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

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Published: 10 March 2023

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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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2

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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7

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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11

Capparelli, Stefano, Arne Meurman, Andrej Primc, and Mirko Primc. "New partition identities from \(C^{(1)}_\ell\)-modules." Glasnik Matematicki 57, no. 2 (December 30, 2022): 161–84. http://dx.doi.org/10.3336/gm.57.2.01.

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In this paper we conjecture combinatorial Rogers-Ramanujan type colored partition identities related to standard representations of the affine Lie algebra of type \(C^{(1)}_\ell\), \(\ell\geq2\), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.

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12

Xia, Limeng, and Dong Liu. "Finite Dimensional Simple Modules over Some GIM Lie Algebras." Mathematics 10, no. 15 (July 28, 2022): 2658. http://dx.doi.org/10.3390/math10152658.

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GIM Lie algebras are the generalizations of Kac–Moody Lie algebras. However, the structures of GIM Lie algebras are more complex than the latter, so they have encountered many new difficulties to study their representation theory. In this paper, we classify all finite dimensional simple modules over the GIM Lie algebra Qn+1(2,1) as well as those over Θ2n+1.

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13

BANICA, TEODOR, and JULIEN BICHON. "HOPF IMAGES AND INNER FAITHFUL REPRESENTATIONS." Glasgow Mathematical Journal 52, no. 3 (August 25, 2010): 677–703. http://dx.doi.org/10.1017/s0017089510000510.

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AbstractWe develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.

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14

IM, BOKHEE, and JONATHAN D. H. SMITH. "REPRESENTATION THEORY FOR VARIETIES OF COMTRANS ALGEBRAS AND LIE TRIPLE SYSTEMS." International Journal of Algebra and Computation 21, no. 03 (May 2011): 459–72. http://dx.doi.org/10.1142/s0218196711006315.

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For a variety of comtrans algebras over a commutative ring, representations of algebras in the variety are identified as modules over an enveloping algebra. In particular, a new, simpler approach to representations of Lie triple systems is provided.

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15

AdamoviĆ, Draen. "A construction of some ideals in affine vertex algebras." International Journal of Mathematics and Mathematical Sciences 2003, no. 15 (2003): 971–80. http://dx.doi.org/10.1155/s0161171203201058.

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We study ideals generated by singular vectors in vertex operator algebras associated with representations of affine Lie algebras of typesAandC. We find new explicit formulas for singular vectors in these vertex operator algebras at integer and half-integer levels. These formulas generalize the expressions for singular vectors from Adamović (1994). As a consequence, we obtain a new family of vertex operator algebras for which we identify the associated Zhu's algebras. A connection with the representation theory of Weyl algebras is also discussed.

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16

Augarten, Tal. "Representation growth of the classical Lie algebras." Communications in Algebra 48, no. 7 (May 12, 2020): 3099–108. http://dx.doi.org/10.1080/00927872.2020.1729364.

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17

ROWE, D. J. "BOSON AND ROTOR EXPANSIONS OF LIE ALGEBRAS IN VECTOR COHERENT STATE THEORY." International Journal of Modern Physics E 02, supp01 (January 1993): 119–35. http://dx.doi.org/10.1142/s0218301393000510.

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A brief overview is given of some of the ways VCS theory can be used to generate boson and rotor expansions of Lie groups. It is demonstrated by examples that such representations are a powerful aid in computing the explicit matrices of the irreducible representations needed in the application of Lie groups and Lie algebras in physics. It is shown that VCS theory is a theory of induced representations and that it has some advantages over other inducing constructions. Boson and rotor expansions are applied to the microscopic theory of nuclear rotations and it is shown that, in addition to providing algorithms for the calculation of the representation matrices needed, these expansions also provide new perspectives on the theory which enable it to be extended to include intrinsic nucleon spin degrees of freedom and the adiabatic mixing of representations.

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18

Martin, Paul P., and David Woodcock. "Generalized Blob Algebras and Alcove Geometry." LMS Journal of Computation and Mathematics 6 (2003): 249–96. http://dx.doi.org/10.1112/s1461157000000450.

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AbstractA sequence of finite-dimensional quotients of affine Hecke algebras is studied. Each element of the sequence is constructed so as to have a weight space labelling scheme for Specht⁄standard modules. As in the weight space formalism of algebraic Lie theory, there is an action of an affine reflection group on this weight space that fixes the set of labelling weights. A linkage principle is proved in each case. Further, it is shown that the simplest non-trivial example may essentially be identified with the blob algebra (a physically motivated quasihereditary algebra whose representation theory is very well understood by Lie-theory-like methods). An extended role is hence proposed for Soergel's tilting algorithm, away from its algebraic Lie theory underpinning, in determining the simple content of standard modules for these algebras. This role is explicitly verified in the blob algebra case. A tensor space representation of the blob algebra is constructed, as a candidate for a full tilting module (subsequently proven to be so in a paper by Martin and Ryom-Hansen), further evidencing the extended utility of Lie-theoretic methods. Possible generalisations of this representation to other elements of the sequence are discussed.

