Tohoku Mathematical Journal

Classification of simple $\mathfrak{q}_2$-supermodules

Volodymyr Mazorchuk

Full-text: Open access

Abstract

We classify all simple supermodules over the queer Lie superalgebra $\mathfrak{q}_2$ up to classification of equivalence classes of irreducible elements in a certain Euclidean ring.

Article information

Source
Tohoku Math. J. (2), Volume 62, Number 3 (2010), 401-426.

Dates
First available in Project Euclid: 15 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1287148620

Digital Object Identifier
doi:10.2748/tmj/1287148620

Mathematical Reviews number (MathSciNet)
MR2742017

Zentralblatt MATH identifier
1276.17005

Subjects
Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 17B35: Universal enveloping (super)algebras [See also 16S30]

Citation

Mazorchuk, Volodymyr. Classification of simple $\mathfrak{q}_2$-supermodules. Tohoku Math. J. (2) 62 (2010), no. 3, 401--426. doi:10.2748/tmj/1287148620. https://projecteuclid.org/euclid.tmj/1287148620


Export citation

References

  • V. Bavula, Generalized Weyl algebras and their representations. (Russian) Algebra i Analiz 4 (1992), 75--97; translation in St. Petersburg Math. J. 4 (1993), 71--92.
  • V. Bavula, Classification of simple $\rm sl(2)$-modules and the finite-dimensionality of the module of extensions of simple $\rm sl(2)$-modules. (Russian) Ukrain. Mat. Zh. 42 (1990), 1174--1180; translation in Ukrainian Math. J. 42 (1990), 1044--1049 (1991).
  • V. Bavula, Filter dimension of algebras and modules, a simplicity criterion of generalized Weyl algebras, Comm. Algebra 24 (1996), 1971--1992.
  • V. Bavula and F. van Oystaeyen, The simple modules of the Lie superalgebra $\rm osp(1,2)$, J. Pure Appl. Algebra 150 (2000), 41--52.
  • E. Behr, Enveloping algebras of Lie superalgebras, Pacific J. Math. 130 (1987), 9--25.
  • A. Beilinson and V. Ginzburg, Wall-crossing functors and $\mathcalD$-modules, Represent. Theory 3 (1999), 1--31.
  • J. Bernstein and S. Gelfand, Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245--285.
  • R. Block, The irreducible representations of the Lie algebra $\mathfraksl(2)$ and of the Weyl algebra, Adv. in Math. 39 (1981), 69--110.
  • J. Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $\mathfrakq(n)$, Adv. Math. 182 (2004), 28--77.
  • J. Brundan and A. Kleshchev, Modular representations of the supergroup $Q(n)$. I, J. Algebra 260 (2003), 64--98.
  • A. Frisk, Typical blocks of the category $\mathcalO$ for the queer Lie superalgebra, J. Algebra Appl. 6 (2007), 731--778.
  • A. Frisk and V. Mazorchuk, Regular strongly typical blocks of $\mathcalO^\mathfrakq$, Commun. Math. Phys. 291 (2009), 533--542.
  • O. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. École Norm. Sup. (4) 14 (1981), 261--302.
  • M. Gorelik, Shapovalov determinants of $Q$-type Lie superalgebras, IMRP Int. Math. Res. Pap. 2006, Art. ID 96895, 71 pp.
  • D. Grantcharov, Coherent families of weight modules of Lie superalgebras and an explicit description of the simple admissible $\mathfraksl(n+1\vert 1)$-modules, J. Algebra 265 (2003), 711--733.
  • J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Ergeb. Math. Grenzgeb. (3) 3, Springer-Verlag, Berlin, 1983.
  • O. Khomenko and V. Mazorchuk, Structure of modules induced from simple modules with minimal annihilator, Canad. J. Math. 56 (2004), 293--309.
  • B. Kostant, On the tensor product of a finite and an infinite dimensional representation, J. Functional Analysis 20 (1975), 257--285.
  • O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), 537--592.
  • V. Mazorchuk, Lectures on $\mathfraksl_2(\mathbbC)$-modules, Imperial College Press, 2009.
  • V. Mazorchuk and C. Stroppel, Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math. 219 (2008), 1363--1426.
  • I. Musson, A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra, Adv. Math. 91 (1992), 252--268.
  • E. Orekhova, Simple weight $\mathfrakq(2)$-supermodules, Master Thesis, Uppsala University, 2009.
  • I. Penkov, Generic representations of classical Lie superalgebras and their localization, Monatsh. Math. 118 (1994), 267--313.
  • I. Penkov, Characters of typical irreducible finite-dimensional $\mathfrakq(n)$-modules, Funktsional. Anal. i Prilozhen. 20 (1986), 37--45.
  • I. Penkov and V. Serganova, Characters of irreducible $G$-modules and cohomology of $G/P$ for the Lie supergroup $G=Q(N)$, Algebraic geometry, 7, J. Math. Sci. (New York) 84 (1997), 1382--1412.
  • L. Ross, Representations of graded Lie algebras, Trans. Amer. Math. Soc. 120 (1965), 17--23.
  • V. Serganova, On representations of the Lie superalgebra $p(n)$, J. Algebra 258 (2002), 615--630.
  • J. Stafford, Homological properties of the enveloping algebra $U(\rm Sl\sb2)$, Math. Proc. Cambridge Philos. Soc. 91 (1982), 29--37.