OCTONION ALGEBRA AND ITS MATRIX REPRESENTATIONS

Transcription

1 Palestine Journal of Mathematics Vol 6(1)(2017) Palestine Polytechnic University-PPU 2017 O/Z p OCTONION ALGEBRA AND ITS MATRIX REPRESENTATIONS Serpil HALICI and Adnan KARATAŞ Communicated by Ayman Badawi MSC 2010 Classications: Primary 17A20; Secondary 16G99 Keywords and phrases: Quaternion algebra Octonion algebra Cayley-Dickson process Matrix representation Abstract Since O is a non-associative algebra over R this real division algebra can not be algebraically isomorphic to any matrix algebras over the real number eld R In this study using H with Cayley-Dickson process we obtain octonion algebra Firstly We investigate octonion algebra over Z p Then we use the left and right matrix representations of H to construct representation for octonion algebra Furthermore we get the matrix representations of O/Z p with the help of Cayley-Dickson process 1 Introduction Let us summarize the notations needed to understand octonionic algebra There are only four normed division algebras R C H and O [2] [6] The octonion algebra is a non-commutative non-associative but alternative algebra which discovered in 1843 by John T Graves Octonions have many applications in quantum logic special relativity supersymmetry etc Due to the non-associativity representing octonions by matrices seems impossible Nevertheless one can overcome these problems by introducing left (or right) octonionic operators and xing the direction of action In this study we investigate matrix representations of octonion division algebra O/Z p Now lets start with the quaternion algebra H to construct the octonion algebra O It is well known that any octonion a can be written by the Cayley-Dickson process as follows [7] a = a + a e; a a H; H = {a = a 0 + a 1 i + a 2 j + a 3 k a 0 a 1 a 2 a 3 R} Here H is the real quaternion division algebra and {1 i j k} is the base for this algebra Where i 2 = j 2 = k 2 = 1 ijk = 1 and e is imaginary unit like i and e 2 = 1 For any elements a b of O the addition and multiplication operations are as follows If a = a + a e and b = b + b e O then and a+b = (a + a e) + (b + b e) = (a + b ) + (a + b )e ab = (a + a e)(b + b e) = (a b b a ) + (a b + b a )e respectively Where a and a denote the conjugates of the quaternions a and a O is a nonassociative but alternative division algebra with an eight-dimension over its center eld R and we note that the cannonical basis of O is e 0 = 1 e 1 = i e 2 = j e 3 = k e 4 = e e 5 = ie e 6 = je e 7 = ke 2 O/Z p Alternative Octonion Algebra In this section we use a prime number p p 2 and e 0 = 1 e 12 = e 22 = = e 72 = 1 Then the set of O/Z p can be written as O/Z p = {k = c 0 e 0 + c 1 e 1 + c 2 e 2 + c 3 e 3 + c 4 e 4 + c 5 e 5 + c 6 e 6 + c 7 e 7 c i Z p 0 i 7}

