# On Functions of Several Split-Quaternionic Variables

Authors: Gueo Grantcharov; Camilo Montoya
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

Alesker studied a relation between the determinant of a quaternionic Hessian of a function and a specific complex volume form. In this note we show that similar relation holds for functions of several split-quaternionic variables and point to some relations with geometry.


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## Body

## 1. Introduction

Quaternions are known to have deep relation to the self-dual 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) self-duality 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 (1)I2=-S2=-1,T=IS=-SI,gIX,IY=-gSX,SY=gX,Y.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 I,S,T 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 ω(X,Y)=g(SX,Y)+ig(TX,Y). Locally the metric arises from a single function, called potential, similar to the Kähler metrics. The function satisfies ∂∘S∘∂¯f=ω. In the quaternionic case, such potentials in multidimensional quaternionic space Hn 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 that ∂∘S∘∂¯f=detfdz1∧⋯∧dz2n, where det is the Moore determinant of the quaternionic Hessian of f. 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 Hsn, 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 f for which det(f)≠0.


## 2. The Split Quaternions and Functions onHsn

### 2.1. The Algebra of the Split Quaternions

The split quaternionsHs are spanned over R by the basis {1,i,s,t} with algebraic relations 12=s2=t2=1, i2=-1, is=-si=t. The inner product ·,· is defined by 1,1=i,i=1 and s,s=t,t=-1 (hence the (2,2) signature). The conjugate of q=x0+x1i+x2s+x3t is q-≔x0-x1i-x2s-x3t, and then q,q=qq-=x02+x12-x22-x32.Another realization ofHs in HC given by the following embedding τ: (2)Hs∋q=z+w·s↪τq≔z¯w¯wz∈EndC2whose usefulness becomes obvious when we notice that it preserves the norm; that is, (3)q2=q,q=zz--ww-=det⁡z¯w¯wz=det⁡τq.The embeddingτ above is extended to vectors in Hsn by the map (4)τ:Hsn↪EndRC2n,where(5)q⟼τq≔τqkk=0,…,n-1=τq0⋮τqk⋮τqn-1,qk≔z2k+z2k+1·s,τqk=τz2k+z2k+1·s≔z¯2kz¯2k+1z2k+1z2kso the embedding looks like(6)Hsn∋q=q0⋮qn-1⟼τq≔τq0⋮τqn-1=z¯0w¯0w0z0⋮z¯n-1w¯n-1wn-1zn-1.Similarly for ann×n split-quaternionic matrix, we define its adjoint matrix as the image of the extension of τ to M2n(C) given by (7)A=Aij0≤i,j≤n-1⟼τA≔τAij0≤i,j≤n-1,where Aij=aij+bij·s and aij,bij∈C.A related homomorphism is that of the Study map, which we denote byμ, which can be generalized for a matrix Mn(Hs)∋M=A+B·s, where A,B∈Mn(C): (8)M=A+B·s⟼μM≔τA¯τB¯τBτA.The Study map