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1.

Let

*I*,*H*,*S*,*P*be the usual class operators on universal algebras. For a class*K*of universal algebras of the same type, let*R*({*K*}) be the class of all algebras isomorphic to a retract of a member of*K*and let*R*denote the corresponding class operator. In this paper the semigroup generated by class operators*I*,*R*,*H*,*S*,*P*and the corresponding partially ordered set are described. Also the standard semigroups of the above operators are determined for some varieties. 相似文献2.

Boža Tasić 《Algebra Universalis》2009,62(4):351-365

Let

*I*,*H*,*S*,*P*_{ u },*P*_{ f }denote the following operators on classes of algebras of the same type:*I*,*H*for isomorphic and homomorphic images of algebras,*S*for subalgebras and*P*_{ u },*P*_{ f }for ultra and filtered products, respectively. In this paper, the monoid generated by the operators*H*,*S*,*P*_{ u },*P*_{ f }with*I*as an identity is described. It turns out that there are 44 different operators such that every composite of*H*,*S*,*P*_{ u },*P*_{ f }coincides with one of them (including the empty composite, the identity operator). 相似文献3.

The Kronecker product of two homogeneous symmetric polynomials

*P*_{1},*P*_{2}is defined by means of the Frobenius map by the formula*P*_{1}o*P*_{2}=*F*(*F*^{−1}*P*_{1})(*F*^{−1}*P*_{2}). When*P*_{1}and*P*_{2}are the Schur functions*S*_{ I },*S*_{ J }then the resulting product*S*_{ I }o*S*_{ J }is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagrams*I*and*J*. Taking the scalar product of*S*_{ I }o*S*_{ J }with a third Schur functions*S*_{ K }gives the so called Kronecker coefficient*c*_{ I,J,K }=<*S*_{ I }*oS*_{ J },*S*_{ K }>. In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for the*c*_{ I,J,K }in terms of some classes of permutations. In Lascoux's work*I*and*J*are restricted to be hooks and in Gessel's both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of <*S*_{ I }*oS*_{ J },*S*_{ K }> for*S*_{ I }a product of homogeneous symmetric functions and*J, K*unrestricted skew shapes. We also show how Gessel's and Lascoux's results are related and show how they can be derived from a special case of our result. Work supported by NSF grant at the University of California, San Diego. 相似文献4.

5.

Let Then (

*F*be a finite field of characteristic not 2, and*S**F*a subset with three elements. Consider the collection**S**={

*S*·

*a*+

*b*|

*a*,

*b*

*F*,

*a*≠0}.

*F*,**S**) is a simple 2-design and the parameter λ of (*F*,**S**) is 1, 2, 3 or 6. We find in this paper the full automorphism group of (*F*,**S**). Namely, if we put*U*={*r*| {0,1,*r*}**S**} and*K*the subfield of*F*generated by*U*, then the automorphisms of (*F*,**S**) are the maps of the form*x***g**(α(*x*))+*b*,*x**F*, where*b**F*, α :*F*→*F*is a field automorphism fixing*U*, and**g**is a linear transformation of*F*considered as a vector space over*K*. 相似文献6.

We describe Noetherian semigroup algebras

*K*[*S*] of submonoids*S*of polycyclic-by-finite groups over a field*K*. As an application, we show that these algebras are finitely presented and also that they are Jacobson rings. Next we show that every prime ideal*P*of*K*[*S*] is strongly related to a prime ideal of the group algebra of a subgroup of the quotient group of*S*via a generalised matrix ring structure on*K*[*S*]/*P*. Applications to the classical Krull dimension, prime spectrum, and irreducible*K*[*S*]-modules are given. 相似文献7.

Let

*A*be a central simple algebra over a field*F*. Denote the reduced norm of*A*over*F*by Nrd:*A** →*F** and its kernel by SL_{1}(*A*). For a field extension*K*of*F*, we study the first Galois Cohomology group*H*^{1}(*K*,SL_{1}(*A*)) by two methods, valuation theory for division algebras and*K*-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series. 相似文献8.

