On the isomorphisms of the finite-dimensional commutative algebras without nilpotent elements over the field of real numbers Текст научной статьи по специальности «Математика»

Научни трудове на Съюза на учените в България - Пловдив. Серия В. Техника и технологии. Том XVII, ISSN 1311 -9419 (Print); ISSN 2534-9384 (Online), 2019. Scientific Works of the Union of Scientists in Bulgaria - Plovdiv. Series C. Technics and Technologies. Vol. XVII., ISSN 1311 -9419(Print); ISSN 2534-9384 (Online), 2019

ВЪРХУ ИЗОМОРФИЗМИТЕ НА КРАЙНОМЕРНИ КОМУТАТИВНИ АЛГЕБРИ БЕЗ НИЛПОТЕНТНИ ЕЛЕМЕНТИ НАД ПОЛЕТО НА

РЕАЛНИТЕ ЧИСЛА Йордан Епитропова, Иванка Градеваа, аПловдивски Университет "П. Хилендарски", Пловдив, България

ONTHE ISOMORPHISMS OF THE FINITE-DIMENSIONAL COMMUTATIVE ALGEBRAS WITHOUT NILPOTENT ELEMENTS OOER THEFIELOOT EEALNUMBERS YordanEpitropov", Evanka Gradeva3, aPlovdiv UniveEsity 'P. Шк^а^Ы', Plovdiv, To 1 garia

Absrtact We give necessary and sufficient conditions for a finite-dimensional commutative algebra without nilpotent elements over the field R of real numbers to be isomorphic as an R -algebra to the group algebra RG of a finite abelian group G . Moreover, we give necessary and sufficient conditions for an one such algebra to be isomorphic as an R -algebra to some group algebra over R . In addition, we derive a criterion for a finite-dimensional commutative algebra without nilpotent elements over an algebraically closed field F to be isomorphic as an F -algebra to a group algebra FG of a finite abelian group G .

Key words: algebra without nilpotent elements; group algebra; finite-dimensional group algebra; commutative group algebra.

1. Introduction

In the 1860s K. Weirstrass and R. Dedekind describe the structure of a finite-dimensional commutative algebra A over the field R of real numbers or over the field C of complex numbers (Kleiner, 2007). More precisely, they prove that A is isomorphic to finite direct sum of copies of the fields R or C .

By using this result, we define in the Section 2 the notion of real cardinality of a finite-dimensional commutative algebra A without nilpotent elements over the field R . We prove (Theorem 3), that the dimension of R -algebra A and its real cardinality determine A up to isomorphism. Through real cardinality we give (Theorem 5) necessary and sufficient conditions for a finite-dimensional commutative algebra without nilpotent elements over the field R to be isomorphic as an R -algebra to group algebra RG of a finite abelian group G . Moreover, we derive (Theorem 7) necessary and sufficient conditions for an one such algebra to be isomorphic as an R -algebra to some group algebra over R .

In addition, we get a criterion (Proposition 9) for a finite-dimensional commutative algebra without nilpotent elements over an algebraically closed field F to be isomorphic as an F -algebra to a group algebra FG of a finite abelian group G.

If K is an arbitrary field, then we shall denote the multiplicative group of K with K* = K \ {0}. If G is a finite multiplicative abelian group, then we shall denote G[2] = {g e GI g2 = l}.

2. Isomorphism of finite-dimensional commutative algebras without nilpotent elements over

the field of real numbers

First, we shall give a more general view the results of K. Weirstrass and R. Dedekind with brief proof in modern terms.

Theorem 1 (Structure). Let A be a finite-dimensional algebra without nilpotent elements over a field K . Then it holds

A = D1 ©D2 ©...©Ds,

s

where Dt are division algebras over K for i = 1,2,...,s and ^dimKDj = dimKA.

i=i

Proof. As the Jacobson radical J (a) is a nilpotent ideal of A , then J (A) = 0 . This implies that A is a semisimple algebra (Knapp, 2008). We apply the Wedderburn's Structure Theorem (Pierce, 1982) to A and we get

A = Mni (D)©Mn2 D)©...©Mn- (Ds), where Dt are division algebras over K , M (Di) are matrix algebras over Di for i = 1,2,...,s and

s

^ n2 dim KDj = dim K A . But if ni > 1 for some i, then M (d,) has non-zero nilpotent elements.

i=1

Therefore ni = 1 for i = 1,2,...,s and A is a finite direct sum of finite-dimensional division algebras over K . #

Corollary 2 (Structure). Let A be a finite-dimensional commutative algebra without nilpotent elements over the field R of real numbers. Then it holds (1) A = R©...©R ©C©...© C,

where C is the field of complex numbers.

