Publications

Preprint Directory

Injective modules for group algebras of locally finite groups
Math. Proc, Camb. Phil. Soc. 84 (1978), no. 2, 247–262
pdf file

Injective modules for group rings of polycyclic groups. I, II
Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 124, 429–448, 449–466.

Irreducible modules for polycyclic group algebras
Canad. J. Math. 33 (1981), no. 4, 901–914.
pdf file
Link to Journal

Integral group rings with residually nilpotent unit groups (with Al Weiss)
Arch. Math. (Basel) 38 (1982), no. 6, 514–530.
Link to Journal

Some examples of modules over Noetherian rings
Glasgow Math. J. 23 (1982), no. 1, 9–13.
Link to Journal

Representations of infinite soluble groups
Glasgow Math. J. 24 (1983), no. 1, 43–52.
Link to Journal

On the structure of certain injective modules over group algebras of soluble groups of finite rank
J. Algebra 85 (1983), no. 1, 51–75.
Link to Journal via Researchgate

Noetherian subrings of quotient rings
Methods in ring theory (Antwerp, 1983), 379–389,
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 129, Reidel, Dordrecht, 1984.
Noetherian subrings of quotient rings

Some rings of differential operators which are Morita equivalent to the Weyl algebra A1
Proc. Amer. Math. Soc. 98 (1986), no. 1, 29–30.
Abstract: For a certain class of semigroup algebras kA, we show that the ring of all differential operators on kA is Morita equivalent to the first Weyl algebra A_1
Link to Journal

Rings of Differential Operators on Invariant Rings of Tori
Transactions of the American Mathematical Society, Vol. 303, No. 2 (Oct., 1987), 805-827
Abstract: Let k be an algebraically closed field of characteristic zero and G a torus
acting diagonally on k^x. For a subset ß of s = (1,2, …,í},set U_beta = {u \in k^s | u¡ \neq 0 if /’ e ß}. Then G acts on 0(U_ß), the ring of regular functions on U_ß, and we study the ring D(O(U_ß)^G) of all differential operators on the invariant-ring. More generally ….
Link to Journal

Injective modules, localization and completion in group algebras
J. Pure Appl. Algebra 57 (1989), no. 3, 265–279.
Abstract: Let k be an uncountable field of characteristic zero and G a polycyclic-by-finite group. If P is a prime ideal of kG and View the MathML source the largest Ore set of elements of kG which are regular mod P, it is shown that the injective hull of any simple kGView the MathML source-module is artinian. The technique used is to construct normal elements in completions of kG at the augmentation ideals of certain abelian normal subgroups. We also show that if N is a normal subgroup of G such that N is torsion free nilpotent of finite rank and G/N is polycyclic-by-finite, then the completion is Noetherian.
Link to Journal

Rings of Differential Operators and Zero Divisors
J. of Algebra, 125 (1989) 489-501.
pdf file

Primative Factors of Enveloping Algebras of Nilpotent Lie Superalberbras (with Allen Bell)
Journal of the London Mathematical Society :1990 vol:42 pg:401-408
Link to Journal

Rings of Differential Operators on One Dimensional Algebras
Pacific J. Math., 147 (1991) 269-290 (with Marc Chamarie).
pdf file

A Classification of Primitive Ideals in the Enveloping Algebras of a Classical Simple Lie Superalgebra
Advances in Mathematics, 91 (1992) 252-268.
pdf file

Enveloping Algebras of Lie Superalgebras – A Survey
Proceedings of a Conference in honor of Prof. G. Azumuya,
Contemporary Mathematics, 124 (1992) 141-149.
Abstract: We survey some recent results on prime and primitive ideals in enveloping algebras of Lie superalgebras.
xerox copy

Differential operators on toric varieties
Journal of Pure and Applied Algebra 95 (1994) 303-315.
Abstract: If X is a toric variety we show that X is isomorphic to a quotient Y//G where G is a torus acting on an affine space k” and Y is a G-invariant open subset of k”. We also show that any ring of differential operators on X twisted by an invertible sheaf is a factor ring of the fixed ring D(Y)’ by an ideal generated by central elements.
xerox copy

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.
pdf file

Conditions for a module to be injective and some applications to Hopf algebra duality
Proc. Amer. Math. Soc. 123 (1995), no. 3, 693–702.
Link to Journal

The Coradical Filtration for Quantized Enveloping Algebras (with William Chin)
J. London Math. Soc. (2) 53 (1996), no. 1, 50–62.
Link to Journal

Multiparameter quantum enveloping algebras (with William Chin)
Contact Franco-Belge en Algèbre (Diepenbeek, 1993).
J. Pure Appl. Algebra 107 (1996), no. 2-3, 171–191.
Abstract: Let A=A(p,lambda) be the multiparameter deformation of the coordinate algebra of nxn matrices as described by Artin et al. Let U be the quantum enveloing algebra which is associated to A, in the sense of FRT. We prove a PBW theorem for U and establish a presentation by generators and relations, when lambda is not a root of unity. Our approach depends on a cocycle twisting method which reduces many arguments to the standard one-parameter deformation.
Link on Researchgate

On the Center of the Enveloping Algebra of a Classical Simple Lie Superalgebra
J. of Algebra 193 (1997) 285-308.
pdf-file