We obtain a family of eight-dimensional unital division algebras over a field

*F*out of a separable quadratic field extension*S*of*F*, a three-dimensional anisotropic hermitian form*h*over*S*of determinant one and an element*c*∈*S*^{×}not contained in*F*. These algebras are not third-power associative. 相似文献9.

10.

V. V. Kapustin 《Journal of Mathematical Sciences》2007,141(5):1538-1542

Let θ be an inner function, let

*K*_{ θ }=*H*^{2}⊖*θH*^{2}, and let S_{θ}: K_{θ}→ S_{θ}be defined by the formula S_{θ}f = P_{θ}zf, where f ∈ K_{θ}is the orthogonal projection of H^{2}onto K_{θ}. Consider the set A of all trace class operators L : K_{θ}→ K_{θ},*L*= ∑(·,u_{n})v_{n}, ∑∥u_{n}∥∥v_{n}∥ < ∞ (u_{n}, v_{n}∈ K_{θ}), such that ∑ū_{n}v_{n}∈*H*_{0}^{1}. It is shown that trace class commutators of the form XS_{θ}− S_{θ}X (where X is a bounded linear operator on K_{θ}) are dense in A in the trace class norm. Bibliography: 2 titles. __________ Translated from*Zapiski Nauchnykh Seminarov POMI*, Vol. 333, 2006, pp. 54–61. 相似文献11.

Let

*K*be a field of characteristic zero. For a torsion-free finitely generated nilpotent group*G*, we naturally associate four finite dimensional nilpotent Lie algebras over*K*, ?_{ K }(*G*), grad^{(?)}(?_{ K }(*G*)), grad^{(g)}(exp ?_{ K }(*G*)), and*L*_{ K }(*G*). Let 𝔗_{ c }be a torsion-free variety of nilpotent groups of class at most*c*. For a positive integer*n*, with*n*≥ 2, let*F*_{ n }(𝔗_{ c }) be the relatively free group of rank*n*in 𝔗_{ c }. We prove that ?_{ K }(*F*_{ n }(𝔗_{ c })) is relatively free in some variety of nilpotent Lie algebras, and ?_{ K }(*F*_{ n }(𝔗_{ c })) ?*L*_{ K }(*F*_{ n }(𝔗_{ c })) ? grad^{(?)}(?_{ K }(*F*_{ n }(𝔗_{ c }))) ? grad^{(g)}(exp ?_{ K }(*F*_{ n }(𝔗_{ c }))) as Lie algebras in a natural way. Furthermore,*F*_{ n }(𝔗_{ c }) is a Magnus nilpotent group. Let*G*_{1}and*G*_{2}be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if*G*_{1}and*G*_{2}are relatively free of finite rank, then they are isomorphic. Let*L*be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set*X*. Give on*L*the structure of a group*R*, say, by means of the Baker–Campbell–Hausdorff formula, and let*H*be the subgroup of*R*generated by the set*X*. We show that*H*is relatively free in some variety of nilpotent groups; freely generated by the set*X*,*H*is Magnus and*L*? ?_{?}(*H*) ?*L*_{?}(*H*) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ?_{ K }and*L*_{ K }are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group*G*, not relatively free, such that ?_{?}(*G*) is not isomorphic to*L*_{?}(*G*) as Lie algebras. 相似文献12.

13.

B. I. Plotkin 《Journal of Mathematical Sciences》2006,139(4):6780-6791

Let Θ be a variety of algebras. In every variety Θ and every algebra

*H*from Θ one can consider algebraic geometry in Θ over*H*. We also consider a special categorical invariant*K*_{Θ}of this geometry. The classical algebraic geometry deals with the variety Θ = Com-*P*of all associative and commutative algebras over the ground field of constants*P*. An algebra*H*in this setting is an extension of the ground field*P*. Geometry in groups is related to the varieties Grp and Grp-*G*, where*G*is a group of constants. The case Grp-*F*, where*F*is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras*H*_{1}and*H*_{2}have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1)*K*_{Θ}(*H*_{1}) and*K*_{Θ}(*H*_{2}) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ^{0}be the category of all algebras*W*=*W*(*X*) free in Θ, where*X*is finite. Consider the groups of automorphisms Aunt(Θ^{0}) for different varieties Θ and also the groups of autoequivalences of Θ^{0}. The problem is to describe these groups for different Θ. 相似文献14.

