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Научни трудове на Съюза на учените в България-Пловдив, серия Б. Естествени и хуманитарни науки, т.ХУ1. Научна сесия „Техника и технологии, естествени и хуманитарни науки", 30-31 Х 2013 Scientific researches of the Union of Scientists in Bulgaria-Plovdiv, series B. Natural Sciences and the Humanities, Vol. XVL,ISSN 1311-9192, Technics, Technologies, Natural Sciences and Humanities Session, 30-31 October 2013
СТРУКТУРА И ИЗОМОРФИЗЪМ НА КРАЙНОМЕРНИ КОМУТАТИВНИ ПОЛУПРОСТИ p -ЦИКЛОТОМИЧНИ АЛГЕБРИ НАД ПОЛЕ ОТ ВТОРИ
РОД
Йордан Епитропов Пловдивски Университет "П. Хилендарски"
STRUCTURE AND ISOMORPHISM OF FINITE DIMENSIONAL COMMUTATIVE SEMISIMPLE p -CYCLOTOMIC ALGEBRAS OVER
FIELD OF SECOND KIND
Yordan Epitropov Plovdiv University 'P. Hilendarski'
Abstract. Let p be a prime number and K - a field, whose characteristic is different from p . In the paper we define the term p -cyclotomic algebra over a field and we study the structure of the finite dimensional commutative semisimple p -cyclotomic algebras over a field K of second kind with respect to the prime number p. We give criteria when one such algebra A is isomorphic as a K -algebra of the group algebra KG to a finite abelian p -group G . We derive a necessary and sufficient condition for a finite dimensional commutative algebra A over the field K of second kind with respect to the prime number p to be isomorphic as a K -algebra of a group algebra.
Key words: finite dimensional algebra; commutative algebra; semisimple algebra; group algebra; structure of algebra; isomorphism of algebras; field of second kind.
1. Introduction. Let p be a prime number, the characteristic of the field K is different from p and sj is a primitive p] -th root of unity in the algebraical closure of K where j is a non-negative whole number. Following Berman [1], the field K will be referred to as field of second kind with respect to the prime number p if the power of the extension K (s1;s2 ,...) of K is finite. Kaprilovsky [2] shows
that (i) if K is a field of second kind with respect to the odd prime number p then K (sj) = K (s1 ) holds for each integer j ; (ii) if K is a field of second kind with respect to the prime number 2, then K (s] ) = K (s2 ) holds for each integer j > 2.
The are many different researches considering group algebras over a field of second kind with respect to a prime number p. The main results are found by Mollov [3, 4], Nachev and Mollov [5, 6], Epitropov, Mollov and Nachev [7] and Nachev [8].
In this paper we define the term p -cyclotomic algebra over a field K and study up to isomorphism the structure of the finite dimensional commutative semisimple p -cyclotomic algebras over the field K of second kind with respect to the prime number p. Thus we define the term real power of a finite dimensional commutative semisimple p -cyclotomic algebra over a field of second kind with respect to p. We give a necessary and sufficient condition for the finite dimensional commutative semisimple p -cyclotomic algebra A over the field K of second kind with respect to the prime number p to be isomorphic as a K -algebra to the group algebra KG to a finite abelian p -group G. Moreover, we derive a necessary and sufficient condition for a finite dimensional commutative algebra A over the field K of second kind with regards to the prime number p to be isomorphic as a K -algebra to a group algebra.
These results summarise the ones given by us in [9].
2. Structure of finite dimensional commutative semisimple p -cyclotomic algebras over a field of second kind.
Definition 2.1. Let p be a prime number, K is a field with characteristic different from p, and L is an expansion of K . The field L is called p -cyclotomic expansion of the field K, if it is derived from K by adjoining only p' -th roots of the unity, where i is a natural number.
Definition 2.2. Let p be a prime number, K is a field with characteristic different from p, and A is an algebra over K . The algebra A is called p -cyclotomic algebra over the field K , if every subfield of A is p -cyclotomic expansion of the field K .
Theorem 2.3. Let p be a prime number, K is a field with characteristic different from p and is a field of second kind with respect to the prime number p. Let A be a finite dimensional commutative semisimple p -cyclotomic algebra over the field K . Then the following decomposition holds
(1)
A = K ©... © K © K (e)©... ©K (e),
where
(i) e is a primitive p -th root of 1 when p # 2 ;
(ii) e is a primitive forth root of 1 when p = 2 .
Proof. Let dimR A = n (n e N ). According to the Structure Theorem of Wedderburn [10, 11], applied to the semisimple algebra A over the field K , we get
(2) A = Mi (D)© Mn2 (D )©... © Mn (Ds),
s
where ^ nj dimK D. = n , and D. are algebras with division over the field K for j = 1,2,..., s . As A
j=1
is a commutative algebra, Mn (D.) are commutative algebras. Thus n. =1 for every j = 1,2,...,s . Moreover, the algebras D. must also be commutative for every j = 1,2,...,s , and so they are fields. As A is a p -cyclotomic algebra over the field K , it follows that the fields D are p -cyclotomic expansions of K . But since K is a field of second kind with respect to the prime number p, the possible p -cyclotomic extensions of K are either of the kind K , or of the kind K (e) with the given for e conditions (i) and (ii).
