E6orE7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal inU(𝔤) whose associated variety is the coadjoint orbit corresponding to e.

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8

Kasjan, Stanisław, and Justyna Kosakowska. "On Lie algebras associated with representation-directed algebras." Journal of Pure and Applied Algebra 214, no. 5 (May 2010): 678–88. http://dx.doi.org/10.1016/j.jpaa.2009.07.012.

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9

Brown, Kenneth A., and Fokko Du Cloux. "On the Representation Theory of Solvable Lie Algebras." Proceedings of the London Mathematical Society s3-57, no. 2 (September 1988): 284–300. http://dx.doi.org/10.1112/plms/s3-57.2.284.

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10

Casas, J. M. "Obstructions to Lie–Rinehart Algebra Extensions." Algebra Colloquium 18, no. 01 (March 2011): 83–104. http://dx.doi.org/10.1142/s1005386711000046.

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The problem of the representation of an action of a Lie–Rinehart algebra on a Lie 𝖠-algebra by means of a homomorphism of Lie–Rinehart algebras is studied. An eight-term exact sequence associated to an epimorphism of Lie–Rinehart algebras for the cohomology of Lie–Rinehart algebras developed by Casas, Ladra and Pirashvili is obtained. This sequence is applied to study the obstruction theory of Lie–Rinehart algebra extensions.

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Dissertations / Theses on the topic "LIE ALGEBRAS, REPRESENTATION THEORY"

1

Carr, Andrew Nickolas. "Lie Algebras and Representation Theory." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1988.

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2

Lemay, Joel. "Valued Graphs and the Representation Theory of Lie Algebras." Thèse, Université d'Ottawa / University of Ottawa, 2011. http://hdl.handle.net/10393/20168.

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Quivers (directed graphs) and species (a generalization of quivers) as well as their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this thesis, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K-species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.

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Cao, Mengyuan. "Representation Theory of Lie Colour Algebras and Its Connection with the Brauer Algebras." Thesis, Université d'Ottawa / University of Ottawa, 2018. http://hdl.handle.net/10393/38125.

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In this thesis, we study the representation theory of Lie colour algebras. Our strategy follows the work of G. Benkart, C. L. Shader and A. Ram in 1998, which is to use the Brauer algebras which appear as the commutant of the orthosymplectic Lie colour algebra when they act on a k-fold tensor product of the standard representation. We give a general combinatorial construction of highest weight vectors using tableaux, and compute characters of the irreducible summands in some borderline cases. Along the way, we prove the RSK-correspondence for tableaux and the PBW theorem for Lie colour algebras.

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4

Muth, Robert. "Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type." Thesis, University of Oregon, 2016. http://hdl.handle.net/1794/20432.

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We study representations of Khovanov-Lauda-Rouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR algebra. For a balanced order, we study imaginary semicuspidal modules by means of `imaginary Schur-Weyl duality'. We then generalize this theory from balanced to arbitrary convex preorders for affine ADE types. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that KLR algebras are properly stratified. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. Finally, working in finite or affine affine type A, we show that skew Specht modules may be defined over the KLR algebra, and real cuspidal modules associated to a balanced convex preorder are skew Specht modules for certain explicit hook shapes.

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5

Lampetti, Enrico. "Nilpotent orbits in semisimple Lie algebras." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23595/.

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This thesis is dedicated to the introductory study of the so-called nilpotent orbits in a semisimple complex Lie algebra g, i.e., the orbits of nilpotent elements under the adjoint action of the adjoint group Gad with Lie algebra g. These orbits have an extremely rich structure and lie at the interface of Lie theory, algebraic geometry, symplectic geometry, and geometric representation theory. The Jacobson and Morozov Theorem relates the orbit of a nilpotent element X in a semisimple complex Lie algebra g with a triple {H,X,Y} that generates a subalgebra of g isomorphic to sl(2,C). There is a parabolic subalgebra associated to this triple that permits to attach a weight to each node of the Dynkin diagram of g. The resulting diagram is called a weighted Dynkin diagram associated with the nilpotent orbit of X. This is a complete invariant of the orbit that one can use in order to show that there are only _nitely many nilpotent orbits in g. The thesis is organized as follows: the first three chapters contain some preliminary material on Lie algebras (Chapter 1), on Lie groups (Chapter 3) and on the representation theory of sl(2,C) (Chapter 2). Chapter 4 and 5 are the heart of the thesis. Namely, Jacobson-Morozov, Kostant and Mal'cev Theorems are proved in Chapter 4 and Chapter 5 is dedicated to the construction of weighted Dynkin diagrams. As an example the conjugacy classes of nilpotent elements in sl(n,C) are described in detail and a formula for their dimension is given. In this case, as well as in the case of all classical Lie algebras, the description of the orbits can be done in terms of partitions and tableaux.

