Infinite Lie Algebras Generated by Supersymmetric Hypermatrices

In this paper we study the structure and properties of complex infinite supersymmetric hypermatrices generated by a semisimple basis, exponential sets of hypermatrices, hypermatrix Lie algebra and elements of the group of complex matrices of order two and determinant one. We study the hypermatrix Lie algebra generated by the polygons on analytic torus of genus g. By using new algebraic tools, namely cubic hypermatrices we study the algebraic structures associated with the hypermatrices of certain Lie algebras e.g. {sl2; f ,∞}; {sl2;∞,∞} and {S L2; f ,∞}; {S L2;∞,∞} and we construct generators of infinite periodic hypermatrix Lie algebraic structures which have classical Lie algebra decomposition; specifically a set of Lie algebras composed of hypermatrices. We study the exponential of a complex analytic Lie algebra, rotations of hypermatrices, and relations between hypermatrix groups, hypermatrix Lie Algebra, Fourier hypermatrices and the Laurent hypermatrix. Finally, as an application we will show that there is an isomorphism of the hypermatrix Lie algebra associated with a set of polygons on the torus of genus g and analytic functions associated with a countable set of solutions of a meromorphic function on the torus. In conclusion we will present a Riemann type isomorphism theorem for hypermatrices on a torus and the convoluted complex plane, generated by holomorphic functions, based on the equivalent relations of the geometry and the algebra of the torus of dimension three and genus g.


Introduction
In this paper we will investigate the algebraic structures associated with infinite hypermatrices and infinite hypermatrix Lie algebra.Hypermatrices were defined in Schreiber (2012a).The paper is based on classical definitions in matrix theory, infinite matrix theory such as described in Cooke (1955), classical Lie algebra see (Humphreys, 1972;Jacobson, 1962;Serre, 1987;Bourbaki, 1980) and previous work we have done on hypermatrices (Schreiber, 2012a;2012b).As application we will study the relations between the field of values associated with a divisor on the torus and its geometry and we will show that there is an isomorphism of hypermatrix Lie algebras structured holomorphically as a set of divisors on the torus and the set of polygons associated with certain convoluted analytic functions represented algebraically on the torus and on a convoluted complex plane (see also Griffith & Harris, 1978).
Definition 1 Lie Algebra of Hypermatrices (Schreiber, 2012a).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: A) WW ∈ W * where (W si , W s j ) ∈ W * , W si a component sheet of W * k , i.e., (W, W) ∈ {×, +, −, W * } ∈ Linear{W * } -a linear combination in W * i sheets.B) 1) the bracket operation is bilinear.
The other possibilities include Laurent type hypermatrices with matrix sheets that have no row or column beginning or end, they will be considered later.

Kojima Conditions
For a given matrix M f,∞ , M ∞, f or M ∞,∞ with components a i, j the necessary and sufficient conditions for transforming every convergent sequence Φ .Φ = Σ ∞ j=1 a i j φ i to another sequence Φ = Σ ∞ j=1 a i j φ i is that a) Σ ∞ j=1 |α i j | ≤ R for all i > i 1 .b) lim i→∞ a i j = α j for fixed j. c) Σ ∞ j=1 |α i j | ≤ M n → α as i → ∞.A special semi infinite example given by the lower triangular Toeplitz matrix T α i, j , α i j = 1/n, 1 ≤ j ≤ i, = 0, j > i.
Definition 3 Let T be an (n × n) matrix (with possible n → ∞) then T α i, j is a Toeplitz matrix, if for all i, j between 1 and n, α i j = 1/n, 1 ≤ j ≤ i, = 0, j > i.
In general a Toeplitz matrix has descending diagonal from left to right and it is constant along the diagonals.
Theorem 1 If W ∞, f ; f is a lower triangular hypermatrix whose sheets have infinite dimension (each sheet is invertible and it satisfies Kojima conditions, see Cooke, 1955), then ∃ a unique right hand hypermatrix W −1 f,∞ which is lower triangular such that ∀W i ∈ {W} if DW = 0 and for all sheets S i of W we have DW si 0, ∀i, D-the determinant of W (Schreiber, 2012b), then W −1 the inverse element of.
Proof.If W is finite then there is a matrix W −1 such that WW −1 = I * (see Schreiber, 2012a) and we note that in the finite case W −1 is lower triangular when W is lower triangular.In the infinite case W −1 is lower triangular because the sequences must converge (by Kojima conditions).As in the finite case also in the infinite case DW = 0 is a necessary condition and another condition is that all the sheets be equal for W to be invertible, but it is not a sufficient condition (Schreiber, 2012a).For each sheet of W it is necessary by the Kojima convergent conditions and the finite conditions for invertibility (defined in Schreiber, 2012a) that DWs must not vanish for the product of sub sheets in WW −1 to result in an identity hypermatrix.If W −1 is finite unique and WW −1 W is pair-wise associative then W −1 is a left inverse.In general the set {W ∞,∞ } is not associative, but an invertible set of hypermatrices is Vol. 4, No. 5;2012 associative, e.g., a set of lower triangular hypermatrics {W ∞, f ; f }, or upper triangular hypermatrics without loss of generality.In short, we have the following Theorem 2 If W ∞, f ; f is an invertible lower triangular hypermatrix satisfying the Kojima conditions there exists an invertible hypermatrix W −1 such that WW −1 = W −1 W = I * .
Notation: If the number of sheets in W is infinite and the sheets in W are infinite (rows and columns) the resulting product of sheets in is an invertible lower triangular hypermatrix (an infinite cubic hypermatrix, and we may replace lower triangular by upper triangular without loss of generality) satisfying the Kojima conditions, there exists an invertible hypermatrix Proof.If we add to Kojima conditions the following hypermatrix conditions Kojima conditions plus hypermatrix conditions, K + .
For a given hypermatrix the matrix sheet M ∞,∞ with components a i, j a necessary and sufficient conditions for transforming every convergent sequence . and require conditions K + for sequences in products and the conditions of theorems one and two above for the invertibility then the theorem follows.Effectively, we add the product conditions for all sub matrices and require convergence under matrix multiplication.