Definition 2.4. Let A be a finite dimensional commutative semisimple p -cyclotomic algebra of second kind over the field K of second kind with respect to the prime number p . The number rA of the direct addends K in the decomposition (1) will be called real power of A .
3. Isomorphism of finite dimensional commutative semisimple p -cyclotomic algebras over a field of second kind.
Theorem 3.1. Let p be a prime number, A is a finite dimensional commutative semisimple p -cyclotomic algebra over the field K of second kind with respect to the prime number p, and G is a finite abelian p -group. Then the algebra A is isomorphic as a K -algebra to the group algebra KG if and only if dim K A = |G|, the real power rA of A is equal to |G [2]| and the characteristic of K is different from p.
Proof. Necessity. Let the algebra A be isomorphic as a K -algebra to the group algebra KG . Then dimK A = dimK KG = |G|.
From the isomorphism it follows that the group algebra KG is semisimple and then using the Theorem of Maschke [10, 12, 13] the characteristic of the field K is coprime with p .
We will prove that the real power rA of the algebra A (i.e. the real power rKG of the group algebra KG ) is equal to |G [2] . From the fact that the group algebra KG is semisimple it can be determined
that KG = ^ KGex , where eX are different minimal idempotents of KG , which correspond to the characters X of the group G .
Let p # 2 . As G is a p -group, the real power rKG of the group algebra KG is equal to the number of those characters x : G ^ K , for which gx = 1 for every g e G , i.e. the real power rKG of KG is 1. On the other hand |G [2] = 1, with which the necessity is proven in this case.
Let p = 2 and G = (g^x...x(gs} is the decomposition of the group G in direct product of cyclic 2-groups. The real power rKG of the group algebra KG is equal to the number of those characters X : G ^ K*, for which gx = ±1 for every g e G . For each direct factor ^g^ there are two different such characters xj0 and xj1 , namely gJXJ0 = 1 and gJXJ1 = -1. Then the number of all wanted characters X of G is 2s = |G [2]|, with which the necessity is proven in this case, too.
Sufficiency. Let dimK A = |G|, the real power rA of the algebra A is equal to |G[2]| and the characteristic of the field K is coprime with p . In order to prove that the algebra A is isomorphic as a K -algebra to the group algebra KG it is enough, according to Theorem 2.3, to prove that dimK A = dimK KG and that the real powers of both algebras are equal, i.e. rA = rKa. The first condition is met by knowing that dimK KG = |G|. The second condition holds because in the necessity we proved that rKG = |G [2]|.
Theorem 3.2. Let A be a commutative algebra over a field K of second kind with respect to the prime number p, dimK A = pn and the characteristic of K is coprime with p. Then A is isomorphic as a K -algebra to some group algebra over K if and only if the following conditions are met:
(i) A is semisimple and p -cyclotomic over K ;
(ii) rA = 2A, where X is a non-negative whole number;
(iii) rA divides dimK A .
Proof. Necessity. Let the algebra A be isomorphic as a K -algebra to the group algebra KG for some group G. As A is commutative algebra and dimK A = pn, then G is a finite abelian p -group. According to the Theorem of Maschke [10, 12, 13] the algebra KG is semisimple, which leads to A being semisimple too. From the isomorphism of the algebras A and KG it follows that A is a p -
cyclotomic algebra over K , i.e. (i) is met.
According to Theorem 3.1 the following holds rA = |G[2] . Therefore rA = 2X for some non-negative whole number X . This proves (ii).
As |G[2]I divides |G|, where |G| = dimK A , and by Theorem 3.1 rA = |G[2] holds, then rA divides dimK A , i.e. (iii) is met. The necessity is proven.
Sufficiency. Let the conditions (i), (ii) and (iii) are met and let p # 2. From condition (i) and by Theorem 2.3 the decomposition (1) follows, i.e. A = K ©... © K © K (s)©... © K (s), where s is a primitive p -th root of 1. According to (ii) the real power rA of the algebra A is 1 = 20. Let G be a random abelian p -group of order pn = dimK A . Then |G [2] = 1 and by Theorem 3.1 it must be true that A is isomorphic as a K -algebra to the group algebra KG .
Let the conditions (i), (ii) and (iii) are met and let p = 2 . From condition (i) and by Theorem 2.3 the decomposition (1) holds, i.e. A = K ©... © K © K (i)©... © K (i), where i is a primitive forth root of 1. According to (ii) the real power of A is rA = 2X. Let G be a random abelian 2-group of order 2n = dimK A , which decomposes in direct product of X cyclic groups. The existence of such a group is given by the conditions (ii) and (iii). When we apply Theorem 3.1 to A and KG we get that A = KG as K -algebras. The proof of the sufficiency is complete.
Acknowledgements
This work was supported by the fund Scientific Research of Plovdiv University, under contract NI13-FMI-002.
The author is indebted to Professor N. Nachev for his helpful discussion of the results of the paper.
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