Superalgebra Structures on    in Characteristic 2

Boboc, Crina

doi:10.1081/agb-120006489

Cited by 6 publications

(8 citation statements)

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“…The first one relies on a result of [7] concerned to graded Clifford theory, while the second one follows from an exact sequence of [2] concerned to graded Brauer groups. This recovers and explains results of [4] in characteristic different from 2, and of [3] in characteristic 2. Gradings of matrix algebras by the cyclic group C n have been investigated in [4] with a different approach under the assumption that the field k contains a primitive nth root of unity.…”

Section: Introductionsupporting

confidence: 87%

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On Gradings of Matrix Algebras and Descent Theory

Caenepeel¹,

Dăscălescu

Năstăsescu

2002

Communications in Algebra

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Section: Introductionsupporting

confidence: 87%

“…Using direct computations, these gradings have been described before in [7] , and classified up to isomorphism in [3] .…”

Section: Now Definementioning

confidence: 99%

On Gradings of Matrix Algebras and Descent Theory

Caenepeel¹,

Dăscălescu

Năstăsescu

2002

Communications in Algebra

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“…In particular all the isomorphism types of C 2 -gradings on M 2 {k) were obtained in [4] in the case where char(fc) ^ 2. For char(fe) = 2, the classification has been completed in [2]. We note that in the case where k is algebraically closed, any C m -grading on M n (k) is isomorphic to a good grading (see [5,10]).…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

“…Now again A = A x ® A C C A^® H implies that A c = H' and we obtain the grading of type (iii')-0 (i) The gradings in (I)(iii) are classified by the factor group k*/(k*) 2 . Indeed, it was proved in [4,5] that two gradings like these, corresponding to 6i, 62 S A; -k 2 , are isomorphic if and only if 61/62 6 k 2 .…”

Section: •W4mentioning

confidence: 90%

“…Indeed, it was proved in [4,5] that two gradings like these, corresponding to 6i, 62 S A; -k 2 , are isomorphic if and only if 61/62 6 k 2 . (ii) The gradings in (II)(iii') are classified by the factor group k/S{k), where S(k) = {a 2 + a I a € k} is an additive subgroup of k. It was proved in [2,5] that two gradings of this type, corresponding to 61,62 € k -S(k) are isomorphic if and only if 61 -b 2 € S(k).…”

Section: •W4mentioning

confidence: 99%

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Group gradings of M₂(K)

Khazal¹,

Boboc²,

Dăscălescu³

2003

Bull. Austral. Math. Soc.

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We describe all group gradings of the matrix algebra M 2 (k), where k is an arbitrary field. We prove that any such grading reduces to a grading of type C 2 , a grading of type C 2 xC 2 , or to a good grading. We give new simple proofs for the description of C 2 -gradings and C 2 x C2-gradings on M 2 (k). INTRODUCTION AND PRELIMINARY RESULTSLet k be a field, G a group and A a /c-algebra. We say that A is G-graded if A = 0 A g , a direct sum of fc-vector subspaces, such that A g A h C A gh for any g,heG. geGThe following general problem was posed by E. Zelmanov (see [8]): find all G-gradings of the matrix algebra M n (k), where G is a group, k a field, and n a positive integer. The answer depends on the structure of G and A;, so it is hard to expect the problem can be solved in the general. However, several results have been obtained in special cases. In [7], gradings on A = M n (k) for which every matrix unit e^ (the matrix having 1 on the (i, j)-position, and zero elsewhere) is a homogeneous element (that is, it belongs to one of the subspaces A g ) have been studied. These gradings have been further investigated in [6], where they were called good gradings. Good gradings are fundamental in the study of all gradings, since as we shall see below, in certain cases any grading is isomorphic to a good grading. The description of all gradings of M 2 (k) by the cyclic group C 2 with two elements was done in [6], by using computational methods and the duality between group actions and group gradings. This was explained in [4] in terms of actions and coactions of Hopf algebras, the basic underlying idea being that a G-grading on an algebra A is precisely a structure of a ftG-comodule algebra on A. This idea was very useful for studying gradings of matrix algebras by cyclic groups, see [4]. In particular all the isomorphism types of C 2 -gradings on M 2 {k) were obtained in [4] in the case where char(fc) ^ 2. For char(fe) = 2, the classification has been completed in [2]. We note that in the case where k is algebraically closed, any C m -grading on M n (k) is isomorphic to a good grading (see [5,10]). This fact led to a different approach to the classification of all gradings of M n (k) over cyclic groups for an arbitrary field k, by using descent theory;

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