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19

Lemay, Joel. "Valued Graphs and the Representation Theory of Lie Algebras." Axioms 1, no. 2 (July 4, 2012): 111–48. http://dx.doi.org/10.3390/axioms1020111.

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20

TJIN, T. "INTRODUCTION TO QUANTIZED LIE GROUPS AND ALGEBRAS." International Journal of Modern Physics A 07, no. 25 (October 10, 1992): 6175–213. http://dx.doi.org/10.1142/s0217751x92002805.

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We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

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21

Aniello, P., C. Lupo, and M. Napolitano. "Exploring Representation Theory of Unitary Groups via Linear Optical Passive Devices." Open Systems & Information Dynamics 13, no. 04 (December 2006): 415–26. http://dx.doi.org/10.1007/s11080-006-9023-1.

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In this paper, we investigate some mathematical structures underlying the physics of linear optical passive (LOP) devices. We show, in particular, that with the class of LOP transformations on N optical modes one can associate a unitary representation of U (N) in the N-mode Fock space, representation which can be decomposed into irreducible sub-representations living in the subspaces characterized by a fixed number of photons. These (sub-)representations can be classified using the theory of representations of semi-simple Lie algebras. The remarkable case where N = 3 is studied in detail.

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22

FARNSTEINER, ROLF, and DETLEF VOIGT. "SCHEMES OF TORI AND THE STRUCTURE OF TAME RESTRICTED LIE ALGEBRAS." Journal of the London Mathematical Society 63, no. 3 (June 2001): 553–70. http://dx.doi.org/10.1017/s0024610701002010.

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Much of the recent progress in the representation theory of infinitesimal group schemes rests on the application of algebro-geometric techniques related to the notion of cohomological support varieties (cf. [6, 8–10]). The noncohomological characterization of these varieties via the so-called rank varieties (see [21, 22]) involves schemes of additive subgroups that are the infinitesimal counterparts of the elementary abelian groups. In this note we introduce another geometric tool by considering schemes of tori of restricted Lie algebras. Our interest in these derives from the study of infinitesimal groups of tame representation type, whose determination [12] necessitates the results to be presented in §4 and §5 as well as techniques from abstract representation theory.In contrast to the classical case of complex Lie algebras, the information on the structure of a restricted Lie algebra that can be extracted from its root systems is highly sensitive to the choice of the underlying maximal torus. Schemes of tori obviate this defect by allowing us to study algebraic families of root spaces. Accordingly, these schemes may also shed new light on various aspects of the structure theory of restricted Lie algebras. We intend to pursue these questions in a forthcoming paper [13], and focus here on first applications within representation theory.

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23

Campoamor-Stursberg, Rutwig. "Some Remarks Concerning the Invariants of Rank One Solvable Real Lie Algebras." Algebra Colloquium 12, no. 03 (September 2005): 497–518. http://dx.doi.org/10.1142/s1005386705000465.

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A corrected and completed list of six dimensional real Lie algebras with five dimensional nilradical is presented. Their invariants for the coadjoint representation are computed and some results on the invariants of solvable Lie algebras in arbitrary dimension whose nilradical has codimension one are also given. Specifically, it is shown that any rank one solvable Lie algebra of dimension n without invariants determines a family of (n+2k)-dimensional algebras with the same property.

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24

Losev, Ivan, and Victor Ostrik. "Classification of finite-dimensional irreducible modules over -algebras." Compositio Mathematica 150, no. 6 (April 7, 2014): 1024–76. http://dx.doi.org/10.1112/s0010437x13007604.

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AbstractFinite $W$-algebras are certain associative algebras arising in Lie theory. Each $W$-algebra is constructed from a pair of a semisimple Lie algebra ${\mathfrak{g}}$ (our base field is algebraically closed and of characteristic 0) and its nilpotent element $e$. In this paper we classify finite-dimensional irreducible modules with integral central character over $W$-algebras. In more detail, in a previous paper the first author proved that the component group $A(e)$ of the centralizer of the nilpotent element under consideration acts on the set of finite-dimensional irreducible modules over the $W$-algebra and the quotient set is naturally identified with the set of primitive ideals in $U({\mathfrak{g}})$ whose associated variety is the closure of the adjoint orbit of $e$. In this paper, for a given primitive ideal with integral central character, we compute the corresponding $A(e)$-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of $A(e)$ introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules for semisimple Lie algebras, the representation theory of $W$-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.

