Quotient Division Ring Research Articles

Weights of semi-invariants of the quotient division ring of an enveloping algebra

Let L L be a finite dimensional Lie algebra over a field k k of characteristic zero, D ( L ) D(L) the quotient division ring of U ( L ) U(L) . It is shown that the weights of the semi-invariants of D ( L ) D(L) form a finitely generated, free abelian group Λ D ( L ) {\Lambda _D}(L) . It follows, among other things, that the semicenter of D ( L ) D(L) is isomorphic to the group algebra of Λ D ( L ) {\Lambda _D}(L) over the center Z ( D ( L ) ) Z(D(L)) of D ( L ) D(L) .

When rings of differential operators are maximal orders

AbstractLet A be a commutative domain, finitely generated as an algebra over a field k of characteristic zero and write (A) for the ring of k -linear differential operators. Then A is an Ore domain with quotient division ring, say Q. Our main result is that A is a maximal order in Q if and only if (i) A = ∩{Ap: height (p) = 1} and (ii) A is geometrically unibranched. In this case A is also a Krull domain with no reflexive ideals. We also determine some conditions under which A is simple.

On the weight spaces of Lie algebra modules and their Jordan kernel

we write V1 and V” instead of V,(L) and V’(L). If V’(S) is nonzero, we call 1 a weight of S in V. Remark that, if V is finite-dimensional, V’(S) is the set of all u E V such that [x - n(x)]” v = 0 for all x E L, where n is the dimension of V. It is clear that V, is a submodule of V, contained in V”. Moreover, if V, is nonzero, then I must be a character of L (i.e., A is linear and ;1( [L, L]) = 0). In [8] Smith asks wether Vi must be nonzero, if V” is non- zero. In case V is finite-dimensional, she answers this question affirmatively in [9] and she also proves that, in that situation, V” is a submodule of I’. These results may be generalized to arbitrary L modules V [Theorem 31. We are able to give a new characterization of these weight spaces [Theorem 11, Proposition 131 which greatly simplifies their actual com- putation. We also take special interest in applying these results to the case where V is either the universal enveloping algebra U(L) of L or its division ring of quotients D(L). In the latter case, each weight vector of D(L) can be written as a quotient of a weight vector of U(L) by a nonzero eigenvec- tor (semi-invariant) of U(L) [Proposition 173. Furthermore, we define the 28

On the semi-center of a universal enveloping algebra

Throughout this paper L will be a nonzero finite-dimensional Lie algebra over a field k of characteristic zero. Let U(L) be the universal enveloping algebra of L with center Z( U(L)), D(L) the division ring of quotients of U(L) with center Z(D(L)). For each 2 E L* we denote by D(L), the set of those UED(L) such that ad x(u) = n(x) u for all XE L. Its elements are called the semi-invariants of D(L) relative to 1. Clearly D(L)$(L), c aL)A+, for all I, p E L*. The sum of the D(L),. is direct and is a subalgebra Sz(D(L)) of D(L) which is called the semi-center of D(L). We put U(L), = U(L) n D(L),. The sum Sz( U(L)) of the U(L), is called the semi-center of U(L). The semi-center Sz(S(L)) of the symmetric algebra S(L) of L can be defined in a similar fashion. The semi-center is a commutative subalgebra of U(L), which never reduces to k [7] and which is stable under all automorphisms of U(L) [16]. From the fact that it is a unique factorization domain in the algebraically closed case [ 131, we are able to deduce that, in the general case, Sz( U(L)) is contained in a factorial subalgebra of U(L) (Theorem 1.2). In particular, any nonzero semi-invariant of U(L) can be factored in a unique way as the product of irreducible semiinvariants (Proposition 1.5). This result proves to be extremely useful throughout the paper. We also obtain some characterizations of the semiinvariants of D(L) (Proposition 1.8). Let A(L) (resp. A,(L)) be the set of all /z E L* such that U(L), #0 (resp. D(L), #O). A(L) is a semi-group which is not always finitely generated [9]. The group n,(L), which is isomorphic to Kharchenko’s group of X-inner automorphisms of U(L) [16], turns out to be the additive subgroup of L* generated by A(L) (Corollary 1.9). Next, the centralizer of Sz( U(L)) in U(L) coincides with U(L,), where L, is the intersection of ker 2, AC/~(L) (Corollary 1.16). L, 324 0021.8693/85 $3.00

