Supersymmetric Hypermatrix Lie Algebra and Hypermatrix Groups Generated by the Dihedral Set D 3

This work is an investigation into the structure and properties of supersymmetric hypermatrix Lie algebra generated by elements of the dihedral group D3. It is based on previous work on the subject of supersymmetric Lie algebra (Schreiber, 2012). In preview work I used several new algebraic tools; namely cubic hypermatrices (including special arrangements of such hypermatrices) and I obtained an algebraic structure associated with the basis of the Lie algebra sl2, and I showed that the basis elements sl2 are generators of infinite periodic hypermatrix Lie algebraic structures with semisimple sub-algebras. The generated algebra has been shown to be an extended Lie hypermatrix algebra that has a classical Lie algebra decomposition composed of hypermatrices with periodic properties. The generators of higher dimensional Lie algebra were shown to be special supersymmetric, anti-symmetric and certain skew-symmetric hypermatrices. The present work takes a different look at the structure of periodic hypermatrix Lie algebra by using elements generating the classical dihedral group D3. Using cubic dihedral symmetric hypermatrices (type: even-even, odd-odd, even-odd odd-even permutation) to generate Lie hypermatrix algebra I show that the extended dihedral algebra is a Lie hypermatrix algebras with special hypermatrix group properties, semisimple, symmetric, skew-symmetric, anti-symmetric, and anti-clockwise symmetric properties.


Introduction
In preview work a hypermatrix Lie algebra generated by the basis elements of sl 2 has been shown to be an extended Lie hypermatrix algebra that has a classical Lie algebra decomposition (Bourbaki, 1980;Humphreys, 1972;Jacobson, 1962;Serre, 1987); specifically a periodic set of Lie algebras composed of hypermatrices has been generated.
In the present work I study the structure of certain periodic hypermatrix Lie algebras by using generators of the classical dihedral group D 3 structured from the simultaneously turning and reflecting pairs of triangles and relations among hypermatrices.Using cubic dihedral symmetric hypermatrices I show that a dihedral D 3 hypermatrices Lie algebra can be extended to higher dimension Lie hypermatrix algebras with special semisimple, symmetric, skew-symmetric, antisymmetric, and anti-clockwise symmetric properties.

The HD 3 Lie algebra of Two Triangles
The group elements associated with the reflection, and turning of a triangle or its permutation are classically represented by using the following set of matrices: Under matrix multiplication it has the non-commutative classical group structure shown in <Table 1>.

The first extension of D 3
We consider the first extension HD 3 Lie hypermatrix algebra (defined below and see also; Schreiber, 2012) of simultaneously turning and reflecting 36 pairs of triangles The triangles are represented by pairs of matrices situated on the back and front of hypermatrix W i, j , such that: ) , and in general all 36 hypermatrices are represented by the relations ) , i = 1, ..., 6; j = 1, ..., 6.
We label each hypermatrix even-even, even-odd, odd-even, odd-odd according to the number of permutations in each W i -hypermatrix components.The matrix t 1 is even, t 2 is even, t 3 is odd, t 4 is even, t 5 is odd, and t 6 is odd.For example t 6 has the permutation (2, 1, 3) with one entry smaller than the first and the sub permutation (1, 3) with no smaller entry than the first element (2), hence t 6 has an odd permutation.Accordingly, W 1,1 is an even-even permutation structured from two identical even permutation matrices.We sum up the 36 elements in four sets, W even−even , W odd−odd , W even−odd , W odd−even as follows: The even elements in D 3 are represented by t 1 , t 2 , t 4 and the odd elements by t 3 , t 5 , t 6 .In general for even elements we have a commutation relation e i e j = e j e i characteristics of the alternating even elements in the symmetric subgroup of S 3 , see also <Table 1>.
Definition 1 Lie Algebra of Hypermetrices (Schreiber, 2012).Consider the space {W} over a field F, with an operation WW ∈ W * .Note that W * is the first extension, e.g., if W i, j is a two sheet hypermatrix W * is a 4-sheet hypermatrix.
Denote by (W i , W j ) the hypermatrix Lie bracket over F; the set {W * } constitutes a Lie hypermatrix algebra if the following conditions are satisfied: The hypermatrix algebra has to be closed in terms of its components, and with respect to the field operations, in the sense that the component sheets {W si ∈ W} are well defined in the extended space, under the bracket operation {WW ∈ W * }.
Next I describe some of the bracket products in the hypermatrix dihedral set (W i, j , W k,l ) in terms of triangle permutation properties (even or odd permutation) and symmetric characteristics of the hypermatrix products.
2.3 Even-even permutation and anti-symmetric sheet arrangement We see that (W 4;2 , W 2;4 ) has trace zero and it has partial sheet skew symmetry in one central plane direction so that (W 4;2 , W 2;4 ) t = −(W 4;2 , W 2;4 ) for inherent sheet components.Sheet elements W s2 and W s4 are skew-symmetric and have trace zero.

