The dimension subalgebra problem for enveloping algebras of Lie superalgebras

Author:
David M. Riley

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2975-2980

MSC:
Primary 17B35; Secondary 16S30

DOI:
https://doi.org/10.1090/S0002-9939-1995-1264829-0

MathSciNet review:
1264829

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Abstract: Let *L* be an arbitrary Lie superalgebra over a field of characteristic different from 2. Denote by the ideal generated by *L* in its universal enveloping algebra . It is shown that for each , where is the *n*th term of the lower central series of *L*. We also prove that is a residually nilpotent ideal if and only if *L* is residually nilpotent. Both these results remain true in characteristic 2 provided we take *L* to be an ordinary Lie algebra.

**[BP]**J. Bergen and D. S. Passman,*Delta methods in enveloping algebras of Lie superalgebras*, Trans. Amer. Math. Soc.**334**(1992), 259-280. MR**1076611 (93a:17008)****[H]**B. Hartley,*The residual nilpotence of wreath products*, Proc. London Math. Soc.**2**(1970), 365-392. MR**0258966 (41:3611)****[J]**S. A. Jennings,*The group ring of a class of infinite nilpotent groups*, Canad. J. Math.**7**(1955), 169-187. MR**0068540 (16:899a)****[K]**M. A. Knus,*On the enveloping algebra and the descending central series of a Lie algebra*, J. Algebra**12**(1969), 335-338. MR**0238914 (39:274)****[L1]**M. Lazard,*Sur les groupes nilpotents et les anneaux de Lie*, Ann. Sci. École Norm. Sup.**71**(1954), 101-190. MR**0088496 (19:529b)****[L2]**-,*Groupes analytiques p-adiques*, Inst. Hautes Études Sci. Publ. Math.**26**(1965). MR**0209286 (35:188)****[Mc]**J. C. McConnell,*The intersection theorem for a class of noncommutative rings*, Proc. London Math. Soc.**17**(1967), 487-498. MR**0210738 (35:1624)****[Q]**D. G. Quillen,*On the associated graded ring of a group ring*, J. Algebra**10**(1968), 411-418. MR**0231919 (38:245)****[R]**D. M. Riley,*Numbers of generators of ideals in graded group rings and p-adic analytic pro-p groups of dimension one*, Arch. Math.**63**(1994), 402-406. MR**1300733 (96c:20056)****[RS]**D. M. Riley and A. Shalev,*Restricted Lie algebras and their envelopes*, Canad. J. Math. (to appear). MR**1319694 (96e:16031)****[Ri]**E. Rips,*On the fourth integral dimension subgroup*, Israel J. Math.**12**(1972), 342-346. MR**0314988 (47:3537)****[S]**M. Scheunert,*The theory of Lie superalgebras*, Lecture Notes in Math., vol. 716, Springer-Verlag, Berlin, 1979. MR**537441 (80i:17005)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1264829-0

Article copyright:
© Copyright 1995
American Mathematical Society