General Infinite Hypermatrices
A finite unitary matrix is a (square) U n×n , or possibly a rectangular U n×m complex matrix or U * m×n matrix satisfying the condition U * U = UU * = I, where I n×m is the identity matrix in n × m dimensions, where U * is the conjugate transpose of U. Note this condition implies that a matrix U is unitary if and only if it has an inverse which is equal to its conjugate transpose U * , U −1 = U * .For infinite hypermatrices we need to adjust the unitary sheets to dimensions n × ∞ or m × ∞; thus an inverse element exists if the product of sequenced sheets in U * U converges.
Definition 5 A hypermatrix composed of sheets which are all lower or upper triangular matrices is said to be a triangular hypermatrix.

Definition 6
The global trace of a hypermatrix W is the sum of the elements along the main diagonals of all the W si sheets of W, gtr(W) = n i=1 trW si .Definition 7 The hypermatrix W is said to be a Hermitian hypermatrix if W h = W.Here h is the transposed complex conjugate operation of matrix theory (I use the notation h instead of the usual notation of * because * is reserved here for the multiplicative extension of hypermatrices).
If for a set of hypermatrices over the complex numbers {W} ∈ C, C the complex field, W h = W for all W then trivially all the Hermitian hypermatrices are normal.By right multiplication we obtain W h W = WW h .We may distinguish among two kinds of Hermitian hypermatrices: a) supersymetric -all component matrix sheets are identical, b) not all component sheet are identical.
Theorem 4 The hypermatrix W is normal if and only if all of its component matrix sheets are identical, and each sheet W S i satisfies W S W h S = W h S W S .Proof.By the definition of multiplication of hypermatrices (Schreiber, 2012a) a necessary and sufficient condition for the existence of a conjugate hypermatrix relation is the equality of all its sub-matrix sheets; hence the necessary requirement that W S W h S = W h S W S follows and it is sufficient for normality.Theorem 5 All invertible hypermatrices are normal.
It follows from (Schreiber, 2012a) that all invertible hypermatrices are normal, because the sheets of the invertible hypermatrix are identical and have non-zero determinant; to see that all normal non-trivial hypermatrices are Definition 9 A hypermatrix is said to be skew Hermitian if W h = −W.We note that W h W = −W 2 = WW h , therefore, the skew Hermitian hypermatrices are normal (see also skew symmetric hypermatrices in Schreiber, 2012a).
By the above work on normal hypermatrices we have the next theorem.
Theorem 6 The hypermatrix W u is unitary if and only if its component sheet matrices are unitary and identical.
Proof.We have seen above that a matrix U is unitary if and only if it has an inverse which is equal to its conjugate transpose U * , U −1 = U * .For W u to be unitary , therefore all sheets in W u must be identical and unitary.
Theorem 7 If W is an invertible hypermatrix with distinct eigenvalues (for each sheet) then there exists a unitary hypermatrix W U with all sheets identical such that W h U WW U = W * * D , W D -diagonal hypermatrix.Proof.The unitary matrix that does it to one sheet of W will be the sheet component of W u , the same W u that satisfies the above theorem.By Schur's theorem (matrix theory) if A ∈ M n , with n-eigenvalues λ i , then there exist a unitary matrix U ∈ M n , such that U h AU = T , T a triangular matrix with λ i strung along the main diagonal, in a prescribed order.
Definition 12 A hypermatrix which is made of triangular sheets, either all upper, or lower triangular sheets is said to be strictly triangular hypermatrix.