2 308 Serpil HALICI and Adnan KARATAŞ Firstly we show that O/Z p is a vector space Then we dene multiplication operation on O/Z p and we obtain that O/Z p forms an alternative division algebra with respect to the dened multiplication Theorem 21 The O/Z p is a vector space over the eld Z p Proof For all k 1 k 2 O/Z p and c i d i Z p 0 i 7 k 1 = c 0 e 0 + c 1 e 1 + c 2 e 2 + c 3 e 3 + c 4 e 4 + c 5 e 5 + c 6 e 6 + c 7 e 7 k 2 = d 0 e 0 + d 1 e 1 + d 2 e 2 + d 3 e 3 + d 4 e 4 + d 5 e 5 + d 6 e 6 + d 7 e 7 The addition operation over the O/Z p is dened as : O/Z p O/Z p O/Z p k 1 k 2 = (c i + d i )e i 0 i 7 So we can easily see that (O/Z p ) is an abelian group For all d Z p and k 1 O/Z p the multiplication operation over the O/Z p is dened as : Z p O/Z p O/Z p d k 1 = d (c 0 e 0 + c 1 e 1 + c 2 e 2 + c 3 e 3 + c 4 e 4 + c 5 e 5 + c 6 e 6 + c 7 e 7 ) d k 1 = (dc i )e i 0 i 7 And we can show the multiplication operation satises the following properties v 1 ) For all d Z p and k 1 k 2 O/Z p v 2 ) For all c d Z p and k O/Z p v 3 ) For all c d Z p and k O/Z p v 4 ) For all k O/Z p Thus {O/Z p Z p } is a vector space d (k 1 k 2 ) = (d k 1 ) (d k 2 ) (c + d) k = (c k) (d k) (cd) k = c (d k) 1 k = k From now on this vector space will be denoted by O/Z p shortly In the following theorem we can give an alternative algebra on this vector space Theorem 22 O/Z p is an alternative algebra over the eld Z p Proof Since O/Z p is a vector space the multiplication of two vectors on this vector space is : O/Z p O/Z p O/Z p for all k 1 k 2 O/Z p k 1 k 2 = (c 0 e 0 + c 1 e 1 + c 2 e 2 + c 3 e 3 + c 4 e 4 + c 5 e 5 + c 6 e 6 + c 7 e 7 ) (d 0 e 0 + d 1 e 1 + d 2 e 2 + d 3 e 3 + d 4 e 4 + d 5 e 5 + d 6 e 6 + d 7 e 7 ) k 1 k 2 = (c 0 d 0 c 1 d 1 c 2 d 2 c 3 d 3 c 4 d 4 c 5 d 5 c 6 d 6 c 7 d 7 )e 0 + (c 0 d 1 + c 1 d 0 + c 2 d 3 c 3 d 2 + c 4 d 5 c 5 d 4 + c 7 d 6 c 6 d 7 )e 1 + (c 0 d 2 + c 2 d 0 + c 3 d 1 c 1 d 3 + c 4 d 6 c 6 d 4 + c 5 d 7 c 7 d 5 )e 2

3 Matrix Representations of O/Z p (c 0 d 3 + c 3 d 0 + c 1 d 2 c 2 d 1 + c 4 d 7 c 7 d 4 + c 6 d 5 c 5 d 6 )e 3 + (c 0 d 4 + c 4 d 0 + c 5 d 1 c 1 d 5 + c 7 d 3 c 3 d 7 + c 6 d 2 c 2 d 6 )e 4 + (c 0 d 5 + c 5 d 0 + c 1 d 4 c 4 d 1 + c 7 d 2 c 2 d 7 + c 3 d 6 c 6 d 3 )e 5 + (c 0 d 6 + c 6 d 0 + c 1 d 7 c 7 d 1 + c 2 d 4 c 4 d 2 + c 5 d 3 c 3 d 5 )e 6 + (c 0 d 7 + c 7 d 0 + c 6 d 1 c 1 d 6 + c 2 d 5 c 5 d 2 + c 3 d 4 c 4 d 3 )e 7 We remark that this multiplication is called as octonion multiplication The octonion multiplication over O/Z p satises the following properties i) For all k 1 k 2 O/Z p and d Z p (d k 1 ) k 2 = d (k 1 k 2 ) ii) For all k 1 k 2 O/Z p k 1 (k 1 k 2 ) = (k 1 k 1 ) k 2 (k 2 k 1 ) k 1 = k 2 (k 1 k 1 ) iii) For all k 1 k 2 k 3 O/Z p k 1 (k 2 k 3 ) = (k 1 k 2 ) (k 1 k 3 ) (k 2 k 3 ) k 1 = (k 2 k 1 ) (k 3 k 1 ) With these properties we obtain that O/Z p forms an alternative algebra It is well known that the property ii) in the above equations is called the alternative property For example let p = 13 d = 11 and if we choose k 1 k 2 as follows then we have k 1 = 3e 0 + 2e 1 + e 2 + 7e 3 + 7e e 5 + 8e 6 + 5e 7 O/Z p k 2 = 1e 0 + 4e e 2 + 2e 3 + 7e e 5 + 6e 6 + 1e 7 O/Z p k 1 k 2 = 4e 0 + 6e 1 + 0e 2 + 9e 3 + 1e 4 + 9e 5 + 1e 6 + 6e 7 O/Z p k 1 k 2 = 6e 0 + 6e 1 + 3e 2 + 1e 3 + 0e e 5 + 6e 6 + 7e 7 O/Z p d k 1 = 7e 0 + 9e e e e 4 + 4e e 6 + 3e 7 O/Z p 3 Matrix Representations and O/Z p It is well known that every nite dimensional associative algebra over an arbitrary eld F is algebraically isomorphic to a subalgebra of a total matrix algebra over the eld F For the real quaternion algebra H there are many studies related with matrix representations[1] [3] and [4] For the properties of left and right matrix representations reader who interested in these matrices may nd useful these papers [3] and [5] In [1] for all a H the matrix form of right multiplication given as φ : H M φ(a) = a 0 a 1 a 2 a 3 a 1 a 0 a 3 a 2 a 2 a 3 a 0 a 1 a 3 a 2 a 1 a 0