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25

Adamović, Dražen, Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi, and Ozren Perše. "An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras." International Mathematics Research Notices 2020, no. 13 (October 22, 2018): 4103–43. http://dx.doi.org/10.1093/imrn/rny237.

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Abstract We discover a large class of simple affine vertex algebras $V_{k} ({\mathfrak{g}})$, associated to basic Lie superalgebras ${\mathfrak{g}}$ at non-admissible collapsing levels $k$, having exactly one irreducible ${\mathfrak{g}}$-locally finite module in the category ${\mathcal O}$. In the case when ${\mathfrak{g}}$ is a Lie algebra, we prove a complete reducibility result for $V_k({\mathfrak{g}})$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k ({\mathfrak{g}})$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one nontrivial ideal.

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26

Sun, Qinxiu, and Zhixiang Wu. "Cohomologies of n-Lie Algebras with Derivations." Mathematics 9, no. 19 (October 2, 2021): 2452. http://dx.doi.org/10.3390/math9192452.

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The goal of this paper is to study cohomological theory of n-Lie algebras with derivations. We define the representation of an n-LieDer pair and consider its cohomology. Likewise, we verify that a cohomology of an n-LieDer pair could be derived from the cohomology of associated LeibDer pair. Furthermore, we discuss the (n−1)-order deformations and the Nijenhuis operator of n-LieDer pairs. The central extensions of n-LieDer pairs are also investigated in terms of the first cohomology groups with coefficients in the trivial representation.

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27

CATTANEO, U., and W. F. WRESZINSKI. "CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS." Reviews in Mathematical Physics 11, no. 10 (November 1999): 1179–207. http://dx.doi.org/10.1142/s0129055x99000374.

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A theory of contractions of Lie algebra representations on complex Hilbert spaces is proposed, based on Trotter's theory of approximating sequences of Banach spaces. Its main distinguishing feature is a careful definition of the carrier space of the limit Lie algebra representation. A set of quite general conditions on the contracting representations, satisfied in all known examples, is proven to be sufficient for the existence of such a representation. In order to show how natural the suggested framework is, the general theory is applied to the contraction of [Formula: see text] into the Lie algebra [Formula: see text] of the 3-dimensional Heisenberg group and to the related study of the limit N→∞ of a quantum system of N identical two-level particles.

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28

Boutonnet, Rémi, and Cyril Houdayer. "Stationary characters on lattices of semisimple Lie groups." Publications mathématiques de l'IHÉS 133, no. 1 (March 2, 2021): 1–46. http://dx.doi.org/10.1007/s10240-021-00122-8.

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AbstractWe show that stationary characters on irreducible lattices $\Gamma < G$ Γ < G of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $\Gamma < G$ Γ < G , the left regular representation $\lambda _{\Gamma }$ λ Γ is weakly contained in any weakly mixing representation $\pi $ π . We prove that for any such irreducible lattice $\Gamma < G$ Γ < G , any Uniformly Recurrent Subgroup (URS) of $\Gamma $ Γ is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices $\Gamma < G$ Γ < G . The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

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29

Cox, Ben. "-Categories and -Functors in the Representation Theory of Lie Algebras." Transactions of the American Mathematical Society 343, no. 1 (May 1994): 433. http://dx.doi.org/10.2307/2154540.

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30

Leznov, A. N. "Graded Lie algebras, representation theory, integrable mappings, and integrable systems." Theoretical and Mathematical Physics 122, no. 2 (February 2000): 211–28. http://dx.doi.org/10.1007/bf02551198.

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31

Feger, Robert, and Thomas W. Kephart. "LieART—A Mathematica application for Lie algebras and representation theory." Computer Physics Communications 192 (July 2015): 166–95. http://dx.doi.org/10.1016/j.cpc.2014.12.023.

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32

MAJID, SHAHN. "SOLUTIONS OF THE YANG-BAXTER EQUATIONS FROM BRAIDED-LIE ALGEBRAS AND BRAIDED GROUPS." Journal of Knot Theory and Its Ramifications 04, no. 04 (December 1995): 673–97. http://dx.doi.org/10.1142/s0218216595000284.

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We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring k[x] of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of S1 are also obtained by the same method.

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33

Wu, Henan. "Finite irreducible representations of map Lie conformal algebras." International Journal of Mathematics 28, no. 01 (January 2017): 1750002. http://dx.doi.org/10.1142/s0129167x17500021.