Diagonal & Triangular Hypermatrices
A hypermatrix which is composed of lower triangular and upper triangular sheets is said to be of mixed type.Any proper (non trivial) mixed type hypermatrix may be decomposed into a sum of hypermatrices as follows: W = W upper−triangular + W lower−triangular + Wd iagonal .
Theorem 8 If W ∞, f is a hypermatrix satisfying the Kojima conditions, and in it each pair of sheets commutes, then there exist a pair of unitary Kojima infinite hypermatrices WU ∞, f , and a infinite hypermatrix Depending on the convergent conditions Ψ T ( f, f ) is triangular hypermatrix, or a hypervector (1 × n, n − times).
Proof.For all matrices M ∈ W ∞, f satisfying Kojima conditions ∃ a unitary matrix U the hypermatrices structured from the matrices satisfy ) sheets are pair-wise associative (see also Schreiber, 2012b).If each pair of matrices in W commutes then they can be simultaneously triangulated and for any two sheets k, l we can write W k W l = W l W k .Therefore, the global trace gtr(W k W l ) = gtr(W l W k ) for any two sheets in W. By Schur theorem the existence of unitary U S matrices satisfying the triangulation of each pair is guaranteed.But in W all the commuting sheets are equal; therefore, by Schur theorem the existence of unitary WU hypermatrices satisfying the triangulation conditions also follows.
Theorem 9 If W ∞, f is an invertible hypermatrix satisfying the Kojima conditions, then there exist a unitary hyper- Where Ψ D( f, f ) is a diagonal hypermatrix or a hypervector.
Proof.Using the above lines of proof note that if all sheets of WT ( f, f ) are invertible they are normal and by Theorem 7 and schurs theorem there exists a unitary matrix such that Ψ D( f, f ) is diagonal hypermatrix, or a vector.

Hypermatrix Lie Algebra Associated with S L 2
S L 2 is the group of complex matrices of order 2 and determinant equal 1.It is a complex Lie group with Lie algebra sl 2 .When we apply to the elements X, Y, H of sl 2 the exponential function it generates sub-groups.We have the standard relations with respect to x, y, h: For example the exponential of e x is The summation of the sequence is conditioned on the convergence of the sequence term by term and on summation of hypermatrix sub-products.The size of the component hypermatrices changes at each step, therefore, summation is generally impossible.Since W x is nilpotent we are actually summing only the first two terms in e W xt .If we consider e ht then summation is conditioned on the completion of W n−1 to W n by adding trivial sheets term by term and deciding where to stop adding terms.So in order to sum e W xt , e Wyt , and e Wht they will have to have the same number of terms in conjunction along the developing sequence.
In general for n = 2 we have a 4-sheet-hypermatrix, but the exponential of a generator in general will result in infinite sum of hypermatrices, in which most are trivial.Similarly for y and h we have: In short, let e Wh(t) be written as e Wh(t) = (., ., ., ., ...) right to left infinitely many time, with most dots standing for trivial matrices.
Open Problems: a) What is the structural relation between the extended Lie hypermatrix algebras of {W S L } and {W sl }? b) What are the characteristic of the extended algebra ELHA?
Proof.We note that in the finite case for two by two matrices A 2×2 ∈ sl 2 we have constructed nine hypermatrices (see Schreiber, 2012a) while in the infinite Hypermatrix construction we have the elements of sl 2 set in an infinite set of cubes {x, y, h : W sl(2,2);∞ }.Which results in {W} ∈ W sl : (2, 2); ∞.The semisimplicity of the algebra {W sl : (2, 2); ∞} follows by induction and induction on dimension for higher dimensional hypermatrices.

The Exponential Complex Analytic Hypermatrix Lie Algebra
I define the exponential of complex hypermatrices as follows: For example the exponential of e iW is In short ∞ were the number of sheets of EXP(W) is finite or infinite depending on whether W is proper or improper, in terms of converging according to Kojima conditions.
Definition 13 A hypermatrix W ∞,∞;∞ is proper if it satisfies the Kojima conditions for webs in W(C) space, otherwise it is improper.

Symmetry Properties of Complex Hypermatrices
A point α in W(C) space is symmetric or conjugate to a point β with respect to some axis If α and β are of the form α n , or β n then the symmetry is with respect to an orthogonal basis in a normalized space C n .Generally, e ±iW does not generate symmetry with respect to the origin.