Odd permutation and anti-symmetric sheet arrangement
We see that (W 3;5 , W 5;3 ) has trace zero and it has partial sheet skew symmetry in one central plane direction so that (W 3;5 , W 5;3 ) t = −(W 3;5 , W 5;3 ) for the sheet components.W 1 and W 4 have trace zero and are skew-symmetric.

Even-odd sheet arrangement
For the Lie bracket operation of the even-odd components hypermatrices W 1;3 , and W 1;5 we have We see that (W 1;3 , W 1;5 ) has trace zero.Sheets W s2 and W s3 are symmetric, and Ws4 is skew-symmetric W s1 = 0.
The multiplication table for the 1296 hypermatrices of D * 3 are represented here according to the evenness and oddness of the permutation of the triangles and the Lie bracket characteristics; it has certain algebraic symmetric characteristics summarized in table 2 for the multiplication of the two triangle hypermatrices and in table 3 for the two triangle antisymmetric hypermatrix product, <Table 2>, <Table 3>.
Theorem 1 A1) Lie bracket products of dihedral antisymmetric D 3 hypermatrices of type (even, even) in the two triangle configuration generate trace-zero hypermatrices with skewsymmetric sheet arrangements of the hypermatrices.
A2) The Lie bracket products of dihedral antisymmetric D 3 hypermatrices of type (even, even) as in the two triangle configuration, generate a semisimple hypermatrix Lie algebra.
B1) The Lie Bracket products of dihedral antisymmetric D 3 hypermatrices of type (odd, odd) result in trace-zero Lie hypermatrices with skewsymmetric sheet arrangements of the hypermatrices.
B2) The Lie bracket products of dihedral antisymmetric D 3 hypermatrices of type (odd, odd) generate a semisimple hypermatrix Lie algebra.
Proof: A1) Let the hypermatrices be composed of even antisymmetric sheets such that, written here horizontally, (W(e i , e j ), W(e j , e i )) = W(e i e j − e j e i , e i e i − e j e j , e j e j − e i e i , e j e i − e i e j ) if I-identity element is not among the e i − s ′ the differences, by table 1, in W are either t 4 − t 2 or t 2 − t 4 therefore the result is since the even sheet elements commute two of the above sheet W S e entries commute and will vanish while the other two are related by W si = −W s j and are skewsymmetric as hypermatrix sheets.If the identity element is among the e i − s ′ only the sign of the sheet will change and, therefore, the result follows.
A2) By Schreiber ( 2012) for antisymmetric matrices we had rules related to the classical semisimpe Lie algebra the antisymmetric hypermatrices with nilpotent components and the antisymmetric hypermatrices with nilpotent and symmetric components )] Generally we have: trace zero hyper−matrix where W * is a trace zero hypermatrix resulting from the product of nilpotent and trace zero elements in antisymmetric hypermetrics arrangement.
Here ⊗ stands for antisymmetric arrangement of any cubic sheet arrangement of the basis elements f (h, y, x).With a mapping, for example (h, y) In conclusion we may sum up with rule 2 stating that: The Lie product of hypermetrics (composed of h, y, x elements) set in antisymmetric hypermatrices, result in skewsymmetric trace zero hypermatrices, or semi skew-symmetric hypermatrices.The generalization to any size hypermatrices follows by induction on dimension.
In order to show that the resulting Lie algebra is semisimple it is required to show that the hypermatrix Lie algebra {W} has no proper abelian ideals other then {0}.
Since the resulting products of antisymmetric even-even hypermatrices have skew-symmetric sheet arrangement the hypermatrices do not commute and the only ideals are the {0} ideal, therefore, the resulting Lie algebra is a semisimple Lie algebra.
B1) Although the odd elements do not commute their product results in an even element (see table 1), therefore, the argument for part B1 is similar to the argument in part A1 and the proof follows in similar lines to those in part A1.
B2) Although the odd elements do not commute their product results in an even element (see table 1), therefore, the argument for part B2 is similar to the argument in part A2 and the proof follows in similar lines to those of part A2.The resulting products constitute semisimple hypermatrix Lie algebra (schriber, 2012).

