Celebratio Mathematica

G. Benkart and M. Ondrus: “Whittaker modules for generalized Weyl algebras,” Represent. Theory 13 (2009), pp. 141–164. MR 2497458 Zbl 1251.16020 ArXiv 0803.3570 article

We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of \( \mathfrak{sl}_2 \) and of Heisenberg Lie algebras, Smith’s generalizations of \( U(\mathfrak{sl}_2) \), various quantum analogues of these algebras, and many others. We show that the Whittaker modules \( V = Aw \) of the generalized Weyl algebra \( A = R(\phi,t) \) are in bijection with the \( \phi \)-stable left ideals of \( R \). We determine the annihilator \( \operatorname{Ann}_A(w) \) of the cyclic generator \( w \) of \( V \). We also describe the annihilator ideal \( \operatorname{Ann}_A(V) \) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant’s well-known results on Whittaker modules and their associated annihilators for \( U(\mathfrak{sl}_2) \).

G. Benkart, S. A. Lopes, and M. Ondrus: “A parametric family of subalgebras of the Weyl algebra, II: Irreducible modules,” pp. 73–98 in Recent developments in algebraic and combinatorial aspects of representation theory (12–16 August 2010, Bangalore, India and 18–20 May 2012, Riverside, CA). Edited by V. Chari, J. Greenstein, K. C. Misra, K. N. Raghavan, and S. Viswanath. Contemporary Mathematics 602. American Mathematical Society (Providence, RI), 2013. Part I was published in Trans. Am. Math. Soc. 367:3 (2015). Part III appeared as a 2014 preprint. MR 3203899 Zbl 1308.16022 ArXiv 1212.1404 incollection

An Ore extension over a polynomial algebra \( \mathbf{F}[x] \) is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra \( \mathbf{A}_h \) generated by elements \( x, y \), which satisfy \( yx - xy = h \), where \( h \in \mathbf{F}[x] \). When \( h \neq 0 \), these algebras are subalgebras of the Weyl algebra \( \mathbf{A}_1 \) and can be viewed as differential operators with polynomial coefficients. In previous work, we studied the structure of \( \mathbf{A}_h \) and determined its automorphism group \( \operatorname{Aut}_{\mathbb{F}}(\mathbf{A}_h) \) and the subalgebra of invariants under \( \operatorname{Aut}_{\mathbb{F}}(\mathbf{A}_h) \). Here we determine the irreducible \( \mathbf{A}_h \)-modules. In a sequel to this paper, we completely describe the derivations of \( \mathbf{A}_h \) over any field.

Samuel Antonio Lopes	Related
Kailash C. Misra	Related
Matthew Ondrus	Related
Vyjayanthi Chari	Related

@incollection {key3203899m,
	AUTHOR = {Benkart, Georgia and Lopes, Samuel A.
		 and Ondrus, Matthew},
	TITLE = {A parametric family of subalgebras of
		 the {W}eyl algebra, {II}: {I}rreducible
		 modules},
	BOOKTITLE = {Recent developments in algebraic and
		 combinatorial aspects of representation
		 theory},
	EDITOR = {Chari, Vyjayanthi and Greenstein, Jacob
		 and Misra, Kailash C. and Raghavan,
		 K. N. and Viswanath, Sankaran},
	SERIES = {Contemporary Mathematics},
	NUMBER = {602},
	PUBLISHER = {American Mathematical Society},
	ADDRESS = {Providence, RI},
	YEAR = {2013},
	PAGES = {73--98},
	DOI = {10.1090/conm/602/12027},
	NOTE = {(12--16 August 2010, Bangalore, India
		 and 18--20 May 2012, Riverside, CA).
		 Part I was published in \textit{Trans.
		 Am. Math. Soc.} \textbf{367}:3 (2015).
		 Part III appeared as a 2014 preprint.
		 ArXiv:1212.1404. MR:3203899. Zbl:1308.16022.},
	ISSN = {0271-4132},
	ISBN = {9780821890370},
}

G. Benkart, S. A. Lopes, and M. Ondrus: “A parametric family of subalgebras of the Weyl algebra, I: Structure and automorphisms,” Trans. Am. Math. Soc. 367 : 3 (2015), pp. 1993–2021. Part II was published in Recent developments in algebraic and combinatorial aspects of representation theory (2013). Part III appeared as a 2014 preprint. MR 3286506 Zbl 1312.16020 ArXiv 1210.4631 article

An Ore extension over a polynomial algebra \( \mathbb{F}[x] \) is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra \( \mathbf{A}_h \) generated by elements \( x,y \), which satisfy \( yx - xy = h \), where \( h\in \mathbb{F}[x] \). We investigate the family of algebras \( \mathbf{A}_h \) as \( h \) ranges over all the polynomials in \( \mathbb{F}[x] \). When \( h \neq 0 \), the algebras \( \mathbf{A}_h \) are subalgebras of the Weyl algebra \( \mathbf{A}_1 \) and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of \( \mathbf{A}_h \) over arbitrary fields \( \mathbb{F} \) and describe the invariants in \( \mathbf{A}_h \) under the automorphisms. We determine the center, normal elements, and height one prime ideals of \( \mathbf{A}_h \), localizations and Ore sets for \( \mathbf{A}_h \), and the Lie ideal \( [\mathbf{A}_h,\mathbf{A}_h] \). We also show that \( \mathbf{A}_h \) cannot be realized as a generalized Weyl algebra over \( \mathbb{F}[x] \), except when \( h\in\mathbb{F} \). In two sequels to this work, we completely describe the irreducible modules and derivations of \( \mathbf{A}_h \) over any field.

Matthew Ondrus	Related
Samuel Antonio Lopes	Related

G. Benkart, S. A. Lopes, and M. Ondrus: “Derivations of a parametric family of subalgebras of the Weyl algebra,” J. Algebra 424 (February 2015), pp. 46–97. MR 3293213 Zbl 1312.16019 article

An Ore extension over a polynomial algebra \( \mathbb{F}[x] \) is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra \( \mathbf{A}_h \) generated by elements \( x,y \), which satisfy \( yx-xy = h \), where \( h\in \mathbb{F}[x] \). When \( h\neq 0 \), the algebra \( \mathbf{A}_h \) is subalgebra of the Weyl algebra \( \mathbf{A}_1 \) and can be viewed as differential operators with polynomial coefficients. This paper determines the derivations of \( \mathbf{A}_h \) and the Lie structure of the first Hochschild cohomology group \[ \mathbf{HH}^1(\mathbf{A}_h) = \operatorname{Der}_{\mathbb{F}}(\mathbf{A}_h)/\operatorname{Inder}_{\mathbb{F}}(\mathbf{A}_h) \] of outer derivations over an arbitrary field. In characteristic 0, we show that \( \mathbf{HH}^1(\mathbf{A}_h) \) has a unique maximal nilpotent ideal modulo which \( \mathbf{HH}^1(\mathbf{A}_h) \) is 0 or a direct sum of simple Lie algebras that are field extensions of the one-variable Witt algebra. In positive characteristic, we obtain decomposition theorems for \( \operatorname{Der}_{\mathbb{F}}(\mathbf{A}_h) \) and \( \mathbf{HH}^1(\mathbf{A}_h) \) and describe the structure of \( \mathbf{HH}^1(\mathbf{A}_h) \) as a module over the center of \( \mathbf{A}_h \).

Matthew Ondrus	Related
Samuel Antonio Lopes	Related

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