The Exponential of Kojima Hypermatrices
If W 1 , W 2 are Kojima and e iW 1 e iW 2 are well defined, then in certain domains, e.g., for F D , we have Choose a numberε > 0 the exponential mapping is one to one on the set {W Kε }.We also take δ > 0 δ < ε, and exp(Wδ) × exp(Wρ) ⊂ exp(W Kε ).Hence the mapping of f defines analytic mapping of , ∀ sufficiently small t and furthermore the series Lemma 1 A continuous finite dimensional representation of a hypermatrix Lie algebra WLr ∈ WK (Kojima hypermatrix Lie algebra) is (real) analytic.
Theorem 10 Let {W Kr } be a Kojima hypermatrix Lie algebra with components {W ∞,∞;n } (Kojima convergent conditions satisfied), W Kr is simply connected, and the representation of W Kr is a real analytic hypermatrix Lie algebra, under the exponentiation {e w ∞,∞;n }.
Proof.By Lemma 1 W Kr is real analytic.Any hypermatrix component is simply connected by Lie's third theorem and induction on dimension the entire algebra is simply connected.Since W Kr is a Kojima hypermatrix algebra and its exponentiation is a finite sheet hypermatrix algebra and a hypermatrix Lie algebra with {e w∞,∞;n } ∈ {W ∞,∞;n }.Therefore, it is (real) analytic as well as simply connected (m ≤ n).
Proof.If {WL r1 } is Kojima hypermatrix Lie algebra then {WL r1 } is simply connected and real analytic by the last theorem.If {WK r2 } is a hypermatrix Lie algebra such that its exponential representation {e k2 } → {W ∞,∞;n } is real analytic.It follows from the last theorem that the algebra {W ∞,∞;n } is a simply connected Kojima hypermatrix Lie algebra.
Kojima hypermatrix Lie algebras which are simply connected and real analytic are homeomorphic.Therefore, among the e kr2 hypermatrix Lie algebra's there is at least one algebra {WK q } = {WK 2 }, i.e., WK 2 ⊂ {e k2 }, and so ∃WK 1 , WK 2 ⊂ {Wk ∞,∞;m }, the invertibility of the mapping defined by (By Definition 7 the hypermatrix W is said to be a Hermitian hypermatrix if W h = W, h is the transposed complex conjugate).

Relations between Hypermatrix Groups, Hypermatrix Lie Algebra and Fourier Hypermatrices
Theorem 14 If G = {W}, × is a finite dimensional matrix group (e.g., D3, see Schreiber, 2012b) and {W n,n;k } is a set of hypermatrices composed of a finite arrangement of all elements of the matrix group G then the components of {W n,n;k } generates a hypermatrix Lie algebra which is isomorphic to the elements of the exponential extension given by {e i f W n,n;k }.
Proof.If the hypermatrices are structured from the elements of a finite group of matrices, or a finite group representation of n-polygon structure (e.g., D3, described in Schreiber, 2012b); the exponential extension of these hypermatrices might terminate after n-steps of the exponential sequence expansion and converge or they may diverge.In any case we may limit the series to a finite number of steps such that the generated set of elements constitutes a clearly defined set and generate a basis for the Lie hypermatrix algebra.Since it is generated from an exponential basis they are 1-1 homeomorphic and isomorphic.An example was given in section 3, and for Vol. 4, No. 5; 2012 non-exponential expansion in Schreiber (2012b).
If the groups is a Kojima hypermatrix group G k , ×, + (composed of Kojima set of hypermatrices), and the hypermatrices are Kojima lower triangular infinite hypermatrices {W T,∞ } the resulting exponentiation is converging, and the resulting hypermatrix algebra will have a bound on the number of sheets, for each hypermatrix.If the sheets are finite {e i f W ∞,n; j }, {e i f W n,∞; j }, we will have a finite converging sequence and well defined set of hypermatrices for all the components of W in the hypermatrix algebra {e i f W n,n;k }.
If the set {W} is Kojima, and we consider {W} with components vectors-series in W Lk then the Fourier series E = E 0 e i f W n,n;k has a finite convergent series.For example if the wave function describes a spin (s), and period ω of an elementary boson particle system then E = E 0 e i f W n,n;k describes the energy for many bosons, e.g., E converges for Kojima hypermatrices, and diverges otherwise.Note that in these Fourier settings of the physical systems the energy and some of the physical characteristics of the system are expending and changing dimensions according to the series expansion we use and convergent properties we might have in a particular problem; it is an open algebraic system in the sense that the constituent elements might change dimensions with the dynamics of the system and with relation to the operations performed.

Laurent Hypermatrices
We consider {W Lk (Ψ z )} to be the set of hypermatrices structured from the Laurent matrix sheets in a Kojima space of all possible arrangements of the hypermatrices.Here W α n are structured from complex numbers and W Lk is doubly infinite (∞, ∞).
Next we consider the associated complex Fourier hypermatrix series structured out of elements hypermatrices ω) where the coefficients are given by C n = 1/ω Proof.The idea of the proof is that if W(Ψ μ ) is bounded and if C n is bounded then the finite sum of a convergent series is convergent on a simply connected convoluted hyperspace C n and therefore f (W K (Ψ z )) converges uniformly.
In the next section we consider applications of holomorphic functions and complex hypermatrices to the representation of the torus T g and properties of the convoluted direct sum series.

