FINITE SEMISIMPLE LOOP ALGEBRAS OF INDECOMPOSABLE RA LOOPS. 1. Introduction

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1 INITE SEMISIMPLE LOOP ALGEBRAS O INDECOMPOSABLE RA LOOPS SWATI SIDANA AND R. K. SHARMA Abstract. There are at the most seven classes of finite indecomposable RA loops upto isomorphism. In this paper, we completely characterize the structure of the unit loop of loop algebras of these seven classes of loops over finite fields of characteristic greater than Introduction A loop is a binary system (L, with an identity element and the property that for each a and b in L, there exist unique elements x and y in L such that a x = b and y a = b. A group is nothing but an associative loop. Let R be a commutative, associative ring with identity. A loop ring R[L] from a loop L can be constructed precisely in the same manner as the group ring R[G] is constructed from a group G. If R =, a field, then we call [L] a loop algebra. An alternative ring is a ring in which x(xy = x 2 y and (yxx = yx 2 are identities. A loop whose loop ring R[L] over some commutative, associative ring R with unity and of characteristic different from 2 is alternative, but not associative is called an RA loop. We refer the reader to [3] for more details. Goodaire [2], has determined the loop of units in the integral alternative loop rings of the six smallest order loops when the loop rings have non-trivial units. The unit loop U(Z[M(Q 8, 2] has been studied by Jespers and Leal [4], where M(Q 8, 2 = M(Q 8, 1, 1 denotes the Moufang Loop obtained from Q 8, the quaternion group of order 8 and the inverse involution on Q 8. Recently the semisimple loop algebras of RA loops have been studied by erraz, Goodaire and Milies [1]. The structure of the unit loops of the loop algebras of RA loops of order 32 and 64 have been determined in [8] Keywords Unit Loop, Loop Algebra, Indecomposable RA Loops Mathematics Subject Classification: 20N05, 17D05. Corresponding author: R K Sharma, address: rksharmaiitd@gmail.com The first author was supported by Council of Scientific and Industrial Research (CSIR, India. 1

2 2 SWATI SIDANA AND R. K. SHARMA Table 1: The seven classes of finite indecomposable RA loops Loop Group Z(L i x 2 y 2 u 2 = g 0 L 1 G 1 t L 2 G 2 t 1 t 1 t 1 t 1 L 3 G 3 t 1 t 2 1 t 2 1 L 4 G 4 t 1 t 2 t 1 t 2 t 1 L 5 G 5 t 1 t 2 t 3 t 2 t 3 1 L 6 G 5 t 1 t 2 t 3 t 2 t 3 t 1 L 7 G 5 w t 1 t 2 t 3 w t 2 t 3 w and [9]. Jespers, Leal and Milies [5] have proved that upto isomorphism, there are seven classes of indecomposable RA loops, given in Table 1. The purpose of this paper is to determine the structure of the unit loops of finite semisimple loop algebras of these seven classes of loops. Throughout denotes the finite field containing q = p n elements with p > 2. Let M(G,, g 0 denote the Moufang loop obtained from a non-abelian group G, g 0 Z(G and the involution on G. Also, we use the following notations: C n cyclic group of order n ξ k primitive k th root of unity n extension field of of degree n over Z(R Zorn s vector matrix algebra over a commutative and associative ring R (with unity GLL(2, R General Linear Loop of degree 2 over R φ(n Euler s phi function ord n (q order of q modulo n (x, y commutator of elements x and y in a loop L 2. Preliminaries In this section, we discuss some useful results. An indecomposable RA loop is 2-loop. rom [3, Ch IV, Th 3.1], a loop L is an RA loop iff L = M(G,, g 0 for a non-abelian group G. A non-abelian group G forms an RA loop M(G,, g 0 if and only if G/Z(G = C 2 C 2. In [3, Ch V, Th 1.2], it has been proved that G/Z(G = C 2 C 2 if and only if G = D C where D

