共查询到20条相似文献,搜索用时 23 毫秒
1.
Charles Lanski 《代数通讯》2013,41(5):1427-1446
ABSTRACT Let D be a division algebra with center F. Consider the group CK 1(D) = D*/F*D′ where D* is the group of invertible elements of D and D′ is its commutator subgroup. In this note we shall show that, assuming a division algebra D is a product of cyclic algebras, the group CK 1(D) is trivial if and only if D is an ordinary quaternion algebra over a real Pythagorean field F. We also characterize the cyclic central simple algebras with trivial CK 1 and show that CK 1 is not trivial for division algebras of index 4. Using valuation theory, the group CK 1(D) is computed for some valued division algebras. 相似文献
2.
George Szeto 《代数通讯》2013,41(12):3979-3985
Let B be a Galois algebra over a commutative ring R with Galois group G such that B H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V B (A) = ?∑ g∈G(A) J g , and the centers of A and B G(A) are the same where V B (A) is the commutator subring of A in B, J g = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}. 相似文献
3.
Let ? be a ring containing a nontrivial idempotent. In this article, under a mild condition on ?, we prove that if δ is a Lie triple derivable mapping from ? into ?, then there exists a Z A, B (depending on A and B) in its centre 𝒵(?) such that δ(A + B) = δ(A) + δ(B) + Z A, B . In particular, let ? be a prime ring of characteristic not 2 containing a nontrivial idempotent. It is shown that, under some mild conditions on ?, if δ is a Lie triple derivable mapping from ? into ?, then δ = D + τ, where D is an additive derivation from ? into its central closure T and τ is a mapping from ? into its extended centroid 𝒞 such that τ(A + B) = τ(A) + τ(B) + Z A, B and τ([[A, B], C]) = 0 for all A, B, C ∈ ?. 相似文献
4.
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 SL1(A). For a field extension K of F, we study the first Galois Cohomology group H 1(K,SL1(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
5.
For an Azumaya algebra A which is free over its centre R, we prove that K-theory of A is isomorphic to K-theory of R up to its rank torsions. We conclude that K i (A, ?/m) = K i (R, ?/m) for any m relatively prime to the rank and i ≥ 0. This covers, for example, K-theory of division algebras, K-theory of Azumaya algebras over semilocal rings, and K-theory of graded central simple algebras indexed by a totally ordered abelian group. 相似文献
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7.
Compact Exceptional Simple Kantor Triple Systems Defined on Tensor Products of Composition Algebras∗
Daniel Mondoc 《代数通讯》2013,41(11):3699-3712
In this article we give the classification of compact exceptional simple Kantor triple systems defined on tensor products of composition algebras A = 1? 2 such that their Kantor algebras ?(φ, A) are real forms of exceptional simple Lie algebras. 相似文献
8.
We show that if A is a representation-finite selfinjective Artin algebra, then every P ? ? K b(𝒫 A ) with Hom K(Mod?A)(P ?,P ?[i]) = 0 for i ≠ 0 and add(P ?) = add(νP ?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B 0, B 1,…, B m = B such that, for any 0 ≤ i < m, B i+1 is the endomorphism algebra of a tilting complex for B i of length ≤ 1. 相似文献
9.
Gábor P. Nagy 《代数通讯》2013,41(2):549-555
ABSTRACT Let (A, ?) be a structurable algebra. Then the opposite algebra (A op , ?) is structurable, and we show that the triple system B op A(x, y, z):=Vopx,y(z)=x(y¯z)+z(y¯x)?y(x¯z), x, y, z ∈ A, is a Kantor triple system (or generalized Jordan triple system of the second order) satisfying the condition (A). Furthermore, if A=𝔸1?𝔸2 denotes tensor products of composition algebras, (?) is the standard conjugation, and (∧) denotes a certain pseudoconjugation on A, we show that the triple systems B op 𝔸1?𝔸2 ( x , y¯∧, z) are models of compact Kantor triple systems. Moreover these triple systems are simple if (dim𝔸1, dim𝔸2) ≠ (2, 2). In addition, we obtain an explicit formula for the canonical trace form for compact Kantor triple systems defined on tensor products of composition algebras. 相似文献
10.
Karolina Mroczyńska 《代数通讯》2013,41(3):916-934
The paper investigates the following problem. Let bimodules N, M yield a stable equivalence of Morita type between self-injective K-algebras A and E. Further, let bimodules S, T yield a stable equivalence of Morita type between self-injective K-algebras B and F. Then we want to know whether the functor M ? A ? ? B S: mod(A ? K B op ) → mod(E ? K F op ) induces a stable equivalence between A ? K B op and E ? K F op . There is given a reduction of this problem to some smaller subcategories for self-injective algebras. Moreover, new invariants of stable equivalences of Morita type are constructed in a general case of arbitrary finite-dimensional algebras over a field. 相似文献
11.