The Classical Construction of Standard Basis for Analytic Torus T g Using a Skew Hermitian Basis on T 2,g and Isomorphism of Infinite Supersymmetric Hypermatrix Lie Algebras
Consider the even 2 dimensional tours (the Riemann surface or an isomorphic one dimensional compact complex connected differential manifold) T 2,g , n = 1, ..., j, j < ∞ and assume that it is smooth, without holes (Griffith & Harris, 1978).We may construct the standard basis for the analytic torus T g using holomorphic functions on T 2n,g .For the Riemann surface with g = 1 the standard construction by all possible holomorphic functions f (z i ) over a point p is such that, mapping from the region Ω to C, f (z) : Ω → C is analytic.The function f (z) over the space of functions Ω f (α) is defined around a local coordinate system of an effective divisor D = Σp i ∈ T g with μ : T g → J and is given by the associated set of holomorphic functions to the Jacobian J(μ) at a point D, denoted by Hartshorne, 1977;Horen & Johnson, 1991).
Given a fixed point, at a disc, and analytic cycles on T 2,g classically there exists a skew Hermitian basis which is Vol. 4, No. 5;2012 given by the matrix By the Toeplitz-Hausdorff theorem the field values for Ξ normal is given by F(Ξ) = Co(σ(Ξ)) where the convex hull Co spectrum of eigenvalues σ(Ξ) determines the resulting convex polygon on C. The field of values of the normalized Hermitian matrices F(Ξ) is a set of complex numbers associated with a set of matrices; it might be a continuum while the spectrum σ(Ξ) is a discrete countable set of values.

The Field of Values Associated with the Torus
The field of values of an nn matrix M g×g is given by the set of complex numbers (g × g) ways on the torus T g .The set of {W g } hypermatrices composed of the M g×g matrices on the torus T g constitutes a skew Hermitian hypermatrix Lie algebra (for the construction of a skew Hermitian hypermatrix Lie algebra see Schreiber, 2012a).It has a set of values F(W g ) possibly continuous such that σ(Ξ w ) ⊆ F(W g ), and if we know the field of values associated with the hypermatrices F({W g 1 }) and F({W g 2 }) we can say that the spectrum of W 1 and W 2 has the following additive property σ({Ξ} , for a discussion on the field of values of matrices see Horen and Johnson (1991).
Consider the following long exact sequence of homeomorphisms of hypermatrices associated with a divisor D on the torus T g and an exact sequence Maclane, 1963) and use it to construct the following exact diagram of hypermatrices and sequences of polygons Each horizontal sequence is exact as F(W g i ) modules or P j groups (For examples, I use modules of polygons and groups of polygons, modules of hypermatrices with {WR hyper }, ×, + being a ring structure with commuting squares in the diagram, and commuting diagrams squares having invertible arrows; see also Schreiber (2012b) for hypermatrix groups G hyper , ×, + from which the extended ring {WR hyper }, ×, + of hypermatrices is structured.
The set of polygons {P j } on a convoluted set of a sum of copies C ∈ C n such that Σ C n ∈ C n constitutes for W g × C ∈ C n a complexified (permuted) hypermatrix Lie algebra with a bracket operation {W g }; w × w ∈ w * ; (w i , w j ) + (w j , w i ) = 0 * ; (w i , (w j , w k )) + (w j , (w k , w i )) + (w k , (w i , w j )) = 0 * * } (Schreiber, 2012a).The algebra of the P i polygons may be set into even, and odd sets of matrices (e.g., see the 3-gon example in Schreiber, 2012b).The dimensions of these hypermatrices are determined by the field of values F(WT 2n,g ) which is defined by the number of vertices on the polygons being used as divisor on the torus.The polygons determine the number of even and odd elements in the Lie hypermatrix algebra.As the dimension and genus of the torus increases the algebraic subdivision of the hypermatrix Lie algebra representation is characterized by a variety of sets of hypermatrices with unique transpositions related to the oddness and evenness of the permuted n-gon sub-structures.These subalgebraic Lie structures are part of the general hypermatrix Lie algebra; they are nested along the main diagonals of each extended hypermatrix algebraic representation (see the tables in Schreiber, 2012a & b).They characterize the hypermatrix algebra together with the minor diagonals, characteristic skew symmetry, symmetry, and some of the sub-structures might be reflected along the diagonal of the extended algebraic representation.In the extended Lie algebra the hypermatrices and sub-Lie algebra could be Hermitian or skew Herrmitian hypermatrices; these mainly characterize the even hypermatrices.The odd hypermatrices and the mixture of odd-even hypermatrices are characterized by interchanges of rows and columns represented by complexified skew-symmetric hypermatrices (see tables two, three and four in Schreiber (2012b) for the 3-gon-triangles).
To sum up, it is possible to represent the characteristics torus T g by a hypermatrix Lie algebra structured from a countable permuted polygon basis T g .It also has a holomorphic representation by a set of functions Ω f (α) defined around a local coordinate system of an effective divisor D = Σp i ∈ T g .