3 INITE SEMISIMPLE LOOP ALGEBRAS O INDECOMPOSABLE RA LOOPS 3 is an indecomposable 2-group and C is a cyclic group which, if nontrivial, is a 2-group. Such a group D is given by D = x, y, Z(D where Z(D = t 1 t 2 t 3, o(t i = 2 m i for i = 1, 2, 3, m 1 1, m 2, 0, s = (x, y = t 2m 1 1 1, x 2 = t α 1 1 tα 2 2, α i 0, and y 2 = t β 1 1 tβ 2 2 tβ 3 3, β i 0. Up to isomorphism, there are at the most five classes of these indecomposable 2-groups. Table 2 contains these groups along with their presentations. Table 2: inite indecomposable groups G with Group Presentation G Z(G = C 2 C 2 G 1 x, y, t 1 x 2, y 2, t 2m 1 1, t 1 central, (x, y = t 2m G 2 x, y, t 1 x 2 = y 2 = t 1, t 2m 1 1, t 1 central, (x, y = t 2m G 3 x, y, t 1, t 2 x 2, y 2 = t 2, t 2m1 1, t 2m 2 2, t 1, t 2 central, (x, y = t 2m G 4 x, y, t 1, t 2 x 2 = t 1, y 2 = t 2, t 2m 1 1, t 2m 2 2, t 1, t 2 central, (x, y = t 2m G 5 x, y, t 1, t 2, t 3 x 2 = t 2, y 2 = t 3, t 2m 1 1, t 2m 2 2, t 2 3, t 1, t 2, t 3 central, (x, y = t 2m urther, these groups form seven non-isomorphic classes of indecomposable RA loops given in Table 1. The following lemma gives the decomposition of semisimple loop algebras of RA loops. Lemma 2.1. [3, Ch VI, Cor 4.8] Let L = M(G,, g 0 be a finite RA loop with commutator-associator subloop L = {1, s} = G and H = L or G. If char K H, then ( 1 + s K[H] = K[H] K[H] 2 ( 1 s 2,

4 4 SWATI SIDANA AND R. K. SHARMA ( ( 1 + s 1 s where K[H] = K[H/H ] is a direct sum of fields and K[H] = 2 2 K (H, H is a direct sum of Cayley-Dickson algebras if H = L and a direct sum of quaternion algebras otherwise. We can determine L/L for an indecomposable RA loop, once we know the center of the loop L using this theorem. Theorem 2.2. [3, Ch XI, Prop 2.3] Let L be a finite indecomposable RA loop. Write Z(L = t 1 t 2 t 3 w = C 2 a C 2 b C 2 c C 2 d, with a 1, b, c, d 0, and L t 1. If L L 1 L 3 L 5 L 7, then L/L = C2 a 1 C 2 b+1 C 2 c+1 C 2 d+1; otherwise L/L = C2 a C 2 b+1 C 2 c+1 C 2 d. The next proposition tells us how to find the center of K (L, L for an indecomposable RA 2-loop L using Z(L. Proposition 2.3. [3, Ch XI, Prop 1.3] Let L be a finite RA 2-loop and K be a field such that char K 2. Write Z(L = t 1 t r, where t i = C 2 n i, for i = 1, 2,, r. Assume that L t 1. Then Z( K (L, L = 2 n 1 1 [K(ξ 2 n 1 : K] K(ξ 2 n 1 [C 2 n 2 C 2 nr ]. Corollary [10, Cor, pp.152] says that every Cayley-Dickson algebra over a finite field is split. rom [3, Ch I, Cor 4.17], any split Cayley-Dickson algebra over a field K of characteristic different from 2 is isomorphic to Zorn s vector matrix algebra Z(K. Also, we have the center of Zorn s vector matrix algebra Z(K is isomorphic to K. Remark 2.4. It follows from above that every non-commutative, nonassociative component in the decomposition of finite semisimple loop algebra [L] is isomorphic to Zorn s vector matrix algebra Z( i, where i denotes the extension field of. The following theorem gives the Wedderburn decomposition of finite semisimple group algebras of abelian groups.

