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In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , there is a correspondence between Verma modules and certain line modules that associates a pair ( h , ϕ ) (\mathfrak {h},\,\phi ) , where h \mathfrak {h} is a 2 2 -dimensional Lie subalgebra of s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) and ϕ ∈ h ∗ \phi \in \mathfrak {h}^* satisfies ϕ ( [ h , h ] ) = 0 \phi ([\mathfrak {h}, \, \mathfrak {h}]) = 0 , to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra s l ( 1 | 1 ) \mathfrak {sl}(1|1) and for a color Lie algebra associated to the Lie algebra s l 2 \mathfrak {sl}_2 .

We study codimension growth of infinite dimensional Lie super- algebras over an algebraically closed field of characteristic zero. We prove that if a Lie superalgebra L L is a Grassmann envelope of a finite dimensional simple Lie algebra, then the PI-exponent of L L exists and is a positive integer.

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r r -matrices. A super dynamical r r -matrix r r satisfies the zero weight condition if [ h ⊗ 1 + 1 ⊗ h , r ( λ ) ] = 0 for all h ∈ h , λ ∈ h ∗ . \begin{equation*} [h\otimes 1 + 1 \otimes h, r(\lambda )] = 0 \text { for all } h \in \mathfrak {h}, \lambda \in \mathfrak {h}^*. \end{equation*} In this paper we explicitly quantize zero-weight super dynamical r r -matrices with zero coupling constant for the Lie superalgebra g l ( m , n ) \mathfrak {gl}(m,n) . We also answer some questions about super dynamical R R -matrices. In particular, we prove a classification theorem and offer some support for one particular interpretation of the super Hecke condition.

We consider Lie superautomorphisms of prime associative superalgebras. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.

The group of automorphisms of the Kac Jordan superalgebra is described, and used to classify the maximal subalgebras., Comment: 10 pages

In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample., Comment: AMS-TeX; 8 Pages; no figures

Let g be a complex semisimple Lie algebra and U(g) be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of U(g) is equal to the dimension of a Borel subalgebra of g.

We present an elementary construction of the 10-dimensional simple Jordan superalgebra K 10 of Kac using the 3-dimensional tiny Kaplansky superalgebra. This new realization of K 10 enables us to derive many of its properties.

Let g be a classical simple Lie superalgebra. We describe the prime ideals P in the enveloping algebra U(g) such that U(g)/P satisfies a polynomial identity. If the factor algebra U(g)/P is not artinian, then it is an order in a matrix algebra over K(z).

Let g {\mathfrak {g}} be the Lie superalgebra o s p ( 1 , 2 r ) osp(1,2r) . We show that there is a surjective homomorphism from U ( g ) U({\mathfrak {g}}) to the r t h r^{th} Weyl algebra A r A_{r} , and we use this to construct an analog of the Joseph ideal. We also obtain a decomposition of the adjoint representation of g {\mathfrak {g}} on A r A_r and use this to show that if A r A_{r} is made into a Lie superalgebra using its natural Z 2 {\mathbb Z}_{2} -grading, then A r = k ⊕ [ A r , A r ] A_{r} = k \oplus [A_{r}, A_{r}] . In addition, we show that if [ A r , A r ] [A_r, A_r] and [ A s , A s ] [A_s, A_s] are isomorphic as Lie superalgebras, then r = s r=s . This answers a question of S. Montgomery.

In this paper we find necessary and sufficient conditions on a finite-dimensional Lie superalgebra under which any associative PI-envelope of L L is finite-dimensional. We also extend M. Scheunert’s result which enables one to pass from color Lie superalgebras to the ordinary ones, to the case of gradings by an arbitrary abelian group.

Let L be an arbitrary Lie superalgebra over a field of characteristic different from 2. Denote by ω u ( L ) \omega u(L) the ideal generated by L in its universal enveloping algebra U ( L ) U(L) . It is shown that L ∩ ω u ( L ) n = γ n ( L ) L \cap \omega u{(L)^n} = {\gamma _n}(L) for each n ≥ 1 n \geq 1 , where γ n ( L ) {\gamma _n}(L) is the nth term of the lower central series of L. We also prove that ω u ( L ) \omega u(L) 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.

Let L L be a free Lie algebra of finite rank n n and R R its arbitrary ideal. A necessary and sufficient condition for n n elements of the Lie algebra L / R 2 L/{R^2} to be a generating set is given. In particular, we have a criterion for n n elements of a free Lie algebra of rank n n to be a generating set which is similar to the corresponding group-theoretic result due to Birman (An inverse function theorem for free groups, Proc. Amer. Math. Soc. 41 (1973), 634-638).

Let K K be a field of characteristic zero and L L a nonabelian Lie algebra over K K . We show that the skew field of fractions of the enveloping algebra of L L over K K contains a free noncommutative K K -algebra when L L is solvable or finite-dimensional.

In this paper it is shown that relative Lie algebra cohomology is related to relative cohomology for restricted Lie algebras by a spectral sequence. Also an interpretation of the relative cohomology groups for Lie algebras in terms of the relative right derived functors of a certain relative derivative functor is given.

We review the extent to which the structure of the universal enveloping algebra of a Lie algebroid over a manifold M resembles a Hopf algebra, and prove a Cartier-Milnor-Moore theorem for this type of structure.

We deal with the reduction theory of local Lie algebra structures in the function space through coadjoint equivariant momentum mappings.