Construction of Hypermatrix Lie Algebra on the Torus T 3ν,g
If we consider the hypermatrix Lie algebra on a compact connected differentiable Torus T 3ν,g ν = 2k + 1, k = 0, 1, ..., g ≥ 1, we find that it could be represented globally by a skew Hermitian sub-Lie algebra with the even Vol.4, No. 5; 2012 elements represented by T even = I n×n sub-matrix and the odd elements are represented classically by the Hermitian matrices The resulting hypermatrix set {W T g } is precisely the even generators of the field of values, representing polygons on a convoluted direct sum of copies of C, Σζ = C 1 C 2 ... C n ∈ C n (see remark 2 below).Multiplying the elements W T i , W T j ∈ {W T g }, we obtain an extension set {W * T g } which was shown to be a hypermatrix Lie algebra, this Lie hypermatrix algebra is infinitely generated, infinitely extended from a relatively simple holomorphic basis, the technique for semisimple extension of Lie hypermatrix algebra is described in Schreiber (2012a).
To construct the initial basis for the extended hypermatrix Lie algebra out of polygons around an effective divisor we use the even-even...even, odd-odd...odd, odd-even..., ...even-odd, ... permutations or an appropriate elements arrangement.The resulting extended hypermatrix Lie algebra is characterized by a complex variety of algebraic properties: symmetric by transposition, by reflection on main diagonal, or by even sheet interchange depending on the basis elements generating the algebra.Skew-symmetric transpositions, row or column interchange characterizes the odd elements in a polygonal hypermatrix Lie algebra (Schreiber, 2012a & b).
As a direct consequence of the above commutative diagram of exact sequences, and the structure of the hypermatrix Lie algebras on the torus T g which has a characteristic Lie structures {W T g }; w × w ∈ w * ; (w i , w j ) + (w j , w i ) = 0 * ; (w i , (w j , w k )) + (w j , (w k , w i )) + (w k , (w i , w j )) = 0 * * , and the polygon algebra associated with {W Pi }, we would want to show the polygon hypermatrix Lie algebra on Σζ and the hypermatrix Lie algebra generated by constant and non-constant holomorphic functions on the torus {W T 3n,g } are isomorphic; stated in the following theorem.
Remark 1 We note that that it follows from ( 13) that ν = 2k + 1, g and j are related by some function Φ(ν, g) = Σ (W T , j) to be determined.Here g determines the distribution of eigenhypermatrix-values on Σ C n (for the definition of eigenhypermatrix-values HW − λI = 0, see Schreiber, 2012b) and the structure of the convex j-polygon, W T determines the j-polygons as well as the associated coefficients.
Remark 2 Classically the Riemann surface of genus 1 in dimension 3 admits a topological surgery, and cuts, in two ways horizontally and vertically such that the resulting surface is isomorphic to the complex plane (see for example Griffith & Harris, 1978).For the torus T 3,g it is possible to construct a similar set of cuts on T 3,g resulting in convoluted complex plane which is isomorphic to a sum of copies of C, denoted Σζ which is also related to Φ by Φ(ν, g) = Σ(ζ, j).
Proof.Consider the 3 dimensional torus of genus g then globally there is a basis for the representation of the holomorphic functions áω ... αω f (ω, z)dω 1 ...dω n in the region Ω ∈ ΣC n given by the following skew Hermitian matrices Locally a hypermatrix basis of the Lie algebra is generated from W(Ξ) = I k * n×n from which we may construct the hypermatrix Lie algebra {W T 3ν }.On Σ C n we consider all j-polygons generating the field of values of F(W T 3ν,g ), a countable set of j-sided polygons over a divisor, from which the hypermatrix Lie algebra is structured.To see the construction a little more clearly we note that for the extended elements Ξ2, and Ξ3 we have , and a T = a These represent a normal typical second extension component sheet; in absolute value (e.g., see Schreiber, 2012b).Note that for The resulting hypermatrix has normal components and a normal representation.We also notice that the left hand side of Equation ( 13) is a representation of the geometry of the T 3ν,g torus by holomorphic functions from which it is possible to generate a hypermatrix Lie algebra; it is related to the distribution of eigen-hyper-matrix-values on the convoluted direct sum of complex connected sheets Σ C n , and has an isomorphic structure to the spectral set of eigen-hyper-matrix-values on T 3n,g .The right hand side is a representation of the torus by a set of j-sided polygons which generates hypermatrix Lie algebra in terms of the field of values on the torus.