also has an associated determinant, wheredetC is the standard determinant of complex matrices, given by (9)Sdet⁡M≔detCμM.Forqk=z2k+z2k+1s, we have τ(q0·q1)=τ(q0)·τ(q1) and we can check that (10)τq0·q1=z0z1+w0w¯1¯z0w1+z¯1w0¯z0w1+z¯1w0z0z1+w0w¯1=z¯0z¯1+w¯0w1z¯0w¯1+z1w¯0z0w1+z¯1w0z0z1+w0w¯1=z¯0w¯0w0z0·z¯1w¯1w1z1=τq0·τq1.So it is natural to expect that this representation extends linearly to matrix groups, that is, the following.Proposition 1.
For any split-quaternionic matricesA and B, we have the following:(1)
τ(AB)=τ(A)τ(B) for A=Ap×m and B=Bm×l, where the superscripts are the respective dimensions.(2)
ForA∈Mn(Hs), τ(A∗)=Tτ(A)∗T, where T≔iτ(i)=idiag-i00i=diag100-1.(3)
ForA∈Mn(Hs), Sτ(A)=τ(A)¯S, where S≔τ(s)=diag0110. Furthermore, if A∈Un(Hs) is unitary, then τ(A)Iτ(A)t=I; that is, A is symplectic, and I≔iτ(-t)=idiag0-ii0=diag01-10.Proof.
LetC=AB. Then Cij=∑k=0p-1‍AikBkj and by the property above we have (11)τCij=τ∑k=0p-1AikBkj=∑k=0p-1τAikBkj=∑k=0p-1τAikτBkj=τAi0,…,τAip-1τB0j⋮τBp-1j=τAikk=0,…,p-1τBkjk=0,…,p-1t=∑k=0p-1τAikτBkjij.Hence(12)τC=τCij=∑k=0p-1τAikτBkjij=τA·τB,τA∗=τAt¯=τA¯ji=τAji¯=τa¯ji0+s·-aji1¯=aji0-aji1¯-aji1aji0¯=Taji0aji1¯aji1aji0¯T=Taij0¯aij1¯aij1aij0t¯T=TτAt¯T=TτA∗T,SτA¯=diag⁡0110aij0¯aij1¯aij1aij0¯=diag⁡0110aij0aij1aij1¯aij0¯=aij1¯aij0¯aij0aij1=aij0¯aij1¯aij1aij0diag⁡0110=τAS.
For the second claim, if the adjoint ofU∈U(n,Hs) is unitary as (2n×2n) matrix, then we show that τ(U) is symplectic; that is, τ(U)Iτ(U)t=I(13)Un,Hs∋U⟺UU∗=Idn⟺τUU∗=τUτU∗=Id2n,but by (12) we have that τ(U∗)=Tτ(U)∗T and combined with (14): (14)τASt=SτAt=SτA¯t=τAt¯S⟹SτAtS=τAt¯we get that(15)Id2n=τUτU∗=τUTτUt¯T=τUISSτUtSIS=τUIτUt-Iand by multiplying the above equation by I on the right we obtain (16)I=Id2nI=τUIτUt-II=τUIτUt-I2=τUIτUtas needed.Following Wan and Wang [13] we define on C2n∋z=(z0,…,z2n-1) the corresponding first-order differential operators that act as partial derivatives with respect to the variable in the (kα)th entry, k=0,…,2n-1, and α=0 or 1, in (6),(17)∇00∇01∇10∇11⋮⋮∇2n-20∇2n-21∇2n-10∇2n-11≔∂z¯0∂z¯1∂z1∂z0⋮⋮∂z¯2n-2∂z¯2n-1∂z2n-1∂z2n-2,and use them to define the analogs of ∂ and ∂¯ in complex analysis for the purposes of our study of functions on Hsn. We will denote by ∂i,∂j¯ the partial derivatives ∂zi and ∂z¯j for notational simplicity.Definition 2 (the Baston operator).
LetD⊂R4n be a domain. The operators d0, d1, and Δ are defined as (18)d0≔ω=∑kfkdzk⟼d0ω≔∑k∇k0ωdzkd1≔f⟼d1f≔∑k∇k1fdzkΔ≔d0d1:f⟼Δf≔d0d1f.Due to the particular embedding we chose, we can simplify notations and calculations, summarized in the following lemma.Lemma 3.
Properties of∇k0, ∇k1, d0, d1, and Δ are as follows: (1)