Proof. We apply Theorem 1 to the finite-dimensional commutative algebra A without nilpotent elements over R . From the Theorem of Frobenius (Bahturin, 2010) it follows, that D, = R or Dt = C , i.e. (1) is fulfilled. #

Definition. Let A be a finite-dimensional commutative algebra without nilpotent elements over the field R of real numbers. We shall call the number rA of the direct summands R in the decomposition (1) real cardinality of A .

Theorem 3 (Isomorphism). If A is commutative algebra without nilpotent elements over the field R of real numbers and dim RA = n (n e N), then the finite-dimensional commutative algebra B without nilpotent elements over R is isomorphic to A as a R -algebra if and only if dim RB = n and rA = rB .

Proof. Due to the Corollary 2, the necessity is evident. Sufficiency follows from the fact that the dimension n of A over the field R of real numbers and the number rA of direct summands R in (1) determine A up to isomorphism. #

Proposition 4. Let RG be a group algebra of a finite abelian group G over the field R of real numbers. Then the real cardinality rRG of RG is equal to |G[2]|.

Proof. The group algebra RG is semisimple. Then the decomposition RG = ©RGez holds, where e are different minimal idempotents of RG, which correspond to the characters % of the group G (Passman, 2011). The real cardinality rRG of RG is equal to the number of those characters %:G ^ R*, for which g% = +1 for each g e G. Let G = {g^ x-.x(gs)x H be the decomposition of G in direct product of primary groups, where (gj are cyclic 2- groups (i = 1,...,s), and 2 does not divide |h| , i.e. |G[2]| = 2s. For the direct factor H there exist exactly

one character x0 with the mentioned properties, namely hx0 = 1 for each h e H . For each of the direct factors (gj there are two different such characters 0 and xi1, namely gXo = 1 and giXi1 = ~1. Therefore the number of all characters x of G with the property gx = +1 for each g e G is 2s = |G[2]|. Since the case G = H is trivial, then rRG = |g[2]| . #

3. Isomorphism of a finite-dimensional commutative algebra without nilpotent elements over

the field of real numbers and a group algebra Theorem 5 (Isomorphism). Let A be a finite-dimensional commutative algebra without nilpotent elements over the field R of real numbers and G be a finite abelian group. The algebra A is

isomorphic as an R -algebra to the group algebra RG if and only if dim RA = |G| and the real cardinality rA of A is equal to |g[2]| .

Proof. Necessity. Let the finite-dimensional commutative algebra A be isomorphic as an R -algebra to the group algebra RG of the finite abelian group G . Then dim R A = dim R RG = |G|.

Since in Proposition 4 we showed that rRG = |G[2]|, then rA = |G[2]|.

Sufficiency. Let dim RA = |G| and the real cardinality rA of the commutative algebra A without nilpotent elements is equal to |g[2]| . According to Theorem 3 in order to prove that A is isomorphic as an R -algebra to the group algebra RG it is sufficient to prove, that dimrA = dimRRG and that the real cardinalities of the two algebras are equal, i.e. rA = rRG. The first condition follows from dim RRG = |G|. The second condition is true, since Proposition 4 implies rRG = |G[2]|. #

Corollary 6. Let A be a finite-dimensional commutative algebra without nilpotent elements over the field R of real numbers and G is a finite abelian p -group, where p is prime. The algebra

A is isomorphic as an R -algebra to the group algebra RG if and only if dim R A = |G| and:

1) if p = 2, then the real cardinality rA of A is equal to |g[2]| ;

2) if p ^ 2, then the real cardinality rA of A is equal to 1.

Theorem 7 (Isomorphism). Let A be a finite-dimensional commutative algebra over the field R of real numbers. Then A is isomorphic as an R -algebra to a group algebra if and only if the following conditions are fulfilled:

(i) A is algebra without nilpotent elements;

(ii) rA = 2', where t is non-negative integer;

(iii) dim rA = rA (2k + l), where k is non-negative integer.

