Ian M. Musson's Papers/Preprints
Complete
Sets of Representations of Classical Simple Lie Superalgebras,
with E.S. Letzter, Letters in Math Physics 31 (1994) 247-253.
Abstract: Descriptions of the complete sets of irreducible highest
weight modules over complex classical Lie superalgebras are recorded. It
is further shown that the finite dimensional irreducible modules over a (not
necessarily classical) finite dimensional complex Lie superalgebra form
a complete set if and only if the even part of the Lie superalgebra is reductive
and the universal enveloping superalgebra is semiprime.
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On
the Center of the Enveloping Algebra of a Classical Simple Lie Superalgebra,
J. of Algebra 193 (1997) 285-308.
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CRYSTAL
BASES FOR U_{q}(osp(1,2r)),
with Yi Ming Zou, Journal of Algebra 210 (1998), 514-534.
Abstract: We construct $Z_{2}$-graded crystal bases for thequantized
universal enveloping algebra of the Lie superalgebra osp(1,2r). We show
that, like the crystal bases in the Liealgebra case, these crystal bases
carry a remarkablecombinatorial structure.
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The
Enveloping Algebra of the Lie Superalgebra osp(1,2r),
Representation Theory 1 (1997), 405-423.
Abstract: In this paper we study the case where g = osp(1,2r) and
obtain a description of Prim U(g) as an ordered set. We also obtain the
multiplicities of composition factor of Verma modules over U(g),and of \widetilde{L}(\lambda)
when regarded as a U(g_{0})-module by restriction. pdf-file dvi-file
Some Lie Superalgebras Associated to the Weyl algebras,
Proc AMS 127 (1999), 2821-2827.
Link
to journal
I. M. Musson and M. Van den Bergh,
Invariants
under tori of rings of differential operators and related topics,
Mem. Amer. Math. Soc. 136 (1998), viii+85.
Abstract: If $G$ is a reductive algebraic group acting rationally
on a smooth affine variety $X$
then it is generally believed that $D(X)^G$ has properties very similar
to those of enveloping
algebras of semisimple Lie algebras. In this paper we show that this is
indeed the case when $G$ is a torus and $X=k^r\times (k^*)^s$. We give a
precise description of the primitive ideals in
$D(X)^G$ and we study in detail the ring theoretical and homological properties
of the minimal primitive quotients of $D(X)^G$. The latter are of the form
$D(X)^G/(\fg-\chi(\fg))$ where ${\mathfrak{g}} = Lie(G)$, $\chi\in {\mathfrak{g}}
^\ast$ and
${\mathfrak{g}} -\chi({\mathfrak{g}} )$ is the set of all $v-\chi(v)$ with
$v\in {\mathfrak{g}} $. They occur as rings of twisted differential operators
on toric varieties. As a side result we prove that if $G$ is a torus acting
rationally on a smooth affine variety then $D(X//G)$ is a simple ring.
Some Lie Superalgebras
Associated to the Weyl algebras,
Proc AMS 127 (1999), 2821-2827
Abstract: Let $\FRAK{g}$ be the Lie superalgebra $osp(1,2r)$.
We show there is a surjective homomorphism from $U(\FRAK{g})$ to the $r^{th}$
Weyl algebra $A_{r}$, and use this to construct an analog of the Joseph
ideal. We also obtain a decomposition of the adjoint representation of $\FRAK{g}$
on $A_r$ and use this to show that if $A_{r}$ is made into a Lie superalgebra
using its natural $\openZ_{2}$-grading, then $A_{r} = k \oplus [A_{r}, A_{r}]$.
In addition we show that if $[A_r, A_r]$ and $[A_s, A_s]$ are isomorphic
as Lie superalgebras then $r=s$. This answers a question of S. Montgomery.
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Down-up
Algebras and their Representation Theory,
with P.A.A.B.Carvalho.
Abstract: A class of algebras called down-up algebras was
introduced by G. Benkart and T. Roby \cite{BenkartRoby}. 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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On
the Goldie Quotient Ring of the Enveloping Algebra of a Classical Simple
Lie Superalgebra.
Abstract: If $\FRAK{g}$ is a classical simple Lie superalgebra
$(\FRAK{g} \neq P(n))$, the enveloping algebra $U(\FRAK{g})$ is a prime
ring and hence has a simple artinian ring of quotients $Q(U(\FRAK{g}))$
by Goldie's Theorem. We show that if $\FRAK{g}$ has Type I then $Q(U(\FRAK{g}))$
is a matrix ring over $Q(U(\FRAK{g}_0))$. On the other hand, if $\FRAK{g}
= osp(1,2r)$ then by extending the center of $U(\FRAK{g})$ we obtain a prime
ring whose Goldie quotient ring is a matrix ring over the quotient division
ring of a Weyl algebra. This is the analog of a result of Gel'fand and Kirillov.
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Associated
Varieties for Classical Simple Lie Superalgebras.
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Finite
Quantum Groups and Pointed Hopf Algebras,
Abstract: We show that under certain conditions a finite dimensional
graded pointed Hopf algebra is an image of an algebra twist of a quantized
enveloping algebra $U_q(\FRAK{b})$ when $q$ is a root of unity. In addition
we obtain a classification of Hopf algebras $H$ such that $G(H)$ has odd
prime order $p$ and $grH$ is of Cartan type.
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P.I.
Envelopes of Classical Simple Lie Superalgebras.
Abstract: Let $\FRAK{g}$ be a classical simple Lie superalgebra.
We describe the prime ideals $P$ in the enveloping algebra $U(\FRAK{g})$
such that $U(\FRAK{g})/P$ satisfies a polynomial identity. If the factor
algebra $U(\FRAK{g})/P$ is not artinian, then it is an order in a matrix
algebra over $K(z)$.
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Hopf
Down-up Algebras,
with Ellen Kirkman.
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Finite
dimensional representations of invariant differential operators,
with Sonia L. Rueda
Abstract: 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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Lie
Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits,
Abstract: 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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Invariant
Differential Operators and FCR factors of Enveloping algebras
Abstract: 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. and 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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Noncommutative
Deformations of Type A Kleinian Singularities and Hilbert Schemes.
Abstract: Let $H_{\mathbf{k}}$ be a symplectic reflection
algebra corresponding to a cyclic subgroup $\Gamma \subseteq SL_2 \C$ oforder
$n$ 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 \begin{itemize} \item[{(1)}] there is an
equivalence of categories $R-\mathrm{qgr} \simeq U_{\bf k}$-mod , \item[{(2)}]
there is an equivalence of categories $gr R-\mathrm{qgr} \simeq \ttt{Coh}(Hilb_\Gamma\mathbb{C}^2)$.
\end{itemize} Here $ \ttt{Coh}(Hilb_\Gamma \mathbb{C}^2)$ is the category
of coherent sheaves on the $\Gamma$-Hilbert scheme. and for a gradedalgebra
$\mathcal{R},$ we write $ \mathcal{R}-\mathrm{qgr}$ forthe quotient category
of finitely generated graded $\mathcal{R}$-modules modulo torsion. \\
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