Book Review
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3090429
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Book Information:
Author:
Ian Musson
Title:
Lie superalgebras and enveloping algebras
Additional book information:
Graduate Studies in Mathematics, Vol. 131,
American Mathematical Society,
Providence, RI,
2012,
xx+488 pp.,
ISBN 978-0-8128-6867-6,
$87.00,
hardcover
I. N. Bernšteĭn and D. A. Leĭtes, A formula for the characters of the irreducible finite-dimensional representations of Lie superalgebras of series $\textrm {Gl}$ and $\textrm {sl}$, C. R. Acad. Bulgare Sci. 33 (1980), no. 8, 1049–1051 (Russian). MR 620836
Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $\mathfrak {g}\mathfrak {l}(m|n)$, J. Amer. Math. Soc. 16 (2003), no. 1, 185–231. MR 1937204, DOI 10.1090/S0894-0347-02-00408-3
Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra ${\mathfrak {q}}(n)$, Adv. Math. 182 (2004), no. 1, 28–77. MR 2028496, DOI 10.1016/S0001-8708(03)00073-2
Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 373–419. MR 2881300, DOI 10.4171/JEMS/306
Jonathan Brundan and Catharina Stroppel, Gradings on walled Brauer algebras and Khovanov’s arc algebra, Adv. Math. 231 (2012), no. 2, 709–773. MR 2955190, DOI 10.1016/j.aim.2012.05.016
Shun-Jen Cheng, Ngau Lam, and Weiqiang Wang, Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Invent. Math. 183 (2011), no. 1, 189–224. MR 2755062, DOI 10.1007/s00222-010-0277-4
S. J. Cheng, N. Lam, W. Wang, Brundan–Kazhdan–Lusztig conjecture for general Lie superalgebras. arXiv:1203.0092v3.
Jonathan Comes and Benjamin Wilson, Deligne’s category $\underline {\rm {Rep}}(GL_\delta )$ and representations of general linear supergroups, Represent. Theory 16 (2012), 568–609. MR 2998810, DOI 10.1090/S1088-4165-2012-00425-3
Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR 1393197, DOI 10.1090/gsm/011
Maria Gorelik, Weyl denominator identity for affine Lie superalgebras with non-zero dual Coxeter number, J. Algebra 337 (2011), 50–62. MR 2796063, DOI 10.1016/j.jalgebra.2011.04.011
Maria Gorelik and Shifra Reif, A denominator identity for affine Lie superalgebras with zero dual Coxeter number, Algebra Number Theory 6 (2012), no. 5, 1043–1059. MR 2968633, DOI 10.2140/ant.2012.6.1043
Caroline Gruson, Finitude de l’homologie de certains modules de dimension finie sur une super algèbre de Lie, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 531–553 (French, with English summary). MR 1450424
Caroline Gruson and Vera Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852–892. MR 2734963, DOI 10.1112/plms/pdq014
V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
V. G. Kac, Characters of typical representations of classical Lie superalgebras, Comm. Algebra 5 (1977), no. 8, 889–897. MR 444725, DOI 10.1080/00927877708822201
Victor G. Kac and Minoru Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 415–456. MR 1327543, DOI 10.1007/978-1-4612-0261-5_{1}5
Ian M. Musson, A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra, Adv. Math. 91 (1992), no. 2, 252–268. MR 1149625, DOI 10.1016/0001-8708(92)90018-G
Ian M. Musson, Enveloping algebras of Lie superalgebras: a survey, Azumaya algebras, actions, and modules (Bloomington, IN, 1990) Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 141–149. MR 1144033, DOI 10.1090/conm/124/1144033
I. B. Penkov, Borel-Weil-Bott theory for classical Lie supergroups, Current problems in mathematics. Newest results, Vol. 32, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 71–124 (Russian). Translated in J. Soviet Math. 51 (1990), no. 1, 2108–2140. MR 957752
I. Penkov and V. Serganova, Characters of irreducible $G$-modules and cohomology of $G/P$ for the Lie supergroup $G=Q(N)$, J. Math. Sci. (New York) 84 (1997), no. 5, 1382–1412. Algebraic geometry, 7. MR 1465520, DOI 10.1007/BF02399196
Manfred Scheunert, The theory of Lie superalgebras, Lecture Notes in Mathematics, vol. 716, Springer, Berlin, 1979. An introduction. MR 537441
Vera Serganova, Kac-Moody superalgebras and integrability, Developments and trends in infinite-dimensional Lie theory, Progr. Math., vol. 288, Birkhäuser Boston, Boston, MA, 2011, pp. 169–218. MR 2743764, DOI 10.1007/978-0-8176-4741-4_{6}
Vera Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}(m|n)$, Selecta Math. (N.S.) 2 (1996), no. 4, 607–651. MR 1443186, DOI 10.1007/PL00001385
A. N. Sergeev, Representations of the Lie superalgebras ${\mathfrak {g}l}(n,\,m)$ and $Q(n)$ in a space of tensors, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 80–81 (Russian). MR 739101
Alexander Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434. MR 1722115, DOI 10.1090/S1088-4165-99-00085-0
References
- I. N. Bernšteĭn and D. A. Leĭtes, A formula for the characters of the irreducible finite-dimensional representations of Lie superalgebras of series $\textrm {Gl}$ and $\textrm {sl}$, C. R. Acad. Bulgare Sci. 33 (1980), no. 8, 1049–1051 (Russian). MR 620836 (82j:17020a)
- Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $\mathfrak {g}\mathfrak {l}(m\vert n)$, J. Amer. Math. Soc. 16 (2003), no. 1, 185–231. MR 1937204 (2003k:17007), DOI 10.1090/S0894-0347-02-00408-3
- Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra ${\mathfrak {q}}(n)$, Adv. Math. 182 (2004), no. 1, 28–77. MR 2028496 (2004m:17018), DOI 10.1016/S0001-8708(03)00073-2
- Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 373–419. MR 2881300 (2012m:17009), DOI 10.4171/JEMS/306
- Jonathan Brundan and Catharina Stroppel, Gradings on walled Brauer algebras and Khovanov’s arc algebra, Adv. Math. 231 (2012), no. 2, 709–773. MR 2955190, DOI 10.1016/j.aim.2012.05.016
- Shun-Jen Cheng, Ngau Lam, and Weiqiang Wang, Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Invent. Math. 183 (2011), no. 1, 189–224. MR 2755062 (2012f:17011), DOI 10.1007/s00222-010-0277-4
- S. J. Cheng, N. Lam, W. Wang, Brundan–Kazhdan–Lusztig conjecture for general Lie superalgebras. arXiv:1203.0092v3.
- Jonathan Comes and Benjamin Wilson, Deligne’s category $\underline \textrm {{Rep}}(GL_\delta )$ and representations of general linear supergroups, Represent. Theory 16 (2012), 568–609. MR 2998810, DOI 10.1090/S1088-4165-2012-00425-3
- Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR 1393197 (97c:17010)
- Maria Gorelik, Weyl denominator identity for affine Lie superalgebras with non-zero dual Coxeter number, J. Algebra 337 (2011), 50–62. MR 2796063 (2012c:17038), DOI 10.1016/j.jalgebra.2011.04.011
- Maria Gorelik and Shifra Reif, A denominator identity for affine Lie superalgebras with zero dual Coxeter number, Algebra Number Theory 6 (2012), no. 5, 1043–1059. MR 2968633, DOI 10.2140/ant.2012.6.1043
- Caroline Gruson, Finitude de l’homologie de certains modules de dimension finie sur une super algèbre de Lie, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 2, 531–553 (French, with English summary). MR 1450424 (98b:17024)
- Caroline Gruson and Vera Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852–892. MR 2734963 (2012a:17010), DOI 10.1112/plms/pdq014
- V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 0486011 (58 \#5803)
- V. G. Kac, Characters of typical representations of classical Lie superalgebras, Comm. Algebra 5 (1977), no. 8, 889–897. MR 0444725 (56 \#3075)
- Victor G. Kac and Minoru Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 415–456. MR 1327543 (96j:11056)
- Ian M. Musson, A classification of primitive ideals in the enveloping algebra of a classical simple Lie superalgebra, Adv. Math. 91 (1992), no. 2, 252–268. MR 1149625 (93c:17022), DOI 10.1016/0001-8708(92)90018-G
- Ian M. Musson, Enveloping algebras of Lie superalgebras: a survey, Azumaya algebras, actions, and modules (Bloomington, IN, 1990) Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 141–149. MR 1144033 (93b:17012), DOI 10.1090/conm/124/1144033
- I. B. Penkov, Borel-Weil-Bott theory for classical Lie supergroups, Current problems in mathematics. Newest results, Vol. 32, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 71–124 (Russian). Translated in J. Soviet Math. 51 (1990), no. 1, 2108–2140. MR 957752 (90f:22018)
- I. Penkov and V. Serganova, Characters of irreducible $G$-modules and cohomology of $G/P$ for the Lie supergroup $G=Q(N)$, J. Math. Sci. (New York) 84 (1997), no. 5, 1382–1412. Algebraic geometry, 7. MR 1465520 (98i:17010), DOI 10.1007/BF02399196
- Manfred Scheunert, The theory of Lie superalgebras. An introduction, Lecture Notes in Mathematics, vol. 716, Springer, Berlin, 1979. MR 537441 (80i:17005)
- Vera Serganova, Kac-Moody superalgebras and integrability, Developments and trends in infinite-dimensional Lie theory, Progr. Math., vol. 288, Birkhäuser Boston Inc., Boston, MA, 2011, pp. 169–218. MR 2743764 (2011m:17056), DOI 10.1007/978-0-8176-4741-4_6
- Vera Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}(m\vert n)$, Selecta Math. (N.S.) 2 (1996), no. 4, 607–651. MR 1443186 (98f:17007), DOI 10.1007/PL00001385
- A. N. Sergeev, Representations of the Lie superalgebras ${\mathfrak {g}l}(n,\,m)$ and $Q(n)$ in a space of tensors, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 80–81 (Russian). MR 739101 (86b:17005)
- Alexander Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434 (electronic). MR 1722115 (2000j:20021), DOI 10.1090/S1088-4165-99-00085-0
Review Information:
Reviewer:
Vera Serganova
Affiliation:
University of California, Berkeley
Email:
serganov@math.berkeley.edu
Journal:
Bull. Amer. Math. Soc.
50 (2013), 691-696
DOI:
https://doi.org/10.1090/S0273-0979-2013-01418-6
Published electronically:
May 1, 2013
Review copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