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Rakotoarisoa, Andriamananjara Tantely. "The Bala-Carter Classification of Nilpotent Orbits of Semisimple Lie Algebras." Thesis, Université d'Ottawa / University of Ottawa, 2017. http://hdl.handle.net/10393/36058.

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Conjugacy classes of nilpotent elements in complex semisimple Lie algebras are classified using the Bala-Carter theory. In this theory, nilpotent orbits in g are parametrized by the conjugacy classes of pairs (l,pl) of Levi subalgebras of g and distinguished parabolic subalgebras of [l,l]. In this thesis we present this theory and use it to give a list of representatives for nilpotent orbits in so(8) and from there we give a partition-type parametrization of them.

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7

O'Dell, Connor. "Non-Resonant Uniserial Representations of Vec(R)." Thesis, University of North Texas, 2018. https://digital.library.unt.edu/ark:/67531/metadc1157650/.

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The non-resonant bounded uniserial representations of Vec(R) form a certain class of extensions composed of tensor density modules, all of whose subquotients are indecomposable. The problem of classifying the extensions with a given composition series is reduced via cohomological methods to computing the solution of a certain system of polynomial equations in several variables derived from the cup equations for the extension. Using this method, we classify all non-resonant bounded uniserial extensions of Vec(R) up to length 6. Beyond this length, all such extensions appear to arise as subquotients of extensions of arbitrary length, many of which are explained by the psuedodifferential operator modules. Others are explained by a wedge construction and by the pseudodifferential operator cocycle discovered by Khesin and Kravchenko.

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8

Meinel, Joanna [Verfasser]. "Affine nilTemperley-Lieb algebras and generalized Weyl algebras: Combinatorics and representation theory / Joanna Meinel." Bonn : Universitäts- und Landesbibliothek Bonn, 2016. http://d-nb.info/1122193874/34.

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9

Lemay, Joel. "Geometric Realizations of the Basic Representation of the Affine General Linear Lie Algebra." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32866.

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The realizations of the basic representation of the affine general linear Lie algebra on (r x r) matrices are well-known to be parametrized by partitions of r and have an explicit description in terms of vertex operators on the bosonic/fermionic Fock space. In this thesis, we give a geometric interpretation of these realizations in terms of geometric operators acting on the equivariant cohomology of certain Nakajima quiver varieties.

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10

Leonardi, Davide. "Kac-Moody algebras and representations of quivers." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/20796/.

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La tesi introduce il lettore alla teoria delle rappresentazioni di algebre di Lie e quivers. Viene studiata la teoria delle algebre di Lie semisemplici e delle algebre di Kac-Moody su un campo algebricamente chiuso di caratteristica zero. Si introduce la teoria delle rappresentazioni di quiver e si dimostra un criterio per decidere qualora una data rappresentazione di un dato quiver sia assolutamente indecomponibile o meno.

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Books on the topic "LIE ALGEBRAS, REPRESENTATION THEORY"

1

Humphreys, James E. Introduction to Lie algebras and representation theory. 7th ed. New York: Springer, 1997.

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2

Introduction to Lie algebras and representation theory. 6th ed. New York: Springer-Verlag, 1994.

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3

Geometric representation theory and extended affine Lie algebras. Providence, R.I: American Mathematical Society, 2011.

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Neher, Erhard. Geometric representation theory and extended affine Lie algebras. Providence, R.I: American Mathematical Society, 2011.

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5

William, Fulton. Representation theory: A first course. New York: Springer-Verlag, 1991.

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6

Yoshiyuki, Koga, ed. Representation theory of the Virasoro algebra. London: Springer, 2011.

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1918-, Coleman A. John, Futorny V, and Pollack Richard D, eds. Modern trends in Lie algebra representation theory: Conference proceedings. Kingston, Ont: Queen's University Press, 1994.

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William, Fulton. Representation theory: A first course. 3rd ed. New York: Springer, 1996.

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9

Fuji-Kawaguchiko Conference on Representation Theory of Lie Groups and Lie Algebras. (1990 Fuji-Kawaguchiko, Japan). Representation theory of Lie groups and Lie algebras: The proceedings of Fuji-Kawaguchiko Conference on Representation Theory of Lie Groups and Lie Algebras, Fuji-Kawaguchiko, Aug 31-Sep 3, 1990. Edited by Kawazoe T, Oshima T, and Sano S. Singapore: World Scientific, 1992.

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10

1936-, Kirillov A. A., and Olshanskiǐ G. I, eds. Kirillov's seminar on representation theory. Providence, R.I: American Mathematical Society, 1998.

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Book chapters on the topic "LIE ALGEBRAS, REPRESENTATION THEORY"

1

Woit, Peter. "Lie Algebras and Lie Algebra Representations." In Quantum Theory, Groups and Representations, 55–71. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-64612-1_5.