Krull and global dimensions of Weyl algebras over division rings

where the xi commute among themselves, the 0, commute among themselves, and Bixj - xjei = 6,. In the case where R is a field of characteristic zero, these algebras are intimately connected with the representation theory of enveloping algebras of solvable Lie algebras [ 1, 31, while in the case where R is a right noetherian ring they provide a very interesting class of examples of right noetherian rings. Scattered throughout the literature and folklore of noncommutative ring theory are various results concerning the Krull and global dimensions of Weyl algebras, usually over fields, but occasionally over more general coef- ficient rings. The earliest results appeared before the introduction of noncom- mutative Krull dimension and so concern global dimension. The first result in the literature was apparently an observation of Rinehart [20] that a more general theorem due to Hochschild, Kostant, and Rosenberg [ 1 l] established the extreme bounds IZ < gl.dim(A,(F)) < 2n, for any field F. Rinehart showed that gl.dim(A,(l;)) = 2n for any field F of positive characteristic, and that gl.dim(A r(F)) = 1 for any field F of charac- teristic zero. In the latter case, he also improved the upper bound for gl.dim(A,(F)) to 2n - 1. The obvious conjecture, that gl.dim(A,(F)) = n for any field F of characteristic zero, was verified by Roos [22]. In the meantime, Hart [ 101 had shown that gl.dim(A r(Di)) = 2, where D, is the quotient division ring of the first Weyl algebra over a field of characteristic 334

The Center of the Quotient Division Ring of the Universal Envelope of a Lie Algebra

The Center of the Quotient Division Ring of the Universal Envelope of a Lie Algebra

Dimensions of division rings

Letk be a field. WriteD(G) for the quotient division ring of the group ringkG of a torsion-free, polycyclic-by-finite groupG, andD(g) for the quotient ring of the enveloping algebra of a finite-dimensional Lie algebrag overk. In this note we show that the Hirsch numberh(G) and dim k g are invariants for the respective division rings, by calculating the Krull and global dimensions ofD(G)⊗ k D(G) andD(g)⊗ k D(g).

The center of the quotient division ring of the universal envelope of a Lie algebra

Let L L be a finite dimensional Lie algebra over a field k k of characteristic zero, D ( L ) D(L) the quotient division ring of U ( L ) U(L) . We compare the center Z ( D ( L ) ) Z(D(L)) with Z ( D ( H ) ) Z(D(H)) where H H is an ideal of L L of codimension one.

$X$-inner automorphisms of filtered algebras

We continue earlier work and compute the X-inner automorphisms of the ring of differential polynomials in one variable over an arbitrary domain. This is then applied to iterated Ore extensions. We also show that the ring of generic matrices has no nonidentity automorphisms which fix the center. In this paper we continue the study of X-inner automorphisms of filtered algebras begun in [7], which will be referred to as (I) in what follows. In (I) we described the X-inner automorphisms of the enveloping algebra of a Lie algebra, and of the ring of differential polynomials A = R[x; d] in one variable over a commutative domain R. Here we first show that the ring of generic matrices has only the trivial X-inner automorphism; equivalently, any automorphism fixing the center is the identity automorphism. We then extend our result from (I) on differential polynomials, allowing an arbitrary prime ring R as the coefficient ring. When R is a domain, we explicitly determine the group of X-inner automorphisms of A = R[x; d] in terms of R and d. We then apply these results to certain iterated Ore extensions; in particular, if the original coefficient ring is commutative, then the group of all X-inner automorphisms is abelian. In what follows, A will always denote a prime ring. Recall that a E Aut(A) is X-inner if it becomes inner when extended to the (right) Martindale quotient ring QO(A) of A; when A is an Ore domain, a is X-inner if and only if it becomes inner on the quotient division ring D of A. We shall need the following properties of QO(A): it is the (right) quotient ring of A with respect to the filter IF of all nonzero two-sided ideals (that is, QO(A) = lim ,HomA('A, A)), it is a prime ring with center C C(A) a field, called the extended center of A, and A may be imbedded in QO(A) as left multiplications. By construction, for any 0 #d x E A, there exists a nonzero I of A so that 0 =# xI c A. Fundamental in what follows is an internal characterization of X-inner automorphisms [8]. LEMMA 1. If a E Aut(A) is X-inner, say r' = s-'rs, all r E A for some s E QO(A), then there exist nonzero 'a, b E A such that sa = b and arb = b0r0a, all r E A. Conversely if arb = b0r0a, 'all r E A, for some nonzero a, b E A, then there exists s E QO(A) with sa b which induces a. Received by the editors April 23, 1982 and, in revised form, July 19, 1982. 1980 Mathematics Subject Classification. Primary 16A72, 16A38. ' Research supported in part by NSF grant No. MCS 81-01730. ? 1983 American Mathematical Society 0002-9939/82/0000-0756/$02.75

Transcendental division algebras and simple Noetherian rings

In this paper we prove the following theorem: Let D be a division ring with center the field k, and let k (x 1, …, xn) denote the rational function field in n variables over k. If D contains a maximal subfield which has transcendence degree at least n over k, then D ⊗k k (x1, …, xn) is a simple Noetherian domain of Krull and global dimensions n. Rather surprisingly, the preceding result can be used to determine the maximum transcendence degrees of the commutative subalgebras of several classically studied division rings. Using the theorem we prove, for example, that in the division ring of quotients of the Weyl algebra,A n, every maximal subfield has transcendence degree at mostn over the center.