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 Lie algebra case, these crystal bases carry a remarkable combinatorial structure.
pdf-file

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

Invariants under tori of rings of differential operators and related topics (with Michel Van den Bergh)
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.
pdf-file
Link to Journal

Down-up Algebras and their Representation Theory (with P.A.A.B. Carvalho)
J. Algebra 228 (2000), 286-310.
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.
pdf-file

On the Goldie Quotient Ring of the Enveloping Algebra of a Classical Simple Lie Superalgebra
J. Algebra 235 (2001), 203–213.
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.
pdf-file

Associated Varieties for Classical Simple Lie Superalgebras
Hopf algebras and quantum groups (Brussels, 1998), 177–188, Lecture Notes in Pure and Appl. Math., 209, Dekker, New York, 2000.
pdf-file

Finite Quantum Groups and Pointed Hopf Algebras
Quantum groups and Lie theory (Durham, 1999), 191–205, London Math. Soc. Lecture Note Ser., 290, Cambridge Univ. Press, Cambridge, 2001.
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.
pdf-file

Noetherian Down-up Algebras with Ellen Kirkman and Donald S. Passman
Proceedings of the American Mathematical Society Volume 127, Number 11, Pages 3161-3167.
Abstract: The down-up algebra A = A(alpha,beta,gamma) was introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that beta \neq 0 is equivalent to A being right (or left) Noetherian, and also to A being a domain. Furthermore, when this occurs, we show that A is Auslander-regular and has global dimension 3
Link to Journal

P.I. Envelopes of Classical Simple Lie Superalgebras
Proc. Amer. Math. Soc. 130 (2002), 3185–3191.
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 $K(z)$.
pdf-file
Link to Journal

Hopf Down-up Algebras with Ellen Kirkman
J. Algebra 262 (2003), no. 1, 42-53.
pdf-file

Finite dimensional representations of invariant differential operators (with Sonia L. Rueda)
Trans Amer Math Soc, 357 (2004), 2739-2752.
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 $\geq 3$.
pdf-file
Link to Journal

Lie Superalgebras, Clifford Algebras, Induced Modules and Nilpotent Orbits
Advances in Mathematics, 207 (2006), 39-72.
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 ${\mathfrak g}$.
pdf-file

Invariant Differential Operators and FCR factors of Enveloping algebras
Comm. Algebra 36 (2008), no. 10, 3759-3777.
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.
pdf-file

Noncommutative Deformations of Type A Kleinian Singularities and Hilbert Schemes
Journal of Algebra 293 (2005) 102-129.
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. \\
pdf-file

Faithful cyclic modules for enveloping algebras and Sklyanin algebras
Groups, rings and algebras, 269-276, Contemp. Math., 420, Amer. Math. Soc., Providence, RI, 2006.
Abstract: 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.
pdf-file

Monolithic modules over Noetherian Rings (with Paula Carvalho)
Glasg. Math. J. 53 (2011), no. 3, 683–692.
Abstract: 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 \cite{CarvalhoLompPusat} for down-up algebras. \\
pdf-file

Hochschild cohomology and deformations of Clifford-Weyl algebras (with Georges Pinczon and Rosane Ushirobira)
SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 028
Abstract: We give a complete study of Clifford-Weyl algebras $\Cc(n,2k)$ from Bose-Fermi statistics, including Hochschild cohomology (with coefficients in itself). We show that $\Cc(n,2k)$ is rigid when $n$ is even or when $k \neq 1$. We find all non-trivial deformations of $\Cc(2n+1,2)$ and study their representations. \\
pdf-file

Combinatorics of Character Formulas for the Lie Superalgebra $\fgl(m,n).$ (with Vera V. Serganova)
Transform. Groups 16 (2011), no. 2, 555–578.
Abstract: 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, \cite{S2} and by J. Brundan \cite{Br}. 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.
Link to Journal
pdf-file

Finitely generated, non-artinian monolithic modules
Dedicated to Ken Goodearl on the occasion of his 65th birthday.
New trends in noncommutative algebra, 211–220, Contemp. Math., 562, Amer. Math. Soc.
Abstract: 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 which was begun in [CLPY10] and [CM].
pdf-file

The lattice of submodules of a multiplicity free module
arXiv:1310.4083
Abstract: In this paper we determine, under some mild restrictions, the lattice of submodules 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 of and show that if Ext is acyclic, then Ext 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 superalgebra in the deformed case.

Coefficients of Šapovalov elements for simple Lie algebras and contragredient Lie superalgebras
arXiv:1311.0570
Abstract: We provide upper bounds on the degrees of the coefficients of Šapovalov elements for a simple Lie algebra. If $\fg$ is a contragredient Lie superalgebra and $\gc$ is a positive isotropic root of $\fg,$ we prove the existence and uniqueness of the Šapovalov element for $\gc$ and we obtain upper bounds on the degrees of their coefficients. For type A Lie superalgebras we give a closed formula for Šapovalov elements.Often the coefficients of Šapovalov elements are products of linear factors, and we provide some reasons for this coming from representation theory. We also explore the relationships between Šapovalov elements coming from different roots, and their behavior when the Borel subalgebra is changed.