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1. CJM Online first

Mackaaij, Marco; Tubbenhauer, Daniel
Two-color Soergel calculus and simple transitive 2-representations
In this paper we complete the ADE-like classification of simple transitive $2$-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to give an explicit construction of a graded (non-strict) version of all these $2$-representations. Moreover, we give simple combinatorial criteria for when two such $2$-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations. Finally, our construction also gives a large class of simple transitive $2$-representations in infinite dihedral type for general bipartite graphs.

Keywords:$2$-representation theory, categorification, Soergel bimodule, Kazhdan--Lusztig theory, Hecke algebras for dihedral groups, zigzag algebra
Categories:20C08, 17B10, 18D05, 18D10, 20F55

2. CJM 2016 (vol 69 pp. 453)

Marquis, Timothée; Neeb, Karl-Hermann
Isomorphisms of Twisted Hilbert Loop Algebras
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a ``minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.

Keywords:locally affine Lie algebra, Hilbert-Lie algebra, positive energy representation
Categories:17B65, 17B70, 17B22, 17B10

3. CJM 2015 (vol 68 pp. 258)

Calixto, Lucas; Moura, Adriano; Savage, Alistair
Equivariant Map Queer Lie Superalgebras
An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) $X$ to a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that $\Gamma$ is abelian and acts freely on $X$. We show that such representations are parameterized by a certain set of $\Gamma$-equivariant finitely supported maps from $X$ to the set of isomorphism classes of irreducible finite-dimensional representations of $\mathfrak{q}$. In the special case where $X$ is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

Keywords:Lie superalgebra, queer Lie superalgebra, loop superalgebra, equivariant map superalgebra, finite-dimensional representation, finite-dimensional module
Categories:17B65, 17B10

4. CJM 2013 (vol 65 pp. 1287)

Reihani, Kamran
$K$-theory of Furstenberg Transformation Group $C^*$-algebras
The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism
Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20

5. CJM 2009 (vol 62 pp. 382)

Lü, Rencai; Zhao, Kaiming
Verma Modules over Quantum Torus Lie Algebras
Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras $\mathfrak{L}_q$. The center of $\mathfrak{L}_q$ now is generally infinite dimensional. In this paper, $\mathbb{Z}$-graded Verma modules $\widetilde{V}(\varphi)$ over $\mathfrak{L}_q$ and their corresponding irreducible highest weight modules $V(\varphi)$ are defined for some linear functions $\varphi$. Necessary and sufficient conditions for $V(\varphi)$ to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules $\widetilde{V}(\varphi)$ to be irreducible are obtained.

Categories:17B10, 17B65, 17B68

6. CJM 2008 (vol 60 pp. 88)

Diwadkar, Jyotsna Mainkar
Nilpotent Conjugacy Classes in $p$-adic Lie Algebras: The Odd Orthogonal Case
We will study the following question: Are nilpotent conjugacy classes of reductive Lie algebras over $p$-adic fields definable? By definable, we mean definable by a formula in Pas's language. In this language, there are no field extensions and no uniformisers. Using Waldspurger's parametrization, we answer in the affirmative in the case of special orthogonal Lie algebras $\mathfrak{so}(n)$ for $n$ odd, over $p$-adic fields.

Categories:17B10, 03C60

7. CJM 2004 (vol 56 pp. 293)

Khomenko, Oleksandr; Mazorchuk, Volodymyr
Structure of modules induced from simple modules with minimal annihilator
We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category $\Oo$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

Keywords:parabolic induction, generalized Verma module, simple module, Ha\-rish-\-Chand\-ra bimodule, equivalent categories
Categories:17B10, 22E47

8. CJM 2001 (vol 53 pp. 225)

Britten, D. J.; Lemire, F. W.
Tensor Product Realizations of Simple Torsion Free Modules
Let $\calG$ be a finite dimensional simple Lie algebra over the complex numbers $C$. Fernando reduced the classification of infinite dimensional simple $\calG$-modules with a finite dimensional weight space to determining the simple torsion free $\calG$-modules for $\calG$ of type $A$ or $C$. These modules were determined by Mathieu and using his work we provide a more elementary construction realizing each one as a submodule of an easily constructed tensor product module.


9. CJM 1998 (vol 50 pp. 1323)

Morales, Jorge
L'invariant de Hasse-Witt de la forme de Killing
Nous montrons que l'invariant de Hasse-Witt de la forme de Killing d'une alg{\`e}bre de Lie semi-simple $L$ s'exprime {\`a} l'aide de l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de $L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines positives), et des invariants associ{\'e}s au groupe des sym{\'e}tries du diagramme de Dynkin de $L$.

Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66

10. CJM 1998 (vol 50 pp. 816)

Mazorchuk, Volodymyr
Tableaux realization of generalized Verma modules
We construct the tableaux realization of generalized Verma modules over the Lie algebra $\sl(3,{\bbd C})$. By the same procedure we construct and investigate the structure of a new family of generalized Verma modules over $\sl(n,{\bbd C})$.

Categories:17B35, 17B10

11. CJM 1998 (vol 50 pp. 266)

Britten, D. J.; Lemire, F. W.
The torsion free Pieri formula
Central to the study of simple infinite dimensional $g\ell(n, \Bbb C)$-modules having finite dimensional weight spaces are the torsion free modules. All degree $1$ torsion free modules are known. Torsion free modules of arbitrary degree can be constructed by tensoring torsion free modules of degree $1$ with finite dimensional simple modules. In this paper, the central characters of such a tensor product module are shown to be given by a Pieri-like formula, complete reducibility is established when these central characters are distinct and an example is presented illustrating the existence of a nonsimple indecomposable submodule when these characters are not distinct.


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