Symmetric Hypermatrices Dihedral Type
The symmetric set of hypermatrices

Lie supersymmetric hypermatrix group
Definition 2 A multiplicative cubic-hypermatrix group is a set {W} of hypermatrices with a multiplicative operation and the following properties: a) Closure in the extended hypermatrix set.W i × W j ∈ {W * }, ∀W i , W j ∈ {W}, W * -the higher dimensional extended cubichypermatrix.Closure is maintained with respect to the field and components matrix sheets.The hypermatrix size and the number of sheets do not remain constant.b) Identity element.∃ an element Associativity of hypermatrices is maintained when the pair order multiplication of hypermatrices is maintained, and when pair order is maintained the associativity property of hypermatrices is derived from matrix associativity.
Definition 3 A subset of hypermatrix Lie algebra is a hypermatrix subgroup if it is a hypermatrix group as a set, i.e., it satisfies all the conditions of a hypermatrix group.

The Determinant of Hypermatrices and Conditions for Invertability
The necessary and sufficient conditions for the existence of an invertible hypermatrix W −1 to an hypermatrix W ∈ {W}, are given by the following theorem in which DW is the determinant of W.
Theorem 2 ∀W i ∈ {W} if DW = 0 and for all sheets S i of W DW si 0, ∀i then W −1 the inverse element of W exist and Summation is over all arrangements of (S 1 , ..., S σn ) sub-sheets in the cubic hypermatrices and sheets.

Proof of theorem 2
The idea of the proof is as follows: since the general case of a hypermatrix determinant has to reduce to the case of a single matrix in terms of conditions for invertability.The individual sheet sub-matrices must be non-trivial and invertible because their product with the inverse hypermatrix results with identity sheet sub-matrices.If the hypermatrix is a symmetric structure by the definition of the determinant of a hypermatrix the determinant of an (eveneven) dihedral hypermatrix has value zero and an inverse.When the hypermatrix W with the above characteristics exists it must have an inverse since its individual components has inverses i.e., WW −1 = (I, I, I, I, I, ..., I) = I * * for all W, the elements product in the brackets are arranged right to left.
Lemma 1 Assume DW 0 and the sheets of W are identical then W is not necessarily invertible.
Proof: In the proof I consider one of the matrix sheets in W and without loss of generality if DW s2 0, W s1 W s2 and if W s2 is invertible W is not invertible because I * cannot exist, this follows because WW −1 = I * if and only if W s1 = W s2 and both sheets are not trivial or nilpotent.This could be checked directly and the lemma holds by induction for any finite dimension of W.
Therefore, DW = 0 is a necessary condition for the invertability of W but not a sufficient condition.Generalization for higher dimension of W is by induction and induction on dimension of hypermatrices.
Lemma 2 If DW = 0 and W si W s j , ∀i, j W is not necessarily invertible.
Proof: To get DW = 0 we might choose one invertible sheet e.g., W si and one trivial sheet W s j = 0 or two nilpotent nonidentical sheets then possibly D(W ma jor ) D(W minor ) and I * cannot exist because WW −1 = I * if and only if W s1 = W s2 and they must be invertible as matrices.
In conclusion we have shown that in order for the hypermatrix W to have an inverse element W −1 all the components sheets of the hypermatrix need to be identical non-trivial and invertible sub-matrices of the hypermatrix W. The determinant of the hypermatrix W has to vanish, i.e., DW = 0. We sum up the bracket multiplication of the (even, even), (odd, odd) elements and their properties in <Table 5>.
Theorem 3 The invertible (even, even) dihedral set elements of any order constitute a commutative multiplicative hypermatrix group.
Proof: By table 1 the multiplication is closed in the matrix set, each element has an inverse in an hypermatrix set, and multiplication is commutative, here associativity follows from the commutativity of the hypermatrix (..., even, even...even) set.
Theorem 4 The invertible (odd, odd) dihedral set elements of any order constitute a skew-symmetric Lie hypermatrix algebra.
Proof: By table 3 the invertible (odd, odd) dihedral set is Lie hypermatrix algebra and by induction on dimension it is a skew-symmetric Lie hypermatrix algebra of the invertible (odd, odd) dihedral set.

