arXiv subjects

Ian M. Musson

Publications and source records attributed to Ian M. Musson.

29 records · Page 2Linked to original sources

arXiv2013

The lattice of submodules of a multiplicity free module

In this paper we determine, under some mild restrictions, the lattice of submodules $\gL$ of a module $M$ all of whose composition factors have multiplicity one. Such a lattice is distributive, and hence determined by its poset of down-sets $P$. We define a directed Ext graph $\Ext_\gL$ of $\gL$ and show that if $\Ext_\gL$ is acyclic, then $\Ext_\gL$ determines $P$. The result applies to multiplicity free indecomposable modules for finite dimensional algebras with acyclic Ext graph. It also applies to some deformed Verma modules which arise in the Jantzen sum formula basic classical simple Lie superalgebras in the deformed case.

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arXiv2011

Combinatorics of Character Formulas for the Lie Superalgebra $\fgl(m,n).$

Let $\fg$ be the Lie superalgebra $\fgl(m,n).$ Algorithms for computing the composition factors and multiplicities of Kac modules for $\fg$ were given by the second author in 1996, and by J. Brundan in 2003. We give a combinatorial proof of the equivalence between the two algorithms. The proof uses weight and cap diagrams introduced by Brundan and C. Stroppel, and cancelations between paths in a graph $\mathcal{G}$ defined using these diagrams. Each vertex of $\mathcal{G}$ corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If $\mathcal{E}$ is the subgraph of $\mathcal{G}$ obtained by deleting all edges of positive weight, then $\mathcal{E}$ is the graph that describes non-split extensions between simple highest weight modules. We also give a procedure for finding the composition factors of any Kac module, without cancelation. This procedure leads to a second proof of the main result.

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arXiv2011

Finitely generated, non-artinian monolithic modules

We survey noetherian rings $A$ over which the injective hull of every simple module is locally artinian. Then we give a general construction for algebras $A$ that do not have this property. In characteristic 0, we also complete the classification of down-up algebras with this property.

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arXiv2010

Monolithic modules over Noetherian Rings

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained in [arXiv:0906.2930] for down-up algebras.

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arXiv2009

Hochschild Cohomology and Deformations of Clifford-Weyl Algebras

We give a complete study of the Clifford-Weyl algebra ${\mathcal C}(n,2k)$ from Bose-Fermi statistics, including Hochschild cohomology (with coefficients in itself). We show that ${\mathcal C}(n,2k)$ is rigid when $n$ is even or when $k \neq 1$. We find all non-trivial deformations of ${\mathcal C}(2n+1,2)$ and study their representations.

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arXiv2007

Invariant Differential Operators and FCR factors of Enveloping algebras

If $\fg$ is a semisimple Lie algebra, we describe the prime factors of $\mcU(\fg)$ that have enough finite dimensional modules. The proof depends on some combinatorial facts about the Weyl group which may be of independent interest. We also determine, which finite dimensional $\mcU(\fg)$-modules are modules over a given prime factor. As an application we study finite dimensional modules over some rings of invariant differential operators arising from Howe duality.

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arXiv2005

Faithful Cyclic Modules for Enveloping Algebras and Sklyanin Algebras

Let $U$ be the enveloping algebra of a finite dimensional nonabelian Lie algebra $\mathfrak{g}$ over a field of characteristic zero. We show that there is an open nonempty open subset $X$ of $U_1 = \mathfrak{g}\oplus K$ such that $U/Ux$ is faithful for all $x \in X$. We prove similar results for homogenized enveloping algebras and for the three dimensional Sklyanin algebras at points of infinite order. It would be interesting to know if there is a common generalization of these results.

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arXiv2005

Noncommutative Deformations of Type A Kleinian Singularities and Hilbert Schemes

Let $H_{\mathbf{k}}$ be a symplectic reflection algebra corresponding to a cyclic subgroup $Γ\subseteq SL_2 \C$ of order $n$ and $U_{\mathbf{k}} = eH_{\mathbf{k}} e$ the spherical subalgebra of $H_{\mathbf{k}}$. We show that for suitable ${\mathbf{k}}$ there is a filtered $\Z^{n-1}$-algebra $R$ such that (1) there is an equivalence of categories $R-\mathrm{qgr} \simeq U_{\bf k}$-mod, (2) there is an equivalence of categories $gr R-\mathrm{qgr} \simeq \ttt{Coh}(Hilb_Γ\mathbb{C}^2)$. Here $ \ttt{Coh}(Hilb_Γ\mathbb{C}^2)$ is the category of coherent sheaves on the $Γ$-Hilbert scheme. and for a graded algebra $\mathcal{R},$ we write $ \mathcal{R}-\mathrm{qgr}$ for the quotient category of finitely generated graded $\mathcal{R}$-modules modulo torsion.

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arXiv2004

Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits

Let $\FRAK{g}$ be a classical simple Lie superalgebra. To every nilpotent orbit $\cal O$ in $\FRAK{g}_0$ we associate a Clifford algebra over the field of rational functions on $\cal O$. We find the rank, $k(\cal O)$ of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a $U(\FRAK{g})$-module with $\cal O$ or an orbital subvariety of $\cal O$ as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant $k(\cal O)$ is in many cases, equal to the odd dimension of the orbit $G\cdot\cal O$ where $G$ is a Lie supergroup with Lie superalgebra ${\mathfrak g.}$

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arXiv2003

Finite dimensional representations of invariant differential operators

Let $k$ be an algebraically closed field of characteristic 0, $Y=k^{r}\times {(k^{\times})}^{s}$ and let $G$ be an algebraic torus acting diagonally on the ring of differential operators $\cD (Y)^G$. We give necessary and sufficient conditions for $\cD (Y)^G$ to have enough simple finite dimensional representations, in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero. As an application we show that if $K\longrightarrow GL(V)$ is a representation of a reductive group $K$ and if zero is not a weight of a maximal torus of $K$ on $V$, then $\cD (V)^K$ has enough finite dimensional representations. We also construct examples of FCR- algebras with any GK dimension $\geq 3$.

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arXiv1999

Down-Up Algebras and Representation Theory

A class of algebras called down-up algebras was introduced by G. Benkart and T. Roby. We classify the finite dimensional simple modules over Noetherian down-up algebras and show that in some cases every finite dimensional module is semisimple. We also study the question of when two down-up algebras are isomorphic.

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