Here we actually take a different approach in order to show the isomorphism of the two algebras.In order to show that two extended hypermatrix Lie algebras are isomorphic it is enough to show that they have a 1-1 homeomorphisms at each extended state and that there exist a map kernel( f ) =basis between the two algebras at each stage of extending the hypermatrix algebras.The structure of the homomorphism and kernel map depending on the type of the Lie algebra (semisimple, polygonal and the unique symmetries and asymmetries of the sub hypermatrix algebras: symmetric, skew-symmetric etc', see Schreiber, 2012a & b for characteristic symmetries of Lie hypermatrices).In certain cases, it is sufficient to show that in the first and last stage of algebraic extension there is an isomorphic, or that just in certain stages of algebraic extensions are isomorphic.Here the two algebras have the same extensions at each stage because they are generated by all the symmetries of polygons on the convoluted complex space in one situation and by the counting and permutations of all possible cycles of holomorphic functions on a countable infinite set of points (or around polygons) situated on the three torus.As the hypermatrix extension gets larger there is a greater technical difficulty to show such isomorphism in practice, however, in certain cases we may look at the limit of the extension, the kernel of the extension, and their algebraic structures.
We obtain: lim Ker[Even cycles T (3ν, g)] → even Ext→∞ 0 At the infinite extension limit polygons behave just like circles and cycles, therefore, one can check that the homomorphism holds in the first and second extensions because each permuted set of hypermatrices is structured from one of the classes: even-even...even, odd-odd...odd, odd-even,..., even-odd,...permutations or an appropriate elements arrangement on the convoluted Conv(C p j , Σζ) space and functional meromorphic cycles on the torus T (3ν, g), for any countable divisor D = Σp i ∈ T g .They have a homeomorphic structure for analytic/meromorphic functions.If we consider the even cycles all permutations we find that limit of their extended kernel Lie hypermatrix algebra vanishes.It is a universal property of the polygonal hypermatrix Lie algebra.For intermediate extensions of the other permuted set of hypermatrices the task of establishing the isomorphism is more difficult and requires a careful analysis of the components in each sub-Lie algebra.For example odd-odd elements tend to be symmetric, skew-symmetric or skew-Hermitian in the complex plane.Instead of checking all the elements we could work with each class and establish the isomorphism by showing that the representation exists in each algebraic extension.
Outline of the proof We will show that (17) holds in three steps: a) If the functions f n (z) are analytic and non-zero in a region Ω ∈ Conv(C f n (z)), and if f n (z) converges to f (z), uniformly on every compact subset of Ω, then f (z) is either identically zero or never equal to zero in Ω. b) Moreover, if we consider the set of countable zeros of f n (z) then from the relation of the geometry of T 3,g to the hypermatrix Lie algebra of polygons on Conv(C f n (z)) we find that on any open convoluted region, say Conv n (P j , Σζ), we may apply to the zero points set of solutions of f (z) the associated Lie hypermatrix algebra of holomorphic functions, on Ση at points of the divisor D generated by the associated Lie algebra of polygons (see Schreiber, 2012b).Next we consider elements of the hypermatrix algebra of holomorphic functions f (z) ∈ T 3,g and we will show that the set of zero solutions is in one continuous region of space on the Conv n (P j , Ση) space and as part of the Geometry of T 3,g , it has a single countable representation in an extendable hypermatrix Lie algebra (see Schreiber, 2012a & b, Tables 2 & 4, respectively).c) Geometrically the real coordinates of the zero solution set of holomorphic functions type ζ(s) on Conv(C n ) corresponds homeomorphically to a countable set of zero solutions of ζ(s) on T 3,g .To show this we use the isomorphism relations in (15).
Algebraically the basic important structures enabling the proof of the isomorphism theorems and relation ( 17) is the structure of the diagonal elements (all elements of the form (w, w), w ∈ {W} and the associated sub-Lie algebras classes arranged by even-odd permutations and semisimplicity on the extended Lie hypermatrix extended representation; generally, if in each extended algebraic representation {(w, w)} Conv(ΣC n ) {(w, w)} T 3,g there is a homomorphism of diagonal structures and a one to one arrangement of hypermatrix sub-algebras and kernel map will suffice to clinch the isomorphism theorem, see also schreiber, 2012a & b) representation of the infinitely extended hypermatrix algebra associated with holomorphic and meromorphic functions on T 3,g and Conv(ΣC n ).
Proof.a) Extend the Hurwitz theorem to the torus T 3,g covered with holomorphic functions around points p i ∈ D, the divisor D, and similarly on the convoluted space Conv(ΣC n ) used as a basis for the hypermatrix Lie algebra structured by a set of polygons around the countable divisor of a set of zeros of ζ(s).b) Construct the standard basis for analytic torus T 3,g using holomorphic functions around a set of countable points p i of a divisor D on T 3,g .For the Riemann surface with g ≥ 1 the standard construction by all possible holomorphic functions f (z i ) over a point p i of a divisor D p is such that the mapping from the region Ω to C, f (z) : Ω → C is analytic.The function f (z) over the region Ω f is defined around a local coordinate system of an effective divisor D = Σp i ∈ T g with μ : T g → J and it is given by the associated set of holomorphic functions with the Jacobian J(μ) at a point p i ∈ D. The extension of these Ω f functional construction to Σζ, using a generated Lie hypermatrix