The Extended Hypermatrix Lie Algebra Associated with Antisymmetric and Symmetric Hypermatrices
The (even, even), (even,even) hypermatrix products of the dihedral algebra in table 4 with identical symmetric sheet elements constitute a hypermatrix group and a commutative Lie hypermatrix algebra.Each extension of the set of group elements constitutes a group structure in a higher cubic hypermatrix set.The set of elements with trace zero is a sub-Lie hypermatrix algebra which generates a higher cubic dimension semisimple Lie hypermatrix algebra, which has global skew-symmetric properties (Schreiber, 2012).Next we consider the third extension of the symmetric elements.

The third extension
Next, consider the extension of the elements in table 4 under the Lie bracket operation: Skew-symmetric elements type (odd, odd, odd, odd) Consider the products of the (odd, odd, odd, odd) elements W 3,3;5,5 and W 3,3;6,6 .These elements are symmetric and their product is skew-symmetric.
Hence the higher extensions set of the symmetric hypermatrices associated with the dihedral D 3 group arrangement of triangles have a zero trace if the hypermatrics have all even or all odd elements and if they are arranged anti-symmetrically their product results in symmetric sheets and symmetric Lie hypermatrix algebra.
Theorem 5 a) The extended dihedral D * 3 Lie algebra associated with symmetric and (odd, odd, odd, odd) anti-symmetric, trace zero type, four dimensional W * hypermatrices generates a semisimple Lie hypermatrix algebra W * * with symmetric sheets.
b) The extended dihedral D * 3 Lie algebra associated with skew-symmetric trace zero four dimensional W*hypermatrices type W * (odd, odd, odd, odd) generates a semisimple Lie hypermatrix algebra W * * with trivial sheets.

c) The extended dihedral D *
3 Lie algebra associated with symmetric trace zero four dimensional W * hypermatrices type W * (odd, odd, even, even) or W*(even, even, odd, odd) generates a trace-zero Lie hypermatrix algebra if the sheets are symmetric and the hypermatrices are anti-symmetric and generates an acs Lie hypermatrix algebra if the sheets are symmetric and the hypermatrices are of general type.Proof: a) I have shown above and it can be seen from the resulting products in table 4 that the (even, even) symmetric element and the (odd, odd) anti-symmetric elements of the second extension W * have trace zero as hypermatrices so when we apply the Lie bracket operation to symmetric hypermetrics we obtain the following W * * extension:

{(WH
trace zero hypermatrix algebra .In table 4 if the generators elements {t i } have trace zero, there are two possibilities: either W * has symmetric sheets, or (odd, odd ,odd, odd) type antisymmetric sheets, as noted above the hypermatrices W * * are globally symmetric because they are generated by symmetric elements type W i,i; j, j and have symmetric trivial sheets if generated by even hypermatrices W(t) and symmetric trivial sheets W * * ((α, β), ..., (α, β)) = W(0) * * if generated by odd hypermatrices W(t).
So the extended dihedral Lie algebra associated with W * skew-symmetric four dimensional hypermatrices generates the semisimple Lie hypermatrix algebra W * * with trivial sheets.
The product of the elements in W i,i; j, j and W k,k;l,l are symmetric trace-zero sheets, their Lie product is either trivial or has an anti-clockwise-symmetric (acs) sheet structure because the constituent sub-matrices are pair-wise-symmetric and acs.We may note that the odd elements W 5 , W 6 are symmetric and we also have the relations t acs 5 = t 6 , t acs 4 = t 2 .But, their product with acs elements results in a acs type sheet and, therefore, all resulting products must be acs type sheets, and so are the resulting hypermatrices.That follows because the product of acs type sheet with another acs type sheet is an acs type sheet.We note that a linear combination of acs matrices is also an acs matrix, and generally for α, β type acs we have α acs = α, β acs = β, therefore, (βα) = (α acs β acs ) acs .Also every symmetric matrix is an acs matrix but not every acs matrix is symmetric.Hence the Lie product of acs hypermatrices produces an acs Lie hypermatrix algebra.
If the elements are symmetric and the hypermatrices are anti-symmetric the resulting product is trivial and the resulting Lie hypermatrix algebra has trace zero (Schreiber, 2012); it is not commutative hence it is semisimple.

Open Questions
Describe the extended dihedral D * ×n n Lie hypermatrix algebra and all iterations associated with symmetric, anti-symmetric, and skew-symmetric sheets composing the Lie hypermatrices algebra of D 5 , W 6,6 } composed from the dihedral group elements t i , i = 1, ..., 6 has special unique properties.Under bracket multiplication it has the non-commutative classical group structure as shown in <Table 4>.The matrices with even permutation elements (t 1 , t 4 , t 2 ) constitute a normal alternating subgroup of the dihedral group D 3 .Therefore, the hypermatrices with even-even permutation elements {W 1,1 , W 2,2 , W 4,4 } constitute a normal alternating hypermatrix subgroup (defined below) of the dihedral Lie hypermatrix algebraD * 3 {W 1,1 , W 2,2 , W 3,3 , W 4,4 , W 5,5 , W 6,6 } symmetric .The Lie algebra D * 3 has trace zero, and it is skew symmetric.By definition D * 3 is the extended Lie hypermatrix algebra of D 3 , while D * ×n n is the nth-extended Lie hypermatrix algebra of D n .
The determinant of an n-dimensional cubic hypermatrix with k−sheets W s1,s2,,sk ∈ {W * ×n , * × n is the nth extension} is defined by the signed sum of transversing products ∑ (σ 1 ,...,σ n )∈S n ,(S σ1 ,...,S σn )∈S n sgn ∏ α 1σS σ1 ...α nσnS σn In table 4 if the generators elements {t i } have trace zero, there are two possibilities: a) either W * has symmetric sheets, or b) (odd, odd, odd, odd) type anti-symmetric sheets, in any case the hypermatrices W * * are globally symmetric because they are generated by symmetric elements type W i,i; j, j which result in symmetric sheets if generated by an (even, even) = even ×2 hypermatrices W(t) and skewsymmetric sheets if generated by odd ×2 hypermatrices W(t).b)If we apply the Lie bracket operation to skew-symmetric hypermatrics W * in table 4 we obtain the following W * *

Table 1 .
nth-extended Lie hypermatrix algebra of the elements of the classical group D n .Cayley, A. (1843).In Muir Thomas, Metzler William 1933.A Treatise on the Theory of Determinants.New York: Dover publication, pp., 745-756.Humphreys, E. James.(1972).Introduction to Lie algebra and Representation Theory.Springer-Verlag.Jacobson, Nathan.(1962).Lie Algebra.New York: Dover publication.Multiplication table for the dihedral group D 3The even permutation elements (t 1 , t 4 , t 2 ) constitute a normal subgroup of the dihedral group D 3 .

Table 4 .
Second extension of symmetric elements of the dihedral group D *