Miriam Cohen 《代数通讯》2013,41(12):4618-4633

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of

*H*as irreducible left*D*(*H*)-modules. For quasitriangular semisimple Hopf algebras*H*, we prove that the product of two class sums is an integral combination of the class sums up to*d*^{?2}where*d*= dim*H*. We show also that in this case the character table is obtained from the**S**-matrix associated to*D*(*H*). Finally, we calculate explicitly the generalized character table of*D*(*kS*_{3}), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums. 相似文献15.

B. V. Karpov 《Mathematical Notes》1998,64(5):600-606

Let

*S*be a smooth projective surface, let*K*be the canonical class of*S*and let*H*be an ample divisor such that*H • K*< 0. We prove that for any rigid sheaf*F*(Ext^{1}(*F, F*) = 0) that is Mumford-Takemoto semistable with respect to*H*there exists an exceptional set (*E*_{ 1 },..., E_{ n }) of sheaves on*S*such that*F*can be constructed from {*E*_{ i }} by means of a finite sequence of extensions. Translated from*Matematicheskie Zametki*, Vol. 64, No. 5, pp. 692–700, November, 1998. The author wishes to express his gratitude to S. A. Kuleshov for useful discussions and to A. N. Rudakov and A. L. Gorodentsev for their attention to the present work. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01323 and by the INTAS Foundation. 相似文献16.

Kevin Hutchinson 《K-Theory》1990,4(2):181-200

We give a proof of Matsumoto's theorem on

*K*_{2}of a field using techniques from homological algebra. By considering a complex associated to the action of GL(2,*F*) on*P*_{1}(*F*) (*F*a field), we derive the Matsumoto presentation for*H*_{0}(*F*^{.},*H*_{2}(SL(2,*F*))) and, by considering the action of GL(*n + 1, F*) on**P**^{ n }(F), we prove the stability part of the theorem; namely, that*H*_{0}(*F*^{.},*H*_{2}(SL(2,*F*))) is isomorphic to*H*_{2}(SL(*F*)) =*K*_{2}(F). 相似文献17.

Christian Le Merdy 《Integral Equations and Operator Theory》2000,37(1):72-94

This paper is devoted to dual operator algebras, that is

*w*^{*}-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebras*A*with the following weak^{*}similarity property: for every Hilbert space*H*, any*w*^{*}-continuous unital homomorphism from*A*into*B(H)*is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras, in analogy with some recent work of Pisier on the similarity problem on operator algebras. 相似文献18.

Let

*σ*be an automorphism of a field*K*with fixed field*F*. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras*K*[*t*;*σ*]∕*fK*[*t*;*σ*] obtained when the twisted polynomial*f*∈*K*[*t*;*σ*] is invariant, and were first defined by Petit. We compute all their automorphisms if*σ*commutes with all automorphisms in*Aut*_{F}(*K*) and*n*≥*m*?1, where*n*is the order of*σ*and*m*the degree of*f*, and obtain partial results for*n*<*m*?1. In the case where*K*∕*F*is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over*F*. We also briefly investigate when two such algebras are isomorphic. 相似文献19.

20.

Palle E. T. Jorgensen 《Israel Journal of Mathematics》1986,56(2):129-142

Let

*H*be an infinite-dimensional separable Hilbert space, and let*S*=(*S*_{ij})∈te*B(H)*⊗*M*_{2}be a unitary 2 × 2 matrix with operator entries. We study the*C**-algebra generated by the operators*S*_{ij}, and show that the study of unitary dilations of isometries*T*in*H*reduces to the special case where*S*_{11}=*T*, and*S*_{21}= 0. We use*C**-algebraic techniques to obtain detailed results about the set of all unitary dilations of*T*. Work supported in part by NSF. 相似文献