The isomorphisms of unitary groups over noncommutative domains

The isomorphisms between projective unitary congruence groups are known when the underlying Witt indices are 33, the underlying spaces are finite dimensional, and the underlying integral domains are commutative [15, 161. Here we extend these results to noncommutative domains possessing a division ring of quotients and to arbitrary dimensions. Our development allows unitary and symplectic groups to be treated simultaneously, applies to collinear groups, and unifies the known theories over commutative domains and noncommutative fields. We consider the class of collinear (unitary or symplectic) groups having “enough projective transvections,” i.e., at least one on each isotropic line (see Sections 2A and 1B). The chief hurdle, as in the commutative case, is to show that in such groups projective transvections are preserved under isomorphism. From this we get a correspondence of isotropic lines to which the Fundamental Theorem of Projective Geometry can be applied. Then it is easy to show that the isomorphism is of the expected form, i.e., induced by an orthogonality-preserving semilinear bijection (reflexive collinear transformation) between the two underlying spaces; in particular, a unitary group is not isomorphic to a symplectic group.

On Lie algebras with primitive envelopes, supplements

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 $k$.

Crossed products of simple rings

Let A be the algebra of all q X q matrices over division ring D. Suppose there is given group G of n automorphisms of A such that (i) the fixed subring S of G is simple ring such that [A: S] = n and (ii) GA, coincides with the totality of all homomorphisms of the S-left module A to itself where Ar is the ring of right multiplication of elements of A. Suppose also that there is given factor system {ca} (o-, r&G) in the center K of A. Then crossed product of A and G is defined via the same formulae as in the commutative case. (See [2 ].) The purpose of this note is to investigate the splitting property of factor system by an extension of S as well as of A. This is generalization of wellknown theorem for the commutative case as well as of result given in [2]. To begin with, we shall consider purely transcendental extension of A as follows. Let xi, * * *, Xm be m variables. Let D [xl, . . ., xm] be the polynomial ring of xi, * * *, Xm over D. We suppose xi lie in the center of the ring. Then the quotient division ring of D [xi, ..., xm ] is denoted by D (xi, * * * , xm). The existence of the quotient ring is clear from general theory of quotient ring, or it can be proved directly as follows. Generally let r be ring with no divisor of zero. Moreover suppose that for any nonzero elements and of r there exist nonzero elements a', b', a and b such that aa' = bb' and aa=bb. We consider the set of formal elements a-lb and cd-1 (a, b, c, dEr and #5O, d # O). Define aT lb1 = aj 'b2 if and only if there exist nonzero elements c and d such that aic =bid and a2c= b2d. It is good exercise to show that the above equivalent relation is well defined. Similarly, define biaT' = b2a' 1 if and only if there exist nonzero elements c and d such that ca1 = db1 and ca2=db2. Also define a-lb = cd-1 if and only if ac = bd. To verify that this is well defined is also good exercise. How to define the algebraic operations in this set is now almost clear. For example, (a-lb) (c-ld) = (c'a)-1b'd, where bc= c'-lb'. Also, a-lb +c-'d = (ca)'(cb + ad), where ac-' = c`-la. The set is division ring called quotient ring of r and it is uniquely determined up to isomorphism. To apply this general theory to our case, we must verify that the above mentioned conditions of r are satisfied for D [Xi, . . . , Xm]. To see it, take D [xi] first. The above

Lattice-Ordered Rings of Quotients

R. E. Johnson (10), Utumi (18), and Findlayand Lambek (7) have defined for each ring R a unique maximal "ring of right quotients" Q. When R is a commutative integral domain (in this paper an integral domain need not be commutative) or an Ore domain, then Q is the usual division ring of quotients of R. Moreover, it is well known that in these special cases, if R is totally ordered, then so is Q.The main purpose of this paper is to study the ring of quotients Q, and in particular its order properties, for certain lattice-ordered rings R.

Self-Injective Rings

Historically, the first example of a ring of quotients was the quotient field of an integral domain. Later on, conditions were found under which a noncommutative integral domain has a quotient division ring. More recently, R.E. Johnson [4], Y. Utumi [5], and G.D. Findlay and J. Lambek [3] have discussed the existence and structure of a maximal ring of quotients of any ring.The present paper uses the methods of Findlay and Lambek to recast the results of Johnson on the quotient ring of a ring with zero singular ideal. It is also shown that such a ring has a unique left-right maximal ring of quotients.