On Functions of Several Split-Quaternionic Variables

Quaternions are known to have deep relation to the selfdual Yang-Mills equations in mathematical physics. It is also known that the self-duality equations have an “indefinite” version, the basic conformal metric used in their definition has signature (2, 2). Such equations are not elliptic, but it is known for more than 30 years that many of the integrable systems arise as a reduction of the indefinite self-duality equations [1]. It is also known that the geometry of a superstring with N = 2 supersymmetry was shown in [2, 3] to be described by a space-time with a pseudo-Kähler metric of signature (2, 2), whose curvature satisfies the (anti) selfduality equations. The split quaternions are indefinite analog of quaternions and play similar role in the indefinite self-duality equations as the quaternions play in the positive definite case. It is well known that spaces with quaternionic-like structures (e.g., quaternionic-kähler and hyperkähler) form an active area of research. One topic in it is developing the notion of quaternionic plurisubharmonic functions [4]. Similar to the quaternionic case, there are geometric structures on manifolds of dimension greater than four, related to the split quaternions. Mathematically, these structures are described by quadruples (g, I, S, T), where g is a signature (2, 2) metric and I, S, T are parallel endomorphisms of the tangent bundle with respect to the Levi-Civita connection of g, such that


Introduction
Quaternions are known to have deep relation to the selfdual Yang-Mills equations in mathematical physics.It is also known that the self-duality equations have an "indefinite" version, the basic conformal metric used in their definition has signature (2,2).Such equations are not elliptic, but it is known for more than 30 years that many of the integrable systems arise as a reduction of the indefinite self-duality equations [1].It is also known that the geometry of a superstring with  = 2 supersymmetry was shown in [2,3] to be described by a space-time with a pseudo-Kähler metric of signature (2,2), whose curvature satisfies the (anti) selfduality equations.
The split quaternions are indefinite analog of quaternions and play similar role in the indefinite self-duality equations as the quaternions play in the positive definite case.It is well known that spaces with quaternionic-like structures (e.g., quaternionic-kähler and hyperkähler) form an active area of research.One topic in it is developing the notion of quaternionic plurisubharmonic functions [4].Similar to the quaternionic case, there are geometric structures on manifolds of dimension greater than four, related to the split quaternions.Mathematically, these structures are described by quadruples (, , , ), where  is a signature (2, 2) metric and , ,  are parallel endomorphisms of the tangent bundle with respect to the Levi-Civita connection of , such that  2 = − 2 = −1,  =  = −,  (, ) = − (, ) =  (, ) . (1) In the literature such structures are called hypersymplectic [5], neutral hyperkähler [6], parahyperkähler [7,8], pseudohyperkähler [9], and so forth.A more general condition is when , ,  are parallel with respect to a connection with skew-symmetric torsion; such structures are considered in [10].One of the features is the existence of a nondegenerate (2, 0)-form given by (, ) = (, ) + (, ).Locally the metric arises from a single function, called potential, similar to the Kähler metrics.The function satisfies  ∘  ∘  = .In the quaternionic case, such potentials in multidimensional quaternionic space H  correspond to a quaternionic plurisubharmonic functions and were considered from analytical view point first by Alesker [4].
The aim of this paper is to provide an analog of the results in [4] for functions of split-quaternionic variables.Although it is unlikely to find an appropriate definition of plurisubharmonic function because of the indefiniteness, a meaning of determinant of a split-quaternionic-Hermitian matrix can be given.As a main result in the paper we show 2 Advances in Mathematical Physics that  ∘  ∘  = det() 1 ∧ ⋅ ⋅ ⋅ ∧  2 , where det is the Moore determinant of the quaternionic Hessian of .The proof is similar to the proof in [4] and relies on a linear change of variables formula and the density of the delta functions of split-quaternionic hyperplanes in H   , which is proven by Graev [11].We notice also that split-Lagrangian calibrations of [12] can be defined naturally for metrics arising from such functions  for which det() ̸ = 0.

The Split
whose usefulness becomes obvious when we notice that it preserves the norm; that is, The embedding  above is extended to vectors in H   by the map where . . .
Similarly for an  ×  split-quaternionic matrix, we define its adjoint matrix as the image of the extension of  to  2 (C) given by where A related homomorphism is that of the Study map, which we denote by , which can be generalized for a matrix   (H  ) ∋  =  +  ⋅ , where ,  ∈   (C): The Study map also has an associated determinant, where det C is the standard determinant of complex matrices, given by Sdet () fl det C ( ()) . ( For   =  2 +  2+1 , we have ( 0 ⋅  1 ) = ( 0 ) ⋅ ( 1 ) and we can check that So it is natural to expect that this representation extends linearly to matrix groups, that is, the following.Proposition 1.For any split-quaternionic matrices  and , we have the following: (1) () = ()() for  =  × and  =  × , where the superscripts are the respective dimensions.
(3) For  ∈   (H  ), () = (), where  fl () = diag ( 0 1 1 0 ).Furthermore, if  ∈   (H  ) is unitary, then ()()  = ; that is,  is symplectic, and and by the property above we have Hence For the second claim, if the adjoint of  ∈ (,H  ) is unitary as (2 × 2) matrix, then we show that () is symplectic; that is, ()( but by (12) we have that ( * ) = () *  and combined with ( 14): we get that and by multiplying the above equation by  on the right we obtain as needed.
Following Wan and Wang [13] we define on C 2 ∋  = ( 0 , . . .,  2−1 ) the corresponding first-order differential operators that act as partial derivatives with respect to the variable in the ()th entry,  = 0, . . ., 2 − 1, and  = 0 or 1, in (6), ) ) ) ) and use them to define the analogs of  and  in complex analysis for the purposes of our study of functions on H   .We will denote by   ,   the partial derivatives    and    for notational simplicity.
Definition 2 (the Baston operator).Let  ⊂ R 4 be a domain.The operators  0 ,  1 , and Δ are defined as

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Due to the particular embedding we chose, we can simplify notations and calculations, summarized in the following lemma.
To demonstrate (2), we use commutation of Proving the final claim is done in two nearly identical calculations, depending on the parity of  and .We will prove one case, where without loss of generality we assume  is even and we still have  < .This means that (in the other case if  was odd we would have the completely symmetric situation, with 2 + 1's becoming 2's, etc.).So we calculate (26) Definition 4 (mixed Baston product).For  1 , . . .,   ∈  2 we define the mixed Baston product of  1 , . . .,   as where  = {1, 2, . . ., 2} and is defined to be the sign of the permutation from In particular, for  1 =  2 = ⋅ ⋅ ⋅ =   = , the mixed Baston product coincides with the -times wedged Baston of ; that is, The results and definitions above will allow us to translate the -times wedged Baston of  in terms of the "splitquaternionic Hessian" of , defined in terms of the Moore determinant, which are precisely the next sections.

Split-Quaternionic Determinants.
Due to the noncommutativity of the multiplication in H  (just as in H) trying to construct an effective definition of determinants is complicated.There are several ways to define them.The main results in this direction follow the work of E. Study, J. Dieudonné, and E. H. Moore, as outlined by Aslaksen in [14].However, the problem becomes much simpler if we are restricted to hyperhermitian matrices; that is,  ∈ GL  (H  ) such that  =  * ; then we can define a simple and useful determinant following the work of E. H. Moore, called the Moore determinant.This is done by specifying a certain ordering of the factors in ! terms in the sum over permutations of the symmetric group   .
Definition 5 (the Moore determinant, see [14] or [15]).For a permutation  ∈   , write  as a product of disjoint cycles such that the smallest number is at the front of each factor and then sort the disjoint cycles in decreasing order according to the first number of each factor.In other words, write where, for , we have  1 >   for all  > 1, and  11 >  21 > ⋅ ⋅ ⋅ >  1 .Then we define the Moore determinant of a hyperhermitian matrix  = (  ) [denoted by det()] as Another equivalent definition of the Moore determinant is the inductive one (see [15]), defined as, for a hyperhermitian  ×  matrix  = (  ), the inductive definition is given as follows: for  = 1, we have det() fl  11 and for  > 2 for  ∈  = {1, 2, . . ., },   = +1 if  =  and   = −1 if  ̸ = , and (, ) the hyperhermitian ( − 1) × ( − 1) matrix obtained by interchanging the th and th columns and then deleting both the th row and column of the corresponding matrix.For any matrix  ∈   (H  ), it can be easily checked that  ⋅  * =  * ⋅  is also hyperhermitian, which leads to the equalities The Moore determinant is related to the Study determinant from (8) as which is given by the middle equality which can be seen easily by noticing that () and () are similar matrices (having the same exact entries in different arrays, except that the former consists of 4 -blocks and the latter consists of  2 2-blocks) and differ by only elementary operations (shuffling some rows, columns, and signs) so that their complex determinants are equal.
Again focusing on hyperhermitian matrices, we can manipulate them to get what are also known as self-adjoint matrices, for which the Pfaffians (Pf) can be defined (again see [15]).They are defined on 2 × 2 skew-symmetric matrices, so for a hyperhermitian matrix  and matrix (and endomorphism)  defined in Proposition 1, we define the map  by   →  () fl  ⋅  () . (34) From this follows the well-known equalities proved by Dyson [15]: This allows us to prove det is a homomorphism and, in particular, the following corollary.
Corollary 6.For any hyperhermitian matrix  and any split-quaternionic matrix , the matrix  * ⋅  ⋅  is also hyperhermitian and Proof.Using the identities (32), (33), and (35) above, Proposition 1, and the multiplicative properties of complex determinants and , a direct calculation shows and the corollary is proved.
This immediately implies that the mixed partials calculation is since  ⋅   =   ⋅  which extends from the fact that for  ∈ C and  ∈ H  ,  ⋅  =  ⋅ .This also shows that the matrix ( 2 /    ) ,=0,...,−1 is hyperhermitian; that is, Definition 8 (the "split-quaternionic Hessian" of a function ).The "split-quaternionic Hessian" (denoted H  ) of a  2 function defined on a domain  in H   ≅ C 2 is defined analogously to the complex Hessian of a function, only with respect to split-quaternionic variables.For ,  ∈ {0, 1, . . .,  − 1} and H  (), the split-quaternionic Hessian of , is defined as We now turn to the Monge-Amperè operator.As Alesker in [4] defined the mixed Monge-Amperè operator in quaternionic space of a  2 function  is defined as MA () :   → det (H  ()) . (44) Generalized further, for  2 functions  1 , . . .,   , the mixed discriminant ds is defined as the Moore determinant of the respective quaternionic Hessian matrices; we follow this construction to define a similar Monge-A-Amperè operator to split-quaternionic functions, denoted by )) . ( Note also that, for  1 =  2 = ⋅ ⋅ ⋅ =   = , the mixed Monge-Amperè operator is equal to the regular Monge-Amperè operator: det () fl det (, . . ., ) = det H  () .

Linear Change of Variables.
In this section we prove a split-quaternionic change of variables formula for linear transformations.Since the split quaternions can be represented by real (2 × 2) matrices, this endeavour is done easier via a real representation of the matrix algebra.In this light we define  R to be the following embedding (also a homomorphism like , see Proposition 1) for a splitquaternionic vector  = (  ) = ( 4 + 4+1 + 4+2 + 4+3 ) ∈ H   and matrix  =  0 +  1  +  2  +  3 , where   ∈   (R): and satisfy the split-quaternionic relations (1).Moreover, if we define for  1 =  R ,  2 =  R ,  3 =  R , then one can calculate that  R () commutes with I  ; that is For a  1 function  : H   → H  ,  =  0 +  1  +  2  +  3 , and the real representation denoted by  R = ( 0 ,  1 ,  2 ,  3 )  , the partial derivative of  with respect to   can be written as the (4 × 4) differential operator and the derivative in the direction  ∈ H   as For a linear transformation  ∈ GL  (H  );  = (  ) : H   → H   ;   →   fl  and a  1 function   : H   → H  we define the pullback via  of   as () fl   () [=   (  )] and their corresponding real representation denoted as   R and  R .Let  =  R () and  =  R (  ), so that  =  R (), and Proposition 11.With the same setup as above, we have that is, Proof.Denote by (   )  the th column of the functional operator    :   → (   ) R , for  = 1, 2, 3, 4, and then by the definitions of (   )  and   it follows directly that with the understanding that  0 = Id 4×4 for the  = 1th column.Hence we have where (D  )  is the th column of the functional operator D  :   → (D  ) R .Since  =  R (), then by definition we have   = ∑ 4−1 =0 ( R ())    so that by the chain rule for functions of several variables directly in the first column, that is,

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D  is a linear operator, we use the commutation relations (53) to calculate for  = 2, 3, 4. (62) And hence and (57) follows.
Corollary 12 (change of variables under split-quaternionic linear transformations).If  is a real-valued  2 function, then Proof.If  is real-valued, then     =    .Then by ( 58) and taking the conjugate of both sides gives us Then applying (58) to the LHS of the line above where   () fl (/  )() =    (), and hence (64) follows and the corollary is proved.

The Main Result of This Paper
Theorem 13.Let  ⊂ R 4 ≅ H   be a domain and  :  → H  .Then where  = {0, 1, . . ., 2 − 1} is a multi-index and 3.2.Proof of Theorem 13 for the Case  = 2. First we prove Theorem 13 for the case  = 2 (base case) and then proceed by induction.
Proof.Let  = 2 and consider the embedding  from Section 2.1: ) . ( The correspoding operators are and we use Lemma 3 to complete the proof: from which it follows that wedging Δ to itself yields where again  = {0, 1, . . ., 2(2) − 1} and 2) .On the other hand (40) tells us how to compute entries in the split-quaternionic Hessian, which works out beautifully: and when combined with Lemma 3, that gives Advances in Mathematical Physics 9 The third row of the above then also implies that so that (Δ) 2 is actually

Proof of Induction.
We now assume that Theorem 13 is true for some  − 1 ∈ N; we want to prove by induction that it holds for .We consider a  2 function ( 0 , . . .,  −1 ) of  variables that has continuous 2nd-order mixed partial derivatives.First we prove a result in functional analysis regarding the density of delta functions (on hyperplanes) in the space of (tempered) distributions, implying that the span of said delta functions contains the set of smooth functions, which are dense in the  2 functions.Lemma 14. Linear combinations of delta functions is dense in the space of generalized functions D(H   ) = (S(H   )) * , where S(H   ) is the Schwartz space of rapidly decreasing functions on H   .
Proof.Consider the Fréchet space S(H   ) with the Fréchet topology and its dual space D(H   ).We wish to show that the Schwartz space S(H   ) is dense in (D(H   )) * , the dual of the distribution space.It is well known that the evaluation map is an injection from a topological vector space  into its double dual  * * ; hence for  = S(H   ) we have a copy of ) is a nuclear Fréchet space which is also barreled (see [16], pg.107, 147) then since the Radon transform (defined on hyperplanes) R  :   → ∫   (for nonzero ) is injective (proved in [11,17]), which is a contradiction since  was assumed to be nonzero.Hence  = (D(H   )) * ; that is, the span of delta functions  is dense in (D(H   )) * .
Proof of Theorem 13 for  > 2. By Lemma 14 and the properties of the mixed Baston product and mixed discriminant, it suffices to prove for  > 2 in the case  1 () =   , where  = { | ∑      = 0} is a split-quaternionic hyperplane, which implies Theorem 13.We proceed by finding a unitary linear transformation  such that  = { |  1 = 0}.We can use the pullback functions    (  ) =   (  ), and by Corollary 12 we have that where  fl   and  = 1, . . ., .Then But letting which means that  2−1  ⋅ Z = ( ⋅ dZ 2−1 ) and similarly  2  ⋅ Z = ( ⋅ dZ 2 ) are exact forms and hence by Stokes' theorem because of the compact support of , and   = {|  | ≤  :  = 3, 4, . . ., 2}.If  = 1 then since  1 and  2 do not appear in dZ 1 and dZ 2 , respectively, the integral and hence (  / 1 )() may not necessarily be zero.Applying the partial derivative with respect to  1 to (84) with  = 1 we get the first entry in the split-quaternionic Hessian matrix for   , and combining with (87) we obtain since  2   /    = 0 if  ̸ = 1 ̸ = , and the second equality is by definition for  = 1.Using Corollary 10 we have where  {1}   = ( Since the domain of integration is  = { 1 = 0}, the integral only depends on no more than 2nd-order derivatives of  2 , . . .,   in the direction of On the other hand, using properties of the mixed Baston product and our inductive hypothesis used in the last equality we have Hence combining (93) with (92) we get that the integrands are equal almost everywhere, but since the functions are continuous, we have equality, and Theorem 13 is proved.

Split Quaternions and Structures on Manifolds
The operator Δ above can be generalized for any manifold with a special structure which we call split-hypercomplex (other known names are parahypercomplex and neutral hypercomplex).Let  be a manifold and let  be a complex structure on it; that is,  :  → ,  2 = − is integrable almost complex structure.Suppose also that there is  :  →  with  2 =  and  = −.If the ±1 eigen-bundles of  are involutive,  is called integrable.When  is integrable,  =  again has  2 =  and it is known that it is integrable.We call such (, , , ) with integrable , ,  split-hypercomplex manifold and (, , )split-hypercomplex structure.Clearly the left multiplication by , ,  in H   provides such a structure.However, unlike the complex manifolds, split-hypercomplex ones do not have nice atlases with "spli-quaternionic-holomorphic" transition functions, so the local considerations of the previous section cannot be extended to an arbitrary manifold.For any function  :  → R, however we can define an analog of the Baston operator Δ. Denote by  and  the standard operators for the structure .Then Δ =  is a globally defined 2-form on , which is of type (2, 0) with respect to .
It is known that when ∘∘ is nondegenerate it defines a pseudo-Riemannian metric  on  of split signature, such that  is an isometry and ,  are anti-isometries of , called split-hyperhermitian. Any split-hyperhermitian structure defines 3 nondegenerate 2-forms by   (, ) = (, ),   (, ) = (, ),   (, ) = (, ), for which   +   is nondegenerate (2, 0)-form with respect to .In particular such metric is necessary of split signature and  has dimension divisible by four.The relation with a function  as above is  ∘  ∘  =   +   and conversely, from nondegenerate form  ∘  ∘  on a split-hypercomplex manifold, one recovers .
However not every hyperhermitian metric arises in such a way.There is an additional integrability condition on  which is obtained as follows: If   +   =  for some , then (  +   ) = 0.The condition is also equivalent to existence of a connection ∇ on  for which ∇ = ∇ = ∇ = ∇ = 0 and ( ∇ (, ), ) is totally skew-symmetric, where  ∇ is the torsion of ∇ [10].On a split-hyperhermitian manifold  admitting such connection with skew-torsion, such function  locally always exists [10] but may not exist globally.
The main result of Section 4 then gives that on where H  is the split-quaternionic Hessian of .In the quaternionic case, this gives rise to the so-called quaternionic Monge-Amperè equation, which arises if we want to find  for which the determinant of the quaternionic Hessian is a given function.The quaternionic Monge-Amperè equation is elliptic.In the split-quaternionic case, however the corresponding equation is ultrahyperbolic and is not well studied.On the other side the reduction of self-duality equations in split signature to two dimensions leads to the equations of [5] describing the deformations of a harmonic map from a Riemann surface into compact Lie group, which are elliptic.In H   natural geometric objects to study are also the split special Lagrangian submanifolds as studied in [12].The description in our terminology is the following.Consider the form Ω =   −   which has values in split-complex numbers D = { +  |  2 = 1}.Then Ω  = Ω 1 + Ω 2 for real nondegenerate 2-forms Ω 1 and Ω 2 .Moreover, when the structure is hypersymplectic, forms Ω 1 and Ω 2 are closed.A split special Lagrangian manifold (of phase zero) then is defined as a submanifold  of H   of real dimension 2, for which the form Ω 2 vanishes on  and Ω 1 is nondegenerate.Such manifold is necessarily complex, since its tangent bundle is preserved by .This is a partial case of split special Lagrangian manifolds, which are analogs of the holomorphic Lagrangian submanifolds in hyperkähler manifold.

) Advances in Mathematical Physics 7 Using
this real embedding the corresponding matrices  R ,  R , and  R are  R fl (