5 INITE SEMISIMPLE LOOP ALGEBRAS O INDECOMPOSABLE RA LOOPS 5 Theorem 2.5 (Perlis-Walker. [7, Th 3.5.4] Let G be a finite abelian group of order n and K[G] be a semi-simple group algebra. Then where a j = K[G] = a j K(ξ j j n n j [K(ξ j : K], n j being the number of elements of order j in G. To calculate [K(ξ j : K], we will use the next theorem. Definition 2.6. Let n be a positive integer. The splitting field K of x n 1 over a field is called the n th cyclotomic field over. Theorem 2.7. [6, Th 2.47] Let K be a finite field containing q = p n elements with gcd(q, n = 1 and Q n (x = (x ξ s denote the n n th s=1 gcd(s,n=1 cyclotomic polynomial. Then Q n factors into φ(n/d distinct monic irreducible polynomials in [x] of the same degree d, K is the splitting field of any such irreducible factor over, and [K : ] = d, where d = ord n (q. 3. The Unit Loops of [L 1 ] and [L 2 ] In this section, we determine the structure of the unit loop of finite semisimple loop algebras of those indecomposable RA loops whose center is isomorphic to C 2 m 1. Lemma 3.1. [C 2 a] = a φ(2 i di, where d i = ord d 2 i(q. i Proof. ollows from Theorem 2.5 and 2.7. Theore.2. The unit loop U( [L 1 ] = 8 m 1 1 where d i = ord 2 i(q. 8 φ(2 i d d i 2m1 1 GLL(2, dm1 i d m1 Proof. We have Z(L 1 = Z(G 1 = t 1 = C 2 m 1. Write Z(L 1 = C 2 m 1 C 2 0 C 2 0 C 2 0 Using Theorem 2.2, we get L 1 /L 1 = C 2 m 1 1 C 2 C 2 C 2.

6 6 SWATI SIDANA AND R. K. SHARMA Note that Z( (L 1, L 1 = Thus, using Remark 2.4, Hence [L 1 ] = 8 m 1 1 Theore.3. The unit loop [L 1 /L 1] = 8 [C2 m 1 1] U( [L 2 ] = 4 m 1 where d i = ord 2 i(q. = 8 m φ(2 i di d i 2 m 1 1 [ (ξ 2 m 1 : ] (ξ 2 m 1 = 2m1 1 d m1 dm1. (L 1, L 1 = 2m 1 1 d m1 Z( dm1. 8 φ(2 i di 2m1 1 Z( dm1. d i d m1 4 φ(2 i d d i 2m1 1 GLL(2, dm1 i d m1 Proof. Note that Z(L 2 = Z(G 2 = t 1 = C 2 m 1. Write Z(L 2 = C 2 m 1 C 2 0 C 2 0 C 2 0 Thus, L 2 /L 2 = C 2 m 1 C 2 C 2. [L 2 /L 2] = 4 [C2 m 1 ] = 4 m 1 4 φ(2 i d i di. Now, Z( (L 2, L 2 2 m 1 1 = [ (ξ 2 m 1 : ] (ξ 2 m 1 = 2m1 1 dm1. d m1 Therefore, (L 2, L 2 = 2m 1 1 Z( dm1. d m1 Hence [L 2 ] = 4 m 1 4 φ(2 i di 2m1 1 Z( dm1. d i d m1 4. The Unit Loops of [L 3 ] and [L 4 ] In this section, we describe the structure of the unit loop of finite semisimple loop algebras of those indecomposable RA loops whose center is isomorphic to C 2 m 1 C 2 m 2.

7 INITE SEMISIMPLE LOOP ALGEBRAS O INDECOMPOSABLE RA LOOPS 7 Lemma 4.1. Let G = C 2 a C 2 b. Then [C 2 a C 2 b] = where d(i, j = ord 2 max{i,j}(q. a b φ(2 i φ(2 j d(i, j d(i,j Proof. Using Theorem 2.5 and 2.7, we have [C 2 a C 2 b] = ( [C2 a][c 2 b] ( = a φ(2 i [ (ξ 2 i : ] (ξ 2 i [C 2 b] = [C 2 b] a φ(2 i [ (ξ 2 i : ] (ξ 2 i[c 2 b] ( = b φ(2 k k=1[ (ξ 2 k : ] (ξ 2 k ( (ξ 2 i b a φ(2 i [ (ξ 2 i : ] j=1 = a φ(2 i [ (ξ 2 i : ] (ξ 2 i b a b j=1 = a φ(2 j [ (ξ 2 i, ξ 2 j : ] (ξ 2 i, ξ 2 j k=1 φ(2 i φ(2 j [ (ξ 2 max{i,j} : ] (ξ 2 max{i,j} b φ(2 i φ(2 j d(i, j d(i,j. φ(2 k [ (ξ 2 k : ] (ξ 2 k Theorem 4.2. The unit loop U( [L 3 ] = 4 m 1 1 where d(i, j = ord 2 max{i,j}(q. 4 φ(2 i φ(2 j d(i, j d(i,j m 2 2 m1 1 φ(2 k GLL(2, d(m 1, k d(m1,k Proof. Note that Z(L 3 = Z(G 3 = t 1 t 2 = C 2 m 1 C 2 m 2. Write Z(L 3 = C 2 m 1 C 2 m 2 C 2 0 C 2 0 Thus, L 3 /L 3 = C 2 m 1 1 C 2 C 2 C 2.