Exact formulas for root space multiplicities in Cartan matrix Lie algebras and their universal enveloping algebras are computed. We go on to determine the number of free generators of each degree of the radicals defining these algebras.

In this paper we determine the linear relations that exist between the free generators of a free Lie algebra L when it is viewed as a module over its enveloping algebra via the adjoint representation. As an application, the annihilator of a homogeneous element of L is determined. 1. Statement of results. Let K be a commutative associative ring with unit, let L be a free Lie algebra over K (cf. [1]), and let U be the enveloping algebra of L. We identify L with its image under the canonical injection of L into U. We are interested in the structure of L as a left U-module via the adjoint representation ad: U-EndK(L). Let xl, . . . , x,, be arbitrary nonzero elements of L. Let el, . . . , en be the usual basis of the (left) U-module U': e* = (di 1, * din) with dik = 1 for k = i and zero otherwise. For 1 < i, j < n, u, v E U, define e,(u), ej(u, v) E Un by e*(u) = (ad(u)xi)ue,, e,1(u, v) = (ad(v)xj)ue, + (ad(u)x,)vej, and let E be the U-submodule of Un generated by the elements e*(u), eij(u, v) with 1 < i,j < n, u, v E U. If xl, . .. , xn are homogeneous for some grading of the Lie algebra L, then the elements u, v above can be taken to be homogeneous for the natural grading of U induced by the grading of L. Hence, in this case, E is a homogeneous submodule for the grading of Un defined by saying that (ul, . .. , uJ) E Un is homogeneous of degree k if and only if ul, . . . , un E U are homogeneous of degree k. If (ul, . . . u un) E E, then we always have the relation ad(ul)xl + * + ad(un)xn = 0. THEOREM 1. If xl,... , xn is a free generating system for L, then ad(ul)xl + * + ad(un)xn = 0 iff (ul, . .. , uX,) E E. Received by the editors November 3, 1976 and, in revised form, May 16, 1977. AMS (MOS) subject classifications (1970). Primary 17B35, 17B65; Secondary 16A06.

An associative unitary (respectively, Lie) algebra is said to be proper in case the intersection of all of its cofinite two-sided (respectively, Lie) ideals is zero. Using the Hopf algebra structure of UL, it is shown that over a field of characteristic zero a Lie algebra is proper if and only if its universal enveloping algebra is proper. (In the finite-dimensional case this provides a new proof of a result of Harish-Chandra.) The analogous result for Lie palgebras and their restricted universal enveloping algebras holds and is proved by the same technique. Notational conventions. Throughout this paper, p will denote a prime number. Barring mention to the contrary, we work over an arbitrary ground field K except that when Lie p-algebras (=restricted Lie algebras of characteristic p) are discussed, the ground field must have characteristic p (cf. [1, p. 78, Exercise 20] or [7, p. 187, Definition 4]). The unmodified words "algebra" and "coalgebra" shall, respectively, mean "associative unitary algebra" and "associative counitary coalgebra." If A is an algebra, L (A) shall denote the Lie algebra whose underlying vector space coincides with that of A, and whose bracket is defined for all elements x and y of A by [x, y] z (A) = x y y . x where " " denotes the multiplication of A; while Lp (A) shall denote the Lie p-algebra whose underlying Lie algebra is t (A) and whose p-mapping, x Fx[P}, is the pth power mapping, x ? 4 xP, of the algebra A. For any algebra map f: A -+ B, C(f ): L (A) -+ L2(B) (respectively, Lp (F): Lp (A) Lp(B)) shall denote the map of Lie algebras (respectively, Lie p-algebras) which coincides with f as a vector space map. For any Lie algebra L, UL shall denote the universal enveloping algebra of L; while for any Lie p-algebra L, VL shall denote the restricted universal enveloping algebra of L; furthermore, i and j shall denote, respectively, the canonical injections i: L -* L(UL) and j: L -k Lp(VL). Occasionally, the vector space whose only element is zero shall be denoted by 0, and, occasionally, the end of proof shall be landmarked by the symbol El. DEFINITION 1. (a) If A is an algebra (respectively, a Lie algebra), denote by C.I.(A) the collection of all cofinite two-sided ideals (respectively, all cofinite Lie Received by the editors December 26, 1979 and, in revised form, August 1, 1980 and June 3, 1986. The author spoke on various stages of development of this paper at Annual Meetings of the AMS on the following dates: January 6, 1980 (Abstract #773-16-13), January 8, 1981 (Abstract #783-16-7), January 27, 1984 (Abstract #809-16-164), January 9, 1986 (Abstract #825-16-363). 1980 Mathematics Subject Classification (1985 Revision). Primary 17B35; Secondary 16A24, 17B10. The author gratefully acknowledges partial support from a SUNY at Buffalo Individual Faculty Development Grant during the preparation of this paper. ? 1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page

We define a restricted simple algebra T T of dimension 18 over an arbitrary field of characteristic 3. From a certain property of its Cartan decomposition, we show T T to be nonisomorphic to any known algebra of identical dimension.

An example is constructed of a Lie algebra whose derived algebra has only nilpotent derivations, thus answering a question of Dixmier and Lister.

In [1], R. H. Cox stated that if x is a square matrix all of whose positive powers lie within a distance a

Let L be a finite dimensional Lie algebra over a field k of characteristic zero, U(L) its universal enveloping algebra and Z(D(L)) the center of the division ring of quotients of U(L). A number of conditions on L are each shown to be equivalent with the primitive of U(L). Also, a formula is given for the transcendency degree of Z(D(L)) over k.