Let S be a nonassociative k-algebra. By using the Lie triple system, we study the subspace I2（S） of the Steinberg Lie algebra st2（S） and give a necessary and sufficient condition for I2（S） = 0.
Sharpe once gave a beautiful 4-term decomposition LPLU for infinite Steinberg group; recently Li Fu-an has given a sufficient and necessary condition for 4-term decomposition LWLU of finite Steinberg group over commut...Sharpe once gave a beautiful 4-term decomposition LPLU for infinite Steinberg group; recently Li Fu-an has given a sufficient and necessary condition for 4-term decomposition LWLU of finite Steinberg group over commutative rings. The present note extends the result of [2] to non-commutative rings and studies the decomposition of a finite Steinberg group over local rings.展开更多
Let A be an associative ring with identity, St_n(A) the Steinberg group of dimension n over A. The stable Steinberg group is the direct limit of the St_n(A) in a natural way. Several authors studied various normal for...Let A be an associative ring with identity, St_n(A) the Steinberg group of dimension n over A. The stable Steinberg group is the direct limit of the St_n(A) in a natural way. Several authors studied various normal forms for elements of St(A) and St_n(A) (cf.[ 3—7]).展开更多
1 PreliminaryIT is a classical problem in the research of classical groups to represent an element of a matrixgroup as a product of matrices of a special nature (such as involution and commutator) and todetermine the ...1 PreliminaryIT is a classical problem in the research of classical groups to represent an element of a matrixgroup as a product of matrices of a special nature (such as involution and commutator) and todetermine the smallest number of the factors in the representation. We have known that everyelernent of SL_nF(= E_nF), the special linear group over a field, can be written as a product展开更多
Abstract The Iwasava decomposition is proved for the Steinberg groups of types 2 A 2l?1, 2 D l , 2 E 6, 3 D 4 over the field of fractions of a principal ideal ring. By using this decomposition, it is described that su...Abstract The Iwasava decomposition is proved for the Steinberg groups of types 2 A 2l?1, 2 D l , 2 E 6, 3 D 4 over the field of fractions of a principal ideal ring. By using this decomposition, it is described that subgroups exist between the Steinberg groups over the rings D and K under some restrictions on the ring D.展开更多
Consider the stable Steinberg group St（K） over a skew field K. An element x is called an involution if x^2 = 1. In this paper, an involution is allowed to be the identity. The authors prove that an element A of GLn...Consider the stable Steinberg group St（K） over a skew field K. An element x is called an involution if x^2 = 1. In this paper, an involution is allowed to be the identity. The authors prove that an element A of GLn（K） up to conjugation can be represented as BC, where B is lower triangular and C is simultaneously upper triangular. Furthermore. B and C can be chosen so that the elements in the main diagonal of B are β1,β2,…,βn,and of C are γ1,γ2,…,γncn,where cn∈[K^n,K^n] and ∏j=1^n βjγj=det A,it is Also proved that every element δin St（K） is a product of 10 involutions.展开更多
The present paper contains two interrelated developments. First, the basic properties of the construction theory over the Steinberg Lie color algebras are developed in analogy with Steinberg Lie algebra case. This is ...The present paper contains two interrelated developments. First, the basic properties of the construction theory over the Steinberg Lie color algebras are developed in analogy with Steinberg Lie algebra case. This is done on the example of the central closed of the Steinberg Lie color algebras. The second development is that we define the first ε-cyclic homology group HC1 （R, ε） of the F-graded associative algebra R （which could be seemed as the generalization of cyclic homology group and the Z/2Z-graded version of cyclic homology that was introduced by Kassel） to calculate the universal central extension of Steinberg Lie color algebras.展开更多
基金The second author is supported in part by the National Natural Science Foundation of China （11101387 and 10971104）, the Anhui Provincial Natural Science Foundation （1208085MA01） and the Fundamental Research Funds for the Central Universities （WK 0010000023）. The third author is supported in part by NSERC of Canada and Chinese Academy of Science.
文摘Let S be a nonassociative k-algebra. By using the Lie triple system, we study the subspace I2（S） of the Steinberg Lie algebra st2（S） and give a necessary and sufficient condition for I2（S） = 0.
文摘Sharpe once gave a beautiful 4-term decomposition LPLU for infinite Steinberg group; recently Li Fu-an has given a sufficient and necessary condition for 4-term decomposition LWLU of finite Steinberg group over commutative rings. The present note extends the result of [2] to non-commutative rings and studies the decomposition of a finite Steinberg group over local rings.
基金Project supported by the National Natural Science Foundation of China.
文摘Let A be an associative ring with identity, St_n(A) the Steinberg group of dimension n over A. The stable Steinberg group is the direct limit of the St_n(A) in a natural way. Several authors studied various normal forms for elements of St(A) and St_n(A) (cf.[ 3—7]).
文摘1 PreliminaryIT is a classical problem in the research of classical groups to represent an element of a matrixgroup as a product of matrices of a special nature (such as involution and commutator) and todetermine the smallest number of the factors in the representation. We have known that everyelernent of SL_nF(= E_nF), the special linear group over a field, can be written as a product
文摘Abstract The Iwasava decomposition is proved for the Steinberg groups of types 2 A 2l?1, 2 D l , 2 E 6, 3 D 4 over the field of fractions of a principal ideal ring. By using this decomposition, it is described that subgroups exist between the Steinberg groups over the rings D and K under some restrictions on the ring D.
基金Project supported by the Key Project of the Ministry of Education of China (No. 03060).
文摘Consider the stable Steinberg group St（K） over a skew field K. An element x is called an involution if x^2 = 1. In this paper, an involution is allowed to be the identity. The authors prove that an element A of GLn（K） up to conjugation can be represented as BC, where B is lower triangular and C is simultaneously upper triangular. Furthermore. B and C can be chosen so that the elements in the main diagonal of B are β1,β2,…,βn,and of C are γ1,γ2,…,γncn,where cn∈[K^n,K^n] and ∏j=1^n βjγj=det A,it is Also proved that every element δin St（K） is a product of 10 involutions.
文摘The present paper contains two interrelated developments. First, the basic properties of the construction theory over the Steinberg Lie color algebras are developed in analogy with Steinberg Lie algebra case. This is done on the example of the central closed of the Steinberg Lie color algebras. The second development is that we define the first ε-cyclic homology group HC1 （R, ε） of the F-graded associative algebra R （which could be seemed as the generalization of cyclic homology group and the Z/2Z-graded version of cyclic homology that was introduced by Kassel） to calculate the universal central extension of Steinberg Lie color algebras.