4 310 Serpil HALICI and Adnan KARATAŞ where M is M = { a 0 a 1 a 2 a 3 a 1 a 0 a 3 a 2 a 2 a 3 a 0 a 1 a 3 a 2 a 1 a 0 a 0 a 1 a 2 a 3 R} Thus H is algebraically isomorphic to the matrix algebra M Similarly the matrix form of left multiplication is a 0 a 1 a 2 a 3 a 1 a 0 a 3 a 2 τ : H M τ(a) = a 2 a 3 a 0 a 1 a 3 a 2 a 1 a 0 Now based on the results related with the real matrix representations of quaternions one of real matrix representation of real octonions as follows Denition 31 Let a = a + a e O; a = a 0 + a 1 i + a 2 j + a 3 k a = a 4 + a 5 i + a 6 j + a 7 k The following 8 8 real matrix is called as left matrix representation of a over R [1] ω(a) = [ φ(a ) τ(a )K 4 φ(a )K 4 τ(a ) ] where K 4 is K 4 = Thus the explicit form of ω(a) is ω(a) = a 0 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 1 a 0 a 3 a 2 a 5 a 4 a 7 a 6 a 2 a 3 a 0 a 1 a 6 a 7 a 4 a 5 a 3 a 2 a 1 a 0 a 7 a 6 a 5 a 4 a 4 a 5 a 6 a 7 a 0 a 1 a 2 a 3 a 5 a 4 a 7 a 6 a 1 a 0 a 3 a 2 a 6 a 7 a 4 a 5 a 2 a 3 a 0 a 1 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 Denition 32 Let a = a + a e O; a = a 0 + a 1 i + a 2 j + a 3 k a = a 4 + a 5 i + a 6 j + a 7 k In the following 8 8 real matrix π(a) π(a) = [ τ(a ) φ(a ) φ(a ) τ(a ) is called as the right matrix representation of a over R [1] The explicit form of π(a) is shown as ]

