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1.
This paper introduces the concept ofn-valued groups and studies their algebraic and topological properties. We explore a number of examples. An important class consists of those that we calln-coset groups; they arise as orbit spaces of groupsG modulo a group of automorphisms withn elements. However, there are many examples that do not arise from this construction. We see that the theory ofn-valued groups is distinct from that of groups with a given automorphism group. There are natural concepts of the action of ann-valued group on a space and of a representation in an algebra of operators. We introduce the (purely algebraic) notion of ann-Hopf algebra and show that the ring of functions on ann-valued group and, in the topological case, the cohomology has ann-Hopf algebra structure. The cohomology algebra of the classifying space of a compact Lie group admits the structure of ann-Hopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also ann-Hopf algebra. In general the group ring of ann-valued group is not ann-Hopf algebra but it is for ann-coset group constructed from an abelian group. Using the properties ofn-Hopf algebras we show that certain spaces do not admit the structure of ann-valued group and that certain commutativen-valued groups do not arise by applying then-coset construction to any commutative group. 相似文献
2.
Automorphism groups of Weyl-type algebras 总被引:2,自引:0,他引:2
Let F be a field of characteristic 0, be n commuting variables over F, and be the field of all rational functions. Let . We have the simple Weyl type algebra . In this paper, the automorphism group of the associative algebra and the automorphism group of the Lie algebra are determined,
and it is proved that .
Received: 4 October 2001 / Revised version: 5 November 2001 相似文献
3.
Stephen P. Humphries 《代数通讯》2013,41(4):1233-1242
In this paper we show that the braid groups B n and the symmetric automorphism groups H(n) of the free group F n,n = 3,4 act in a non-linear way on an algebra with straightening law (ordinal Hodge algebra). We indicate various properties of these rings. 相似文献
4.
M. Katherine Brandl 《Algebras and Representation Theory》2006,9(3):241-258
We examine families of twists by an automorphism of the complex polynomial ring on n generators. The multiplication in the twisted algebra determines a Poisson structure on affine n-space. We demonstrate that if the automorphism has a single eigenvalue, then the primitive ideals in the twist are parameterized by the algebraic symplectic leaves associated to this Poisson structure. Furthermore, in this case all of the leaves are algebraic and can be realized as the orbits of an algebraic group.
Presented by K. Goodearl 相似文献
5.
6.
Dmitri Nikshych 《Advances in Mathematics》2002,170(2):257-286
We study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford's formula for the fourth power of the antipode S, which implies that the antipode has a finite order modulo, a trivial automorphism. We find a sufficient condition in terms of Tr(S2) for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple. 相似文献
7.
The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations
that only depends on the path graph A
n − 1 on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A
n − 1, D
n
, E6, E7, E8). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically
semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical
type. 相似文献
8.
A. S. Khoroshkin 《Journal of Mathematical Sciences》2007,143(1):2816-2830
In this paper, we consider the infinite-dimensional Lie algebra Wn⋉g⊗O
n of formal vector fields on the n-dimensional plane which is extended by formal g-valued functions of n variables. Here g is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of
the Weyl algebra of (gl
n ⊕ g) by the (2n+1)st term of the standard filtration. We consider separately the case of a reductive Lie algebra g. We show how one can use the methods of formal geometry to construct characteristic classes of bundles. For every G-bundle on an n-dimensional complex manifold, we construct a natural homomorphism from the ring A of relative cohomologies
of the Lie algebra Wn⋉g⊗O
n to the ring of cohomologies of the manifold. We show that generators of the ring A are mapped under this homomorphism to
characteristic classes of tangent and G-bundles. Bibliography: 10 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 205–230. 相似文献
9.
We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation
of Hodges’ result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree
2, in terms of isomorphisms of quantum tori, inspired by similar considerations in noncommutative differential geometry. We
study how far this link can be generalized for n ≥ 3. 相似文献
10.
A. T. Gainov 《Algebra and Logic》2002,41(1):30-38
We study NQM algebras A having an orthogonal automorphism of finite order n 3 (called Z
n
-orthograded NQM algebras). The Z
3-orthograded NQM algebras of dimension 7 are treated in more detail. In particular, we find all algebras A which are not bi-isotropic in this class, and for every algebra A, determine an automorphism group Aut,A and an orthogonal automorphism group Ortaut,A. In constructing and classifying (up to isomorphism) NQM algebras, use is made of orthogonal decompositions of the algebras. 相似文献
11.
Xing Tao Wang 《代数通讯》2013,41(4):1133-1140
Over a 2-torsionfree commutative ring R with identity, the algebra of all strictly upper triangular n + 1 by n + 1 matrices is denoted by n 1. In this article, we prove that any Jordan automorphism of n 1 can be uniquely decomposed as a product of a graph automorphism, a diagonal automorphism, a central automorphism and an inner automorphism for n ≥ 3. In the cases n = 1, 2, we also give a decomposition for any Jordan automorphism of n 1. 相似文献
12.
Michael Peretzian Williams 《代数通讯》2013,41(6):1843-1849
We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I 2 nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent. 相似文献
13.
Thiago Fassarella 《Bulletin of the Brazilian Mathematical Society》2011,42(2):321-330
We show that if a Laurent polynomial on the coordinate ring of the complex algebraic torus on n variables has vanishing logarithmic Hessian, then up to an automorphism of the torus, the Laurent polynomial depends on at
most n−1 variables. 相似文献
14.
Yi Zhong 《数学学报(英文版)》1997,13(4):433-442
In this paper we prove that under some natural conditions, the Ore extensions and skew Laurent polynomial rings are injectively
homogeneous or homologically homogeneous if so are their coefficient rings. Specifically, we prove that ifR is a commutative Noetherian ring of positive characteristic, thenA
n (R), then
th Weyl algebra overR, is injectively homogeneous (resp. homologically homogeneous) ifR has finite injective dimension (resp. global dimension). 相似文献
15.
Paul Levy 《Transformation Groups》2009,14(2):417-461
We generalize the basic results of Vinberg’s θ-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify
the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated
to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded
Lie algebra (in zero or odd positive characteristic), confirming a conjecture of Popov in this case. 相似文献
16.
R.M. Green 《Journal of Algebraic Combinatorics》2002,15(3):241-252
We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A
n – 1 by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of W (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations.Using Shi's characterization of the Kazhdan–Lusztig cells in the group W, we use our main result to show that the fully commutative elements of W form a union of Kazhdan–Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan–Lusztig basis of the associated Hecke algebra to be computed combinatorially.We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to GL
n
() 相似文献
17.
Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ø, with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring Mm(D(A ø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c?Asuch that D(A[c-1];ø)?M m(D(A[c-1]ø)). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism. 相似文献
18.
S. A. Amitsur 《Israel Journal of Mathematics》1971,10(2):210-211
The classical result that an algebra which satisfies a polynomial identity satisfies a powerS
2n
[x]
m
=0 of the standard identity, is generalized to arbitrary rings. 相似文献
19.
The Birman–Murakami–Wenzl algebra (BMW algebra) of type D n is shown to be semisimple and free of rank (2 n + 1)n!! ? (2 n?1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n ? 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ?[δ±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D n is a subalgebra of the BMW algebra of the same type. 相似文献
20.
《代数通讯》2013,41(9):2957-2975
ABSTRACT Let F m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) G is not finitely generated. 相似文献