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In this paper, we study the finite representation theory of the map Lie conformal algebra [Formula: see text], where G is a finite simple Lie conformal algebra and A is a commutative associative algebra with unity over [Formula: see text]. In particular, we give a complete classification of nontrivial finite irreducible conformal modules of [Formula: see text] provided A is finite-dimensional.

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34

Campoamor-Stursberg, Rutwig. "On some algebraic formulations within universal enveloping algebras related to superintegrability." Acta Polytechnica 62, no. 1 (February 28, 2022): 16–22. http://dx.doi.org/10.14311/ap.2022.62.0016.

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We report on some recent purely algebraic approaches to superintegrable systems from the perspective of subspaces of commuting polynomials in the enveloping algebras of Lie algebras that generate quadratic (and eventually higher-order) algebras. In this context, two algebraic formulations are possible; a first one strongly dependent on representation theory, as well as a second formal approach that focuses on the explicit construction within commutants of algebraic integrals for appropriate algebraic Hamiltonians defined in terms of suitable subalgebras. The potential use in this context of the notion of virtual copies of Lie algebras is briefly commented.

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Falcón, Óscar J., Raúl M. Falcón, Juan Núñez, Ana M. Pacheco, and M. Trinidad Villar. "Classification of Filiform Lie Algebras up to dimension 7 Over Finite Fields." Analele Universitatii "Ovidius" Constanta - Seria Matematica 24, no. 2 (June 1, 2016): 185–204. http://dx.doi.org/10.1515/auom-2016-0036.

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Abstract This paper tries to develop a recent research which consists in using Discrete Mathematics as a tool in the study of the problem of the classification of Lie algebras in general, dealing in this case with filiform Lie algebras up to dimension 7 over finite fields. The idea lies in the representation of each Lie algebra by a certain type of graphs. Then, some properties on Graph Theory make easier to classify the algebras. As main results, we find out that there exist, up to isomorphism, six, five and five 6-dimensional filiform Lie algebras and fifteen, eleven and fifteen 7-dimensional ones, respectively, over ℤ/pℤ, for p = 2, 3, 5. In any case, the main interest of the paper is not the computations itself but both to provide new strategies to find out properties of Lie algebras and to exemplify a suitable technique to be used in classifications for larger dimensions.

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36

Hadasz, Leszek, and Błażej Ruba. "Airy Structures for Semisimple Lie Algebras." Communications in Mathematical Physics 385, no. 3 (June 24, 2021): 1535–69. http://dx.doi.org/10.1007/s00220-021-04142-7.

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AbstractWe give a complete classification of Airy structures for finite-dimensional simple Lie algebras over $${\mathbb {C}}$$ C , and to some extent also over $${\mathbb {R}}$$ R , up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are $$\mathfrak {sl}_2$$ sl 2 , $$\mathfrak {sp}_4$$ sp 4 and $$\mathfrak {sp}_{10}$$ sp 10 . Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a number of interesting properties of these Airy structures and constructed many examples. Techniques used to derive our results may be described, broadly speaking, as an application of representation theory in semiclassical analysis.

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37

da Silva Macedo, David Levi, and Plamen Koshlukov. "Codimension growth for weak polynomial identities, and non-integrality of the PI exponent." Proceedings of the Edinburgh Mathematical Society 63, no. 4 (July 20, 2020): 929–49. http://dx.doi.org/10.1017/s0013091520000243.

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Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.

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38

Strade, H. "Representations of finitary Lie algebras." Journal of Algebra 257, no. 1 (November 2002): 13–36. http://dx.doi.org/10.1016/s0021-8693(02)00129-1.

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Loday, Jean-Louis, and Teimuraz Pirashvili. "Leibniz Representations of Lie Algebras." Journal of Algebra 181, no. 2 (April 1996): 414–25. http://dx.doi.org/10.1006/jabr.1996.0127.

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Du, Jie, and Bin Shu. "Representations of finite Lie algebras." Journal of Algebra 321, no. 11 (June 2009): 3197–225. http://dx.doi.org/10.1016/j.jalgebra.2008.06.016.

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JING, NAIHUAN, KAILASH C. MISRA, and CARLA D. SAVAGE. "ON MULTI-COLOR PARTITIONS AND THE GENERALIZED ROGERS–RAMANUJAN IDENTITIES." Communications in Contemporary Mathematics 03, no. 04 (November 2001): 533–48. http://dx.doi.org/10.1142/s0219199701000482.

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Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.