8 8 SWATI SIDANA AND R. K. SHARMA Now [L 3 /L 3] = 4 [C2 m 1 1 C 2 ] = 4 m 1 1 Z( (L 3, L 3 = 2 m 1 1 [ (ξ 2 m 1 : ] (ξ 2 m 1 [C 2 m 2 ] = Thus, (L 3, L 3 = m 2 Hence [L 3 ] = m m 1 1 [ (ξ 2 m 1 : ] (ξ 2 m 1 m 2 = m 2 2 m1 1 φ(2 k d(m 1, k d(m1,k. 2 m1 1 φ(2 k d(m 1, k 4 φ(2 i φ(2 j d(i, j φ(2 i φ(2 j d(i, j k=1 Z( d(m1,k. d(i,j 2 m1 1 φ(2 k [ (ξ 2 max{m 1,k} : ] (ξ 2 max{m,k} 1 d(i,j m 2 2 m1 1 φ(2 k Z( d(m 1, k d(m1,k. Theorem 4.3. The unit loop U( [L 4 ] = 2 m 1 where d(i, j = ord 2 max{i,j}(q. 2 φ(2 i φ(2 j d(i, j d(i,j m 2 2 m1 1 φ(2 k GLL(2, d(m 1, k d(m1,k Proof. As Z(L 4 = Z(G 4 = t 1 t 2 = C 2 m 1 C 2 m 2. Write Z(L 4 = C 2 m 1 C 2 m 2 C 2 0 C 2 0 Thus, L 4 /L 4 = C 2 m 1 C 2 C 2. [L 4 /L 4] = 2 [C2 m 1 C 2 ] = 2 m 1 φ(2 i φ(2 j d(i, j Observe that Z(L 4 = Z(L 3. So, by previous theorem, d(i,j Z( (L 4, L 4 = Z( (L 3, L 3 = m 2 2 m1 1 φ(2 k d(m 1, k d(m1,k.

9 INITE SEMISIMPLE LOOP ALGEBRAS O INDECOMPOSABLE RA LOOPS 9 Thus, (L 4, L 4 = m 2 Hence [L 4 ] = 2 m 1 2 m1 1 φ(2 k d(m 1, k Z( d(m1,k. 2 φ(2 i φ(2 j d(i, j d(i,j m 2 2 m1 1 φ(2 k Z( d(m 1, k d(m1,k. 5. The Unit Loops of [L 5 ] and [L 6 ] In this section, we determine the structure of the unit loop of finite semisimple loop algebras of those indecomposable RA loops whose center is isomorphic to C 2 m 1 C 2 m 2 C 2. Working similar to Lemma 4.1, the following lemma is straightforward. Lemma 5.1. Let G = C 2 a C 2 b C 2 c. Then [C 2 a C 2 b C 2 c] = a where d(i, j, k = ord 2 max{i,j,k}(q. b c φ(2 i φ(2 j φ(2 k d(i, j, k d(i,j,k Theorem 5.2. The unit loop U( [L 5 ] = 2 m 1 1 m 2 where d(i, j, k = ord 2 max{i,j,k}(q φ(2 i φ(2 j φ(2 k d(i, j, k d(i,j,k φ(2 j 1 GLL(2, d(m 1, i 1, j 1 d(m1,i 1,j 1 Proof. We have Z(L 5 = Z(G 5 = t 1 t 2 t 3 = C 2 m 1 C 2 m 2 C 2. Write Z(L 5 = C 2 m 1 C 2 m 2 C 2 C 2 0 Thus, L 5 /L 5 = C 2 m 1 1 C 2 C 2 +1 C 2. [L 5 /L 5] = 2 [C2 m 1 1 C 2 C 2 +1] = 2 m φ(2 i φ(2 j φ(2 k d(i, j, k d(i,j,k