Proof. Necessity. Let the algebra A be isomorphic as an R -algebra to the group algebra RG for some group G. Since A is a finite-dimensional and commutative, then G is a finite abelian group. By Maschke's Theorem (Lang, 2005) the algebra RG is semisimple which implies that A is semisimple. Then according to the Wedderburn's Structure Theorem (Pierce, 1982) A decomposes into direct sum of fields, i.e. A is without nilpotent elements and (i) is fulfilled.

By Theorem 5, the equality rA = |G[2]| is holds. Consequently rA = 2' for some non-negative

integer t . In this way (ii) is proved.

Since |G[2]| divides |G|, where |G| = dim RA and rA = |G[2]| holds, then dim RA = rA (2k +1) for some non-negative integer k , i.e. (iii) is fulfilled.

Sufficiency. Let the conditions (i), (ii) and (iii) be fulfilled. We shall construct a group G in such

a way that the algebra A be isomorphic as an R -algebra to the group algebra RG .

The condition (i) and Corollary 2 imply that the decomposition (1) holds for finite-dimensional

commutative algebra A without nilpotent elements over the field R of real numbers, i.e.

A = R ©...© R © C ©...© C, where C is the field of complex numbers. We shall denote n = dim R A. The condition (ii) implies that the number of the direct summands R in the decomposition of A is t and the condition (iii) implies that dim R A = 2' (2k + 1) for the non-negative integers t and k . We construct an abelian group G = (g^x...x(gt}xH of order |G| = 2'(2k + 1), whose 2-component decomposes into a direct product of t cyclic groups and |h| = 2k + 1. Then Proposition 4 for the group algebra RG

implies rRG = 2t and (ii) implies rA = rRG. When we apply Theorem 5 to the algebras A and RG, we get A = RG as R -algebras. #

Corollary 8. Let A be a finite-dimensional commutative algebra without nilpotent elements over the field R of real numbers, G be an abelian group of order |G| = 2t (2k + 1) for some nonnegative integers t and k . Then A is isomorphic as an R -algebra to the group algebra RG if and only if the number of the direct summands R in (1) is 2t and the number of the direct summands C in (1) is 2'k .

4. Isomorphism of a finite-dimensional commutative algebra without nilpotent elements over

an algebraically closed field and a group algebra Proposition 9. Let F be an algebraically closed field, A be a commutative algebra without nilpotent elements over F and dim F A = n (n e N).Then A is isomorphic as an F -algebra to the group algebra FG of some abelian group G of order n, if the characteristic of F does not divide the orders of elements of G .

Proof. Similarly to the first part of the proof of Theorem 1 it is shown that A is semisimple algebra.

For the algebra A over F we apply the Wedderburn's Structure Theorem and we get (Pierce, 1982):

A = Mm (F)©M„2 (F)©...©Mn_ (F), where nf + n^ +... + n] = n. Since A is commutative, then Mn (F) are commutative algebras for each i = 1,2,...,s . Therefore n, = 1, which leads to

A = F © F ©...© F, where the number of the direct summands is n .

On the other hand, according to (Passman, 2011) each group algebra FG of finite abelian group G of order n over a field F with characteristic, which does not divide the orders of elements of G , decomposes into direct sum of n minimal ideals, generated from minimal idempotents, corresponding to all characters of G in the multiplicative group F * of F. Then FG is isomorphic to direct sum of n copies of the field F :

FG = FGe ©FGe ©...©FGe = F©F©...©F .

Xa X\ Xn

Therefore the finite-dimensional commutative algebra A without nilpotent elements is isomorphic as an F -algebra to the group algebra FG . #

References

Y. Bahturin, Basic structures of modern algebra. Springer, 2010. I. Kleiner, A history of abstract algebra. Birkhaeuser, 2007.

A. Knapp, Advanced algebra, Chapter II: Wedderburn-Artin ring theory. Birkhaeuser, 2008.

5. Lang, Algebra. Springer, 2005.

D. Passman, The algebraic structure of group rings. Dover Publications, 2011. R. Pierce, Associative algebras. Springer, 1982.