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1.
Let be a finitely generated non-PI Ore domain and the quotient division algebra of . If is the center of , then .
2.
S. P. Smith 《Israel Journal of Mathematics》1983,46(1-2):33-39
LetR be a factor ring of the enveloping algebra of a finite dimensional Lie algebra over a fieldk. If the centre ofR, Z, consists of non-zero divisors inR, the ringR
z
obtained by localizing at the non-zero elements ofZ becomes a finitely generated algebra over the fieldK which arises as the field of fractions ofZ. The Gelfand-Kirillov dimension of anR-moduleM is denotedd(M). In this paper it is shown that ifR
Z
⊗
R
M ≠ 0 thend(M) ≧d(R
Z
⊗
R
M) + tr. deg
k
Z, whered (R
z
⊗M) is the Gelfand-Kirillov dimension ofR
z
⊗M) viewed as anR
z
-module andR
z
is viewed as a finitely generatedK-algebra (not as ak-algebra). The result is primarily of a technical nature. 相似文献
3.
4.
5.
6.
We prove that every finitely generated Lie algebra containing a nilpotent ideal of class and finite codimension has Gelfand-Kirillov dimension at most . In particular, finitely generated virtually nilpotent Lie algebras have polynomial growth.
7.
C. Martinez 《Transactions of the American Mathematical Society》1996,348(1):119-126
In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.
8.
During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring(in the dual pair setting) and is inspired by Wang's formula for the dimension of a minimal nilpotent orbit. 相似文献
9.
Quanshui Wu 《Israel Journal of Mathematics》1991,73(3):289-296
LetF ⊂K be a field extension,A be aK-algebra. It is proved that, in general, GK dim
F
A≥GK dim
K
A+tr
F
(K). For commutative algebras or Noetherian P.I. algebras, the equality holds. Two examples are also constructed to show that:
(i) there exists an algebraA such that GK dim
F
A=GK dim
K
A+tr
F
(K)+1; (ii) there exists an algebraic extensionF ⊂K and aK-algebraA such that GK dim
F
A=∞, but GK dim
K
A<∞. 相似文献
10.
In this paper, the growth and Gelfand-Kirillov dimension of some primitive abundant semigroups are investigated. It is shown that for certain primitive abundant (regular) semigroup S, S as well as the semigroup algebra K [S] has polynomial growth if and only if all of its cancellative submonoids (subgroups) T as well as K[T] have polynomial growth. As applications, it is shown that if S is a finitely generated primitive inverse monoid having the permutational property, then clK dim K[S] = GK dim K[S] = rk(S). 相似文献
11.
Let R be a finitely generated algebra over a field of characteristic 0 with a locally nilpotent derivation δ ≠ 0. We show that if {ie313-1}, where the invariants {ie313-2} are prime and satisfy a polynomial identity, then {ie313-3}. Furthermore, when R is a domain, the same conclusion holds without the assumption that R is finitely generated. This enables us to obtain a result on skew polynomial rings. These results extend work of Bell and Smoktunowicz on domains with GK dimension in the interval [2, 3). 相似文献
12.
Previous work (Pradines, C. R. Acad. Sci. Paris 263 (1966) 907; Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the 2-dimensional information. 相似文献
13.
Ferran Cedó Eric Jespers Jan Okninski 《Proceedings of the American Mathematical Society》2006,134(3):653-663
We consider algebras over a field presented by generators and subject to square-free relations of the form with every monomial , appearing in one of the relations. It is shown that for 1$"> the Gelfand-Kirillov dimension of such an algebra is at least two if the algebra satisfies the so-called cyclic condition. It is known that this dimension is an integer not exceeding . For , we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators has Gelfand-Kirillov dimension if and only if it is of -type, and this occurs if and only if the multiplicative submonoid generated by is cancellative.
14.
Agata Smoktunowicz 《Journal of Pure and Applied Algebra》2007,209(3):839-851
It is shown that for every countable field K, there is a finitely generated graded Jacobson radical algebra over K of Gelfand-Kirillov dimension two. Examples of finitely generated Jacobson radical algebras of Gelfand-Kirillov dimension two over algebraic extensions of finite fields of characteristic 2 were earlier constructed by Bartholdi [L. Bartholdi, Branch Rings, thinned rings, tree enveloping rings, Israel J. Math. (in press)]. 相似文献
15.
Sérgio Mota Alves 《Rendiconti del Circolo Matematico di Palermo》2009,58(1):109-124
The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic p > 2 little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic;
we showed that the so-called Tensor Product Theorem cannot be extended for infinite fields of positive characteristic p > 2. Furthermore we studied the Gelfand-Kirillov dimension of the relatively free algebras of verbally prime and related
algebras. In this paper we compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the
algebras M
a,a
(E) ⊗ E and M
2a
(E) are not PI equivalent in characteristic p > 2. Furthermore we show that the following algebras are not PI equivalent in positive characteristic: M
a,b
(E) ⊗ M
c,d
(E) and M
ac+bd,ad+cb
(E); and M
a,b
(E) ⊗ M
c,d
(E) and M
e, f
(E) ⊗ M
g,h
(E) when a ≥ b, c ≥ d, e ≥ f, g ≥ h, ac + bd = eg+ f h, ad +bc = eh + fg and ac ≠ eg. Here E stands for the infinite dimensional Grassmann algebra with 1, and M
a,b
(E) is the subalgebra of M
a+b
(E) of the block matrices with blocks a × a and b × b on the main diagonal with entries from E
0, and off-diagonal entries from E
1; E = E
0 ⊗ E
1 is the natural grading on E.
Partially supported by CNPq 620025/2006-9. This paper was written during the author’s PhD study at the UNICAMP, under the
supervision of P.Koshlukov, to whom he expresses his sincere thanks. 相似文献
16.
The famous 1960's construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not locally nilpotent, algebra which has polynomially bounded growth.
17.
Chen-bo Zhu 《Proceedings of the American Mathematical Society》1998,126(10):3125-3130
Let be the reductive dual pair . We show that if is a representation of (respectively ) obtained from duality correspondence with some representation of (respectively ), then its Gelfand-Kirillov dimension is less than or equal to
(respectively ).
(respectively ).
18.
19.
G. Berhuy 《Advances in Mathematics》2005,198(1):128-171
We define and study a numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in Pn-1. 相似文献
20.
Z. Reichstein 《Transformation Groups》2000,5(3):265-304
We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our groupG isS
n
, these objects are field extensions; ifG=O
n
, they are quadratic forms; ifG=PGL
n
, they are division algebras (all of degreen); ifG=G
2, they are octonion algebras; ifG=F
4, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, relating essential dimension to Galois cohomology.Partially supported by NSA grant MDA904-9610022 and NSF grant DMS-9801675 相似文献