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Hilgert, Joachim, and Karl-Hermann Neeb. "Representation Theory of Lie Algebras." In Springer Monographs in Mathematics, 167–226. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-0-387-84794-8_7.

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Lal, Ramji. "Representation Theory of Lie Algebras." In Algebra 4, 127–79. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-0475-1_3.

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Bernstein, Joseph. "Lectures on Lie Algebras." In Representation Theory, Complex Analysis, and Integral Geometry, 97–132. Boston: Birkhäuser Boston, 2011. http://dx.doi.org/10.1007/978-0-8176-4817-6_6.

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Hayashi, Masahito. "Representations of Typical Lie Groups and Typical Lie Algebras." In Group Representation for Quantum Theory, 113–49. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-44906-7_4.

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Hayashi, Masahito. "Representation of General Lie Groups and General Lie Algebras." In Group Representation for Quantum Theory, 201–29. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-44906-7_6.

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Enright, Thomas. "Representation theory of semisimple Lie algebras." In Mathematical Surveys and Monographs, 21–28. Providence, Rhode Island: American Mathematical Society, 1987. http://dx.doi.org/10.1090/surv/024/02.

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Meinrenken, Eckhard. "The spin representation." In Clifford Algebras and Lie Theory, 49–85. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-36216-3_3.

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Casselman, Bill. "Structure constants of Kac–Moody Lie algebras." In Symmetry: Representation Theory and Its Applications, 55–83. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-1590-3_4.

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Lakshmibai, V., and Justin Brown. "Representation Theory of Complex Semisimple Lie Algebras." In Texts and Readings in Mathematics, 103–14. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-1393-6_8.

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Conference papers on the topic "LIE ALGEBRAS, REPRESENTATION THEORY"

1

Kawazoe, T., T. Oshima, and S. Sano. "Representation Theory of Lie Groups and Lie Algebras." In Fuji-Kawaguchiko Conference on Representation Theory of Lie Groups and Lie Algebras. WORLD SCIENTIFIC, 1992. http://dx.doi.org/10.1142/9789814537162.

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LESLIE, JOSHUA A. "ON A SOLUTION TO A GLOBAL INVERSE PROBLEM WITH RESPECT TO CERTAIN GENERALIZED SYMMETRIZABLE KAC-MOODY ALGEBRAS." In Infinite Dimensional Lie Groups in Geometry and Representation Theory. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812777089_0003.

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Patera, J. "Graded contractions of Lie algebras, representations and tensor products." In Group Theory in Physics: Proceedings of the international symposium held in honor of Professor Marcos Moshinsky. AIP, 1992. http://dx.doi.org/10.1063/1.42858.

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KUBO, F. "COMPATIBLE ALGEBRA STRUCTURES OF LIE ALGEBRAS." In 5th China–Japan–Korea International Ring Theory Conference. WORLD SCIENTIFIC, 2008. http://dx.doi.org/10.1142/9789812818331_0020.

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Smirnov, Yu F. "Projection operators for Lie algebras, duperalgebras, and quantum algebras." In The XXX Latin American school of physics ELAF: Group theory and its applications. AIP, 1996. http://dx.doi.org/10.1063/1.50219.

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Leclerc, Bernard. "Cluster Algebras and Representation Theory." In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010). Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All Markets Except in India, 2011. http://dx.doi.org/10.1142/9789814324359_0154.

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Poletaeva, Elena, and Vladimir Dobrev. "On Exceptional Superconformal Algebras." In LIE THEORY AND ITS APPLICATIONS IN PHYSICS: VIII International Workshop. AIP, 2010. http://dx.doi.org/10.1063/1.3460169.

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Campoamor-Stursberg, R., M. Rausch de Traubenberg, and Vladimir Dobrev. "Parafermions, Ternary Algebras and Their Associated Superspace." In LIE THEORY AND ITS APPLICATIONS IN PHYSICS: VIII International Workshop. AIP, 2010. http://dx.doi.org/10.1063/1.3460167.

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Rajan, G. Susinder, and B. Sundar Rajan. "STBCs from Representation of Extended Clifford Algebras." In 2007 IEEE International Symposium on Information Theory. IEEE, 2007. http://dx.doi.org/10.1109/isit.2007.4557141.

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ARAKAWA, TOMOYUKI. "REPRESENTATION THEORY OF W-ALGEBRAS AND HIGGS BRANCH CONJECTURE." In International Congress of Mathematicians 2018. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789813272880_0096.

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Reports on the topic "LIE ALGEBRAS, REPRESENTATION THEORY"

1

Berceanu, Stefan. A Holomorphic Representation of the Semidirect Sum of Symplectic and Heisenberg Lie Algebras. Journal of Geometry and Symmetry in Physics, 2012. http://dx.doi.org/10.7546/jgsp-5-2006-5-13.

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