10 10 SWATI SIDANA AND R. K. SHARMA Z( (L 5, L 2 m = [ (ξ 2 m 1 : ] (ξ 2 m 1 [C 2 m 2 C 2 ] ( 2 = m 1 1 [ (ξ 2 m 1 : ] (ξ 2 m 1 m 2 i 1 =1 [ (ξ 2 m 1, ξ 2 i 1 : ] (ξ 2 m 1, ξ 2 i 1 [C 2 ] = 2 m 1 1 [ (ξ 2 m 1 : ] (ξ 2 m 1 j 1 =1 m 2 i 1 =1 m 2 i 1 =1 = m 2 2 m 1 1 φ(2 i 1 [ (ξ 2 m 1, ξ 2 i 1 : ] (ξ 2 m 1, ξ 2 i 1 k 1 =1 Thus, (L 5, L 5 = m 2 Hence [L 5 ] 2 m 1 1 φ(2 j 1 [ (ξ 2 m 1, ξ 2 j 1 : ] (ξ 2 m 1, ξ 2 j 1 2 m 1 1 φ(2 i 1 φ(2 k 1 [ (ξ 2 m 1, ξ 2 i 1, ξ 2 k 1 : ] (ξ 2 m 1, ξ 2 i 1, ξ 2 k 1 = 2 m 1 1 φ(2 j 1 d(m 1, i 1, j 1 d(m1,i 1,j 1. m 2 φ(2 j 1 Z( d(m 1, i 1, j 1 d(m1,i 1,j φ(2 i φ(2 j φ(2 k d(i, j, k φ(2 j 1 Z( d(m 1, i 1, j 1 d(m1,i 1,j 1. d(i,j,k Theorem 5.3. The unit loop U( [L 6 ] = m 1 m 2 where d(i, j, k = ord 2 max{i,j,k}(q. +1 φ(2 i φ(2 j φ(2 k d(i, j, k d(i,j,k φ(2 j 1 GLL(2, d(m 1, i 1, j 1 d(m1,i 1,j 1 Proof. Note that Z(L 6 = Z(G 5 = t 1 t 2 t 3 = C 2 m 1 C 2 m 2 C 2. Write Z(L 6 = C 2 m 1 C 2 m 2 C 2 C 2 0 Thus, L 6 /L 6 = C 2 m 1 C 2 C 2 +1.

11 INITE SEMISIMPLE LOOP ALGEBRAS O INDECOMPOSABLE RA LOOPS 11 [L 6 /L 6] = [C2 m 1 C 2 C 2 +1] = m 1 +1 As Z(L 6 = Z(L 5, by previous theorem, we have Z( (L 6, L 6 = Z( (L 5, L 5 = m 2 So, (L 6, L 6 = m 2 Hence [L 6 ] = m 1 m 2 φ(2 i φ(2 j φ(2 k d(i, j, k d(i,j,k φ(2 j 1 d(m 1, i 1, j 1 d(m1,i 1,j 1. φ(2 j 1 Z( d(m 1, i 1, j 1 d(m1,i 1,j φ(2 i φ(2 j φ(2 k d(i, j, k φ(2 j 1 Z( d(m 1, i 1, j 1 d(m1,i 1,j 1. d(i,j,k 6. The Unit Loop of [L 7 ] In this section, we determine the structure of the unit loop of finite semisimple loop algebra of indecomposable RA loop whose center is isomorphic to C 2 m 1 C 2 m 2 C 2 C 2 d. The following lemma is useful. Lemma 6.1. Let G = C 2 a C 2 b C 2 c C 2 d. Then [C 2 ac 2 bc 2 cc 2 d] = a where d(i, j, k, l = ord 2 max{i,j,k,l}(q. Theorem 6.2. The unit loop U( [L 7 ] = m 1 1 m 2 b c d l=0 i+j+k+l>0 d k 1 =0 +1 i+j+k+l>0 where d(i, j, k, l = ord 2 max{i,j,k,l}(q. d+1 l=0 φ(2 i φ(2 j φ(2 k φ(2 l d(i, j, k, l d(i,j,k,l φ(2 i φ(2 j φ(2 k φ(2 l d(i, j, k, l d(i,j,k,l φ(2 j 1 φ(2 k 1 GLL(2, d(m 1, i 1, j 1, k 1 d(m1,i 1,j 1,k 1

12 12 SWATI SIDANA AND R. K. SHARMA Proof. We have Z(L 7 = Z(G 5 w = t 1 t 2 t 3 w = C 2 m 1 C 2 m 2 C 2 C 2 d. Thus, L 7 /L 7 = C 2 m 1 1 C 2 C 2 +1 C 2 d+1. [L 7 /L 7] = [C2 m 1 1 C 2 C 2 +1 C 2 d+1] Now = m i+j+k+l>0 d+1 l=0 φ(2 i φ(2 j φ(2 k φ(2 l d(i, j, k, l Z( (L 7, L 7 = 2 m 1 1 [ (ξ 2 m 1 : ] (ξ 2 m 1 [C 2 m 2 C 2 C 2 d] = m 2 So, (L 7, L 7 = m 2 Hence [L 7 ] = m 1 1 m 2 d k 1 =0 d k 1 =0 d k 1 =0 +1 i+j+k+l>0 d(i,j,k,l φ(2 j 1 φ(2 k 1 d(m 1, i 1, j 1, k 1 d(m1,i 1,j 1,k 1. φ(2 j 1 φ(2 k 1 Z( d(m 1, i 1, j 1, k 1 d(m1,i 1,j 1,k 1 d+1 l=0 φ(2 i φ(2 j φ(2 k φ(2 l d(i, j, k, l φ(2 j 1 φ(2 k 1 Z( d(m 1, i 1, j 1, k 1 d(m1,i 1,j 1,k 1 d(i,j,k,l Acknowledgments. The authors are thankful to the anonymous reviewers for their useful comments and suggestions. References [1] erraz, R.A., Goodaire, E.G., Milies, C.P.: Some classes of semisimple group (and loop algebras over finite fields, J. Algebra, 324(12, , (2010. [2] Goodaire, E.G.: Six Moufang loops of units, Canad. J. Math., 44(5, , (1992. [3] Goodaire, E.G., Jespers, E., Milies, C.P.: Alternative loop rings, North-Holland Math. Stud., Vol. 184, Elsevier, Amsterdam, (1996. [4] Jespers, E., Leal, G.: A characterization of the unit loop of the integral loop ring ZM 16(Q, 2, J. Algebra, 155(1, , (1993. [5] Jespers, E., Leal, G., Milies, C.P.: Classifying indecomposable RA loops, J. Algebra, 176(2, , (1995.

13 INITE SEMISIMPLE LOOP ALGEBRAS O INDECOMPOSABLE RA LOOPS 13 [6] Lidl, R., Niederreiter, H.: Introduction to finite fields and applications, Cambridge University Press, Cambridge, (1994. [7] Milies, C.P., Sehgal, S.K.: An introduction to group rings, Vol. 1, Kluwer Academic Publishers, Dordrecht, (2002. [8] Sidana, S., Sharma, R.K.: The unit loop of finite loop algebras of loops of order 32, Beitr Algebra Geom, DOI: /s , (2013. [9] Sidana, S., Sharma, R.K.: inite loop algebras of RA loops of order 64, Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, accepted for publication. [10] Zhevlakov, K.A., Slin ko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are nearly associative, Vol. 104, Academic Press, New York-London, (1982, translated by Harry. Smith. Department Of Mathematics, Indian Institute of Technology Delhi, New Delhi, India address: swatisidana@gmail.com Department Of Mathematics, Indian Institute of Technology Delhi, New Delhi, India address: rksharmaiitd@gmail.com