5 Matrix Representations of O/Z p 311 π(a) = a 0 a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 1 a 0 a 3 a 2 a 5 a 4 a 7 a 6 a 2 a 3 a 0 a 1 a 6 a 7 a 4 a 5 a 3 a 2 a 1 a 0 a 7 a 6 a 5 a 4 a 4 a 5 a 6 a 7 a 0 a 1 a 2 a 3 a 5 a 4 a 7 a 6 a 1 a 0 a 3 a 2 a 6 a 7 a 4 a 5 a 2 a 3 a 0 a 1 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 Using the above denitions for matrix representations over O/Z p we will give the following theorems without proofs Theorem 33 For k = q + q e = q 0 e 0 + q 1 e 1 + q 2 e 2 + q 3 e 3 + q 4 e 4 + q 5 e 5 + q 6 e 6 + q 7 e 7 O/Z p the explicit form of the right matrix representation π(k) can be written as follows π(k) = q 0 (p 1)q 1 (p 1)q 2 (p 1)q 3 (p 1)q 4 (p 1)q 5 (p 1)q 6 (p 1)q 7 q 1 q 0 q 3 (p 1)q 2 q 5 (p 1)q 4 (p 1)q 7 q 6 q 2 (p 1)q 3 q 0 q 1 q 6 q 7 (p 1)q 4 (p 1)q 5 q 3 q 2 (p 1)q 1 q 0 q 7 (p 1)q 6 q 5 (p 1)q 4 q 4 (p 1)q 5 (p 1)q 6 (p 1)q 7 q 0 q 1 q 2 q 3 q 5 q 4 (p 1)q 7 q 6 (p 1)q 1 q 0 (p 1)q 3 q 2 q 6 q 7 q 4 (p 1)q 5 (p 1)q 2 q 3 q 0 (p 1)q 1 q 7 (p 1)q 6 q 5 q 4 (p 1)q 3 (p 1)q 2 q 1 q 0 Theorem 34 For k = q + q e = q 0 e 0 + q 1 e 1 + q 2 e 2 + q 3 e 3 + q 4 e 4 + q 5 e 5 + q 6 e 6 + q 7 e 7 O/Z p the explicit form of the left matrix representation ω(k) can be written as follows ω(k) = q 0 (p 1)q 1 (p 1)q 2 (p 1)q 3 (p 1)q 4 (p 1)q 5 (p 1)q 6 (p 1)q 7 q 1 q 0 (p 1)q 3 q 2 (p 1)q 5 q 4 q 7 (p 1)q 6 q 2 q 3 q 0 (p 1)q 1 (p 1)q 6 (p 1)q 7 q 4 q 5 q 3 (p 1)q 2 q 1 q 0 (p 1)q 7 q 6 (p 1)q 5 q 4 q 4 q 5 q 6 q 7 q 0 (p 1)q 1 (p 1)q 2 (p 1)q 3 q 5 (p 1)q 4 q 7 (p 1)q 6 q 1 q 0 q 3 (p 1)q 2 q 6 (p 1)q 7 (p 1)q 4 q 5 q 2 (p 1)q 3 q 0 q 1 q 7 q 6 (p 1)q 5 (p 1)q 4 q 3 q 2 (p 1)q 1 q 0

6 312 Serpil HALICI and Adnan KARATAŞ For example for p = 17 and k O/Z 17 we take k as k = 5e 0 + 2e 1 + 2e e e e 5 + 4e 6 + 1e 7 O/Z 17 Then it can be calculated that q = 5 + 2i + 2j + 15k and q = i + 4j + k So using q and q we can obtain the matrices below; And φ(q ) = τ(q ) = φ(q ) = τ(q ) = Moreover the left matrix representation of k O/Z 17 is ω(k) = The right matrix representation of k O/Z 17 is π(k) = In conclusion we study octonion algebra over Z p and give their left and right matrix representations according to dened real matrix representation [1] References [1] Tian Yongge Matrix representations of octonions and their applications Advances in Applied Clifford Algebras 101 (2000): [2] Schafer Richard Donald An introduction to nonassociative algebras Vol 22 Courier Corporation 1966 [3] Zhang Fuzhen Quaternions and matrices of quaternions Linear algebra and its applications 251 (1997): [4] G Koru Yucekaya Matrix Representation on Quaternion Algebra Turkish Journal of Mathematics and Computer Science ID

7 Matrix Representations of O/Z p 313 [5] Farid F O Qing-Wen Wang and Fuzhen Zhang On the eigenvalues of quaternion matrices Lin Multilin Alg 594 (2011): [6] Ward Joseph Patrick Quaternions and Cayley numbers: Algebra and applications Vol 403 Springer Science and Business Media 1997 [7] Baez John The octonions Bulletin of the American Mathematical Society 392 (2002): Author information Serpil HALICI and Adnan KARATAŞ Department of Mathematics Pamukkale University Denizli Turkey shalici@pauedutr adnank@pauedutr Received: November Accepted: May