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42

Ibraev, Sh Sh, A. Zh Seitmuratov, and L. S. Kainbayeva. "On simple modules with singular highest weights for so2l+1(K)." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 105, no. 1 (March 30, 2022): 52–65. http://dx.doi.org/10.31489/2022m1/52-65.

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In this paper, we study formal characters of simple modules with singular highest weights over classical Lie algebras of type B over an algebraically closed field of characteristic p ≥ h, where h is the Coxeter number. Assume that the highest weights of these simple modules are restricted. We have given a description of their formal characters. In particular, we have obtained some new examples of simple Weyl modules. In the restricted region, the representation theory of algebraic groups and its Lie algebras are equivalent. Therefore, we can use the tools of the representation theory of semisimple and simply-connected algebraic groups in positive characteristic. To describe the formal characters of simple modules, we construct Jantzen filtrations of Weyl modules of the corresponding highest weights.

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Leznov, A. N. "A new approach to the representation theory of semisimple Lie algebras and quantum algebras." Theoretical and Mathematical Physics 123, no. 2 (May 2000): 633–50. http://dx.doi.org/10.1007/bf02551396.

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Feger, Robert, Thomas W. Kephart, and Robert J. Saskowski. "LieART 2.0 – A Mathematica application for Lie Algebras and Representation Theory." Computer Physics Communications 257 (December 2020): 107490. http://dx.doi.org/10.1016/j.cpc.2020.107490.

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Billig, Yuly. "Representation theory of $$\mathbb {Z}^n$$ Z n -graded Lie algebras." São Paulo Journal of Mathematical Sciences 11, no. 1 (July 11, 2016): 53–58. http://dx.doi.org/10.1007/s40863-016-0044-6.

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Zhelobenko, D. P. "On quantum methods in the representation theory of reductive Lie algebras." Functional Analysis and Its Applications 28, no. 2 (1994): 114–16. http://dx.doi.org/10.1007/bf01076498.

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47

Jiang, Cuipo, and Song Wang. "Extension of Vertex Operator Algebra $V_{\widehat{H}_{4}}(\ell,0)$." Algebra Colloquium 21, no. 03 (June 24, 2014): 361–80. http://dx.doi.org/10.1142/s1005386714000327.

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We classify the irreducible restricted modules for the affine Nappi-Witten Lie algebra [Formula: see text] with some natural conditions. It turns out that the representation theory of [Formula: see text] is quite different from the theory of representations of Heisenberg algebras. We also study the extension of the vertex operator algebra [Formula: see text] by the even lattice L. We give the structure of the extension [Formula: see text] and its irreducible modules via irreducible representations of [Formula: see text] viewed as a vertex algebra.

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48

Dipper, Richard, and Jochen Gruber. "Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 511 (June 25, 1999): 145–91. http://dx.doi.org/10.1515/crll.1999.511.145.

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Abstract We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.

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WU, YONG-SHI, and KENGO YAMAGISHI. "CHERN-SIMONS THEORY AND KAUFFMAN POLYNOMIALS." International Journal of Modern Physics A 05, no. 06 (March 20, 1990): 1165–95. http://dx.doi.org/10.1142/s0217751x90000556.

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We report on a study of the expectation values of Wilson loops in D=3 Chern-Simons theory. The general skein relations (of higher orders) are derived for these expectation values. We show that the skein relations for the Wilson loops carrying the fundamental representations of the simple Lie algebras SO(n) and Sp(n) are sufficient to determine invariants for all knots and links and that the resulting link invariants agree with Kauffman polynomials. The polynomial for an unknotted circle is identified to the formal characters of the fundamental representations of these Lie algebras.

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GĂRĂJEU, DANIELA, and MIHAIL GĂRĂJEU. "MATHEMATICA™ PACKAGES FOR COMPUTING PRINCIPAL DECOMPOSITIONS OF SIMPLE LIE ALGEBRAS AND APPLICATIONS IN EXTENDED CONFORMAL FIELD THEORIES." International Journal of Modern Physics C 14, no. 01 (January 2003): 1–27. http://dx.doi.org/10.1142/s012918310300419x.

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In this article, we propose two Mathematica™ packages for doing calculations in the domain of classical simple Lie algebras. The main goal of the first package, [Formula: see text], is to determine the principal three-dimensional subalgebra of a simple Lie algebra. The package provides several functions which give some elements related to simple Lie algebras (generators in fundamental and adjoint representation, roots, Killing form, Cartan matrix, etc.). The second package, [Formula: see text], concerns the principal decomposition of a Lie algebra with respect to the principal three-dimensional embedding. These packages have important applications in extended two-dimensional conformal field theories. As an example, we present an application in the context of the theory of W-gravity.

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