Mariam Imtiaz 《代数通讯》2013,41(8):3095-3112
Abstract Let R = K[y 1,…,y t ] be an affine domain over a field K and I be a nonzero proper ideal of R. In Sec. 1 of this note, we characterize when (K + I, R) is a Mori pair. In Sec. 2 of this note, we prove the following theorem: Let A ? B be domains such that C/Q is Mori for each subring C of B containing A and for any prime ideal Q of C. Then dim A ? 1 ≤ dim B ≤ dim A + 1 and if dim A > 1 or dim B > 1 then dim A = dim B. 相似文献
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The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB n contains an ideal K n of dimension (2 n ? 1)2, and we construct an explicit isomorphism of K n with the matrix algebra M 2 n ?1(F). 相似文献
14.
George Voutsadakis 《代数通讯》2013,41(8):3093-3112
Let ? = ?F, R, ρ? be a system language. Given a class of ?-systems K and an ?-algebraic system A = ?SEN,?N,F??, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K A of A-systems in K as the collection of all members of K of the form 𝔄 = ? SEN,?N,F?,R 𝔄 ?, for some set of relation systems R 𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K , on the one hand, and local properties of K A and global properties connecting K A and K A′, whenever there exists an ?-morphism ? F,α? : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K A is an algebraic closure system, for every ?-algebraic system A, provided that K is closed under subsystems and reduced products. 相似文献
15.
Guangquan Guo 《代数通讯》2013,41(6):2269-2280
In this article, the notions of a Frobenius pair of functors and Frobenius corings are generalized to an l-QF pair of functors and l-QF corings. We prove that an extension ι:B → A is left quasi-Frobenius if and only if (F 1,G 1) is an l-QF pair of functors, where F 1: A ? → B ? is the restriction of scalars functors, and G 1 = A? B ? : B ? → A ? is the induction functor. For an A-coring , we prove that is an l-QF coring if and only if A → ? is an l-QF extension and A is a finitely generated projective modules if and only if (G 2,F 2) is an l-QF pair of functors, where G 2 = ? A ? : A ? → ? is the induction functor, F 2: ? → A ? is the forgetful functor, the result of Brzezinski is generalized. 相似文献
16.
Let K be an algebraically closed field of arbitrary characteristic and Γ an abelian multiplicative group equipped with a bicharacter ε: Γ × Γ → K*. It is proved that, for any finite-dimensional derivation simple color algebra A over K, there exists a simple color algebra S and a color vector space V such that A? S? Sε(V), where Sε(V) is the ε-symmetric algebra of V. As an application of this result, a necessary and sufficient condition such that a Lie color algebra is semisimple is obtained. 相似文献
17.
J. Cimprič 《代数通讯》2013,41(1):165-178
An involution # on an associative ring R is formally real if a sum of nonzero elements of the form r # r where r ? R is nonzero. Suppose that R is a central simple algebra (i.e., R = M n (D) for some integer n and central division algebra D) and # is an involution on R of the form r # = a ?1 r? a, where ? is some transpose involution on R and a is an invertible matrix such that a? = ±a. In Section 1 we characterize formal reality of # in terms of a and ?| D . In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on D = (K/F, Φ) that extend to a formally real involution on the split algebra D ? F K ? M n (K). Every such involution is formally real but we show that there exist formally real involutions on D which are not of this form. In particular, there exists a formally real involution # for which the hermitian trace form x ? tr(x # x) is not positive semidefinite. 相似文献
18.
《代数通讯》2013,41(5):2357-2379
Abstract Restrictions of irreducible representations of classical algebraic groups to root A 1-subgroups, i.e., subgroups of type A 1 generated by root subgroups associated with two opposite roots, are studied. Composition factors of such restrictions are found in the following cases: for groups of types A n with n > 2 and D n , for groups of type B n , n > 2, and long root subgroups, for groups of type C n , n > 2, and short root subgroups, and for p-restricted representations of A 2(K), C 2(K) (recall that B 2(K) ? C 2(K)), and of B n (K), n > 2, and short root subgroups. Here we assume that p > 2 for G = B n (K) or C n (K). 相似文献
19.
《代数通讯》2013,41(10):5071-5094
Abstract Ternary derivations, ternary Cayley derivations and ternary automorphisms are computed over fields of characteristic ≠ 2, 3 for the algebras A t obtained by the Cayley–Dickson duplication process. While the derivation algebra of A t stops growing after t = 3, the ternary derivation algebra significantly decreases in the step from the octonions A 3 to the sedenions A 4, revealing the symmetry lost on that stage. 相似文献
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