Definition 11
The eigenvalues of W are the solutions of the determinant D(W − λ) = 0, D was defined in Schreiber (2012b) .The solution for D(W − λ) = 0 are the alternating sum of main transversals +DW.D(W − λ) = alternating sum of main transversals +DW = Signed sum of transversals +DW.
W 5 , W 9 are the invertible hypermatrices.The non invertible elements have bracket relations given by ( Denote by R 180• the hypermatrix rotation by 180 degrees along an axis or a diagonal line of the hypermatrix.Denote the transformed hypermatrix W by W R180T or W RT .Theorem 12 If W RT = W then there exists a non trivial unitary hypermatrix W U W U W RT W * U = W T .Where W Tis a triangular hypermatrix.(A unitary hypermatrix satisfies W h U W U = W U W h U = I * , with ( * ) denoting the hypermatrix extension under multiplication, see Theorem 6, Definition 8).Proof.If W/ − RT = W, then W RT W = WW RT , and if W and W RT commute so are their component sheets, thus W has commuting elements, and therefore, if any two elements commute W U W RT W * U = W U W T W * U = W T .Theorem 13 If W RT = W then there exists a unitary hypermatrix W U W U W RT W h U = W D .(By Theorem 7 a W D -diagonal hypermatrix satisfies W h U WW U = W * * D , * * -the second extension).Proof.Rotate W k,k;n along the main diagonal 180 degrees, then for 2 ≤ k ≤ ∞, n -finite if W RT = W and the sheets of W are symmetric to start with because W RT = W, W is Hermitian and W U WW * U = W U W * W * U hence the result follows.
and β * = −β.Vol.4, No. 5;2012 in f inite extension Ker[Even cycles on Conv(C p j , Σζ)] → ζ Ext→∞ 0 Constructing the hypermatrix algebra of {W S L } we find that each element is an element of semisimple hypermatrix Lie algebra {W S L ; (2, 2); ∞ }.That follows from the fact that {W S L ; (2, 2); ∞ , W S L ; (2, 2); ∞ } ∈ {W sl ; (2, 2); ∞ } is an imbedding for all elements of W S L .The integration of {W S L ; (2, 2); ∞ , W S L ; (2, 2); ∞ } with respect to (t) in the complex field C results in (Ahlfors, 1979)inite polygonal hypermatrix Lie algebra structured over analytic manifolds and over countable divisors we can check the isomorphism problem by checking systematically all possibilities, in practice, e.g.An Application of Complex Hypermatrices to the Solution of Holomorphic Functions on the Convoluted Complex Spaces Σζ[Conv(C P j,Σζ )] and Solutions of Meromorphic Functions on the Torus T 3ν,g By the Hurwitz theorem(Ahlfors, 1979)if the functions f n (z) are analytic and nonzero in a region Ω ∈ C, and if f n (z) converges to f (z), uniformly on every compact subset of Ω, then f (z) is either identically zero or never equal to zero on Ω.For example, the infinite analytic series Σ n=1 n −σ converges uniformly for all real σ greater or equal to a fixed σ 0 > 1.It is the majorant of the infinite Riemann ζ series ζ(s) = Σ n=1 n −s , s = (σ + it), which represents an analytic function in the half plane Re(s) > 1. Classically the integral of ζ(s), ζ(s) is convergent in the entire plane and by Cauchy's theorem it's value does not depend on the shape of curve if it does not enclose a multiple of 2πi.The onto mapping of hypermatrix groups G H × G H → G H denoted (g, h) → gh −1 is an analytic mapping of manifolds in an extended higher product space (gh −1 ) * .Theorem 17 Geometrically, the coordinates of the set of zero solutions of the function ζ(s) on the complex convoluted space Conv(ΣC n ) has a representation by a (separated set of points) g-convoluted on a connected line L ΣT i (ζ(s))∈T 3,g .Algebraically, the solution set of holomorphic function represented by ζ(s) on T 3,g is a linear countable set of points which corresponds 1-1 to a countable linear set of solutions of ζ(s) on Conv(ΣC n ).Geometry o f LΣT i (ζ(s)) ∈T 3ν,g Algebra Zeros o f ζ(s) ∈Conv(C P j ,Σζ)