(19)Δf≔d0d1f=∑0≤i<j≤2n-1Δijfdzi∧dzj,where(20)Δij≔∇i0∇j1-∇i1∇j0=det⁡∇i0∇i1∇j0∇j1.(2)
Δij=-Δji for the same pair of indices i and j.(3)
LetI={(2k,2k+1)∣k=0,…,n-1}={(0,1),…,(2n-2,2n-1)} such that (clearly) 2k<2k+1 for all k. Let r≠s∈{0,1,…,n-1}; then without loss of generality we can suppose r<s. Then if we consider the ordered pairs (i0,i1)≔(2r,2r+1) and (j0,j1)≔(2s,2s+1), we have (21)Δi0j1=-Δi1j0¯.Proof.
We computeΔf as (22)Δf≔d0d1f=d0∑j=02n-1∇j1fdzj=∑i=02n-1∇i0dzi∧∑j=02n-1∇j1fdzj=∑i,j=02n-1∇i0∇j1fdzi∧dzj.
Since both indicesi and j run from 0 to 2n-1, the “symmetric terms” ∇i0∇j1fdzi∧dzj and ∇j0∇i1fdzj∧dzi both appear, and by the skew-symmetry of the wedge product (dzi∧dzj=-dzj∧dzi), (23)∇i0∇j1fdzi∧dzj+∇i1∇j0fdzj∧dzi=∇i0∇j1f-∇i1∇j0fdzi∧dzj⟹Δf=∑i,j=02n-1∇i0∇j1fdzi∧dzj=12∑i,j=02n-1∇i0∇j1f-∇i1∇j0ffdzi∧dzjand since any dzk∧dzk=0, the only remaining terms are the terms with 0≤i<j≤2n-1, so we can cancel the 1/2 and write (19) as (24)Δf=∑0≤i<j≤2n-1∇i0∇j1f-∇i1∇j0fdzi∧dzj=∑0≤i<j≤2n-1Δijfdzi∧dzj,where Δij is the coefficient of dzi∧dzjth term in Δf.
To demonstrate(2), we use commutation of ∇kα∇jβ=∇jβ∇kα:(25)-Δji≔-∇j0∇i1-∇j1∇i0=∇j1∇i0-∇j0∇i1=∇i0∇j1-∇i1∇j0=Δij.
Proving the final claim is done in two nearly identical calculations, depending on the parity ofr and s. We will prove one case, where without loss of generality we assume r is even and we still have r<s. This means that ∇i00=∇(2r)0=∂2r¯, ∇i10=∇(2r+1)0=∂2r+1, but ∇i01=∇(2r)1=∂2r+1¯, ∇i11=∇(2r+1)1=∂2r, and similarly ∇j00=∇(2s)0=∂2s¯, ∇j10=∇(2s+1)0=∂2s+1, but ∇j01=∇(2s)1=∂2s+1¯, ∇j11=∇(2s+1)1=∂2s (in the other case if s was odd we would have the completely symmetric situation, with 2r+1’s becoming 2r¯’s, etc.). So we calculate (26)-Δi1j0¯=-Δi1j0¯=Δj0i1¯=∇j00∇i11-∇j01∇i10¯=∂2s¯∂2r-∂2s+1¯∂2r+1¯=∂2s¯∂2r¯-∂2s+1¯∂2r+1¯=∂2r¯∂2s-∂2r+1¯∂2s+1=∇2r0∇2s+11-∇2r1∇2s+10≕Δ2r2s+1=Δi0j1.Definition 4 (mixed Baston product).
Forf1,…,fn∈C2 we define the mixed Baston product of f1,…,fn as (27)Δnf1,…,fn≔Δf1∧Δf2∧⋯∧Δfn≔∑i1,j1,…Δi1j1f1⋯Δinjnfndzi1∧dzj1∧⋯∧dzin∧dzjn=∑i1,j1,…δ1,2,3,4,…,2ni1j1i2j2,…,injnΔi1j1f1⋯ΔinjnfndzI,where I={1,2,…,2n} and δ1,2,3,4,…,2ni1j1i2j2,…,injn is defined to be the sign of the permutation from (i1,j1,…,in,jn) to (1,2,3,…,2n) if {ik,jk∣k=1,2,…,2n}=I and 0 otherwise.
In particular, forf1=f2=⋯=fn=f, the mixed Baston product coincides with the n-times wedged Baston of f; that is, (28)Δnf≔Δnf,…,f=Δfn≔:⋀nΔf≔Δf∧⋯∧Δf︸n-times.The results and definitions above will allow us to translate then-times wedged Baston of f in terms of the “split-quaternionic Hessian” of f, defined in terms of the Moore determinant, which are precisely the next sections.


### 2.2. Split-Quaternionic Determinants

Due to the noncommutativity of the multiplication inHs (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 tohyperhermitian matrices; that is, A∈GLn(Hs) such that A=A∗; 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 n! terms in the sum over permutations of the symmetric group Sn.Definition 5 (the Moore determinant, see [14] or [15]).
For a permutationσ∈Sn, write σ 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 (29)σ=n11⋯n1r1n21⋯n2r2⋯nj1⋯njrj,where, for i, we have ni1>nik for all k>1, and n11>n21>⋯>nj1. Then we define the Moore determinant of a hyperhermitian matrix A=(aij) [denoted by det(A)] as (30)det⁡A≔∑σ∈Snσan11n12⋯an1r1n11an21n22⋯anjrjnj1.
Another equivalent definition of the Moore determinant is the inductive one (see [15]), defined as, for a hyperhermitian n×n matrix A=(aij), the inductive definition is given as follows: for n=1, we have det(A)≔a11 and for n>2(31)det⁡A=∑k=1nϵrkarkdet⁡Ar,kfor r∈I={1,2,…,n}, ϵrk=+1 if k=r and ϵrk=-1 if k≠r, and A(k,r) the hyperhermitian (n-1)×(n-1) matrix obtained by interchanging the kth and rth columns and then deleting both the kth row and column of the corresponding matrix. For any matrix A∈Mn(Hs), it can be easily checked that A·A∗=A∗·A is also hyperhermitian, which leads to the equalities (32)det⁡A∗·A=detCμA.The Moore determinant is related to the Study determinant from (8) as(33)det⁡A∗·A=detCτA=detCμA=Sdet⁡A,which is given by the middle equality which can be seen easily by noticing that μ(A) and τ(A) are similar matrices (having the same exact entries in different arrays, except that the former consists of 4 n-blocks and the latter consists of n22-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 asself-adjoint matrices, for which the Pfaffians (Pf) can be defined (again see [15]). They are defined on 2n×2n skew-symmetric matrices, so for a hyperhermitian matrix A and matrix (and endomorphism) T defined in Proposition 1, we define the map ρ by (34)A⟼ρA≔T·τA.From this follows the well-known equalities proved by Dyson [15]: (35)det⁡A=PfρA=Sdet⁡Ai.e.,det⁡A2=detCτA.This allows us to prove det is a homomorphism and, in particular, the following corollary.Corollary 6.
For any hyperhermitian matrixA and any split-quaternionic matrix C, the matrix C∗·A·C is also hyperhermitian and (36)det⁡C∗·A·C=det⁡C∗·C·det⁡A.Proof.
Using the identities (32), (33), and (35) above, Proposition 1, and the multiplicative properties of complex determinants and τ, a direct calculation shows(37)det⁡C∗·A·C2=detCτC∗·A·C=detCτC∗·τA·τC=detCτC∗·detCτA·detCτC=detCτC∗detCτCdetCτA=detCτC∗·C·detCτA=det⁡C∗·C2·det⁡A2and the corollary is proved.Definition 7 (mixed discriminant of hyperhermitian matrices).
LetA1,…,An be hyperhermitian n×n matrices. Themixed discriminant [denoted by det(A1,…,An)] of A1,…,An is defined to be the coefficient of the monomial x1,…,xn divided by n! in the polynomial given by det(x1A1+⋯+xnAn), where det is again the Moore determinant. Note also that det(A,…,A)=det(A).


### 2.3. The “Split-Quaternionic Hessian” of a Function

Letf be a function of n split-quaternionic variables, where qi≔z2i+z¯2i+1·s. We define the partial derivative with respect to qi as (38)∂∂qi≔∂2i+∂2i+1¯·s;and for qj≔z2j+z2j+1·s, its conjugate is q¯j≔z¯2j-z¯2j+1·s, we similarly define the partial derivative with respect to a conjugate variable as (39)∂∂q¯j≔∂∂qj¯≔