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Semiholonomic jets and induced modules in Cartan geometry calculus

Jan Slovák, Vladimír Souček (2024)

Archivum Mathematicum

The famous Erlangen Programme was coined by Felix Klein in 1872 as an algebraic approach allowing to incorporate fixed symmetry groups as the core ingredient for geometric analysis, seeing the chosen symmetries as intrinsic invariance of all objects and tools. This idea was broadened essentially by Elie Cartan in the beginning of the last century, and we may consider (curved) geometries as modelled over certain (flat) Klein’s models. The aim of this short survey is to explain carefully the basic...

Serre functors for Lie algebras and superalgebras

Volodymyr Mazorchuk, Vanessa Miemietz (2012)

Annales de l’institut Fourier

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $𝒪$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $𝒪$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic...

S-functions and characters of Lie algebras and Lie groups

Ronald C. King (1990)

Banach Center Publications

Singular Blocks of the Category O.

Ronald S. Irving (1990)

Mathematische Zeitschrift

Singular localization of $𝔤$ -modules and applications to representation theory

Erik Backelin, Kobi Kremnitzer (2015)

Journal of the European Mathematical Society

We prove a singular version of Beilinson–Bernstein localization for a complex semi-simple Lie algebra following ideas from the positive characteristic case settled by [BMR06]. We apply this theory to translation functors, singular blocks in the Bernstein–Gelfand–Gelfand category O and Whittaker modules.

Singular vectors corresponding to imaginary roots in verma modules over affine Lie algebras.

F. Malikow (1990)

Mathematica Scandinavica

Some properties of generalized reduced Verma modules over $ℤ$ -graded modular Lie superalgebras

Keli Zheng, Yongzheng Zhang (2017)

Czechoslovak Mathematical Journal

We study some properties of generalized reduced Verma modules over $ℤ$ -graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for $ℤ$ -graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules.

Spherical Functions and a q-Analogue of Kostant's Weight Multiplicity Formula.

Shin-ichi Kato (1982)

Inventiones mathematicae

String functions for affine Lie algebras integrable modules.

Kulish, Petr, Lyakhovsky, Vladimir (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Sugawara operators and Kac-Kazhdan conjecture.

Takahiro Hayashi (1988)

Inventiones mathematicae

Super boson-fermion correspondence

Victor G. Kac, J. W. Van de Leur (1987)

Annales de l'institut Fourier

We establish a super boson-fermion correspondence, generalizing the classical boson-fermion correspondence in 2-dimensional quantum field theory. A new feature of the theory is the essential non-commutativity of bosonic fields. The superbosonic fields obtained by the super bosonization procedure from super fermionic fields form the affine superalgebra $\tilde{g} l_{1 | 1}$ . The converse, super fermionization procedure, requires introduction of the super vertex operators. As applications, we give vertex operator constructions...

Sur certaines représentations locales de l’algèbre de Lie $s o (4, 1)$ et de l’algèbre de Lie du groupe de Poincaré

Christiane Martin (1974)

Annales de l'I.H.P. Physique théorique

Sur la représentation adjointe d'une algèbre de Lie libre. II

Alexandros Patsourakos (1994)

Annales de l'institut Fourier

Soit $N$ le noyau de l’application $Γ$ de l’idéal d’augmentation de l’algèbre enveloppante de $L (X)$ sur $L (X)$ , l’algèbre de Lie libre sur $X$ , définie par $Γ (x_{1} ... x_{n}) = [... [x_{n - 1}, x_{n}] ...]]$ pour $x_{1}, ..., x_{n} \in X$ . Si $L (X)$ est munie de la représentation adjointe, alors un ensemble de générateurs de $N$ comme module sur l’algèbre enveloppante est déterminé en termes des ensembles de Hall relatifs à $X$ .

Sur la représentation coadjointe des algèbres de Lie résolubles.

J. Dixmier, J.-Y. Charbonnel (1981)

Journal für die reine und angewandte Mathematik

Sur la représentation coadjointe d'une algèbre de Lie

J. Dixmier, M. Duflo, M. Vergne (1974)

Compositio Mathematica

Sur les représentations de dimension finie de la super algèbre de Lie $𝔤𝔩 (m, n)$

Caroline Gruson (2005/2006)

Séminaire Bourbaki

La catégorie des modules de dimension finie sur la super algèbre de Lie $𝔤𝔩 (m, n)$ n’est pas semi-simple. Elle se décompose en une infinité de blocs, dont on cherche depuis les travaux de Kac en 1977 à comprendre la structure. Vera Serganova apporte une réponse presque complète à ce problème, formulée selon le cercle d’idées introduites par Bernstein, Gelfand et Gelfand pour étudier la catégorie $𝒪$ dans le cas classique ; ne disposant pas pour $𝔤𝔩 (m, n)$ d’analogues des théorèmes de Kostant et de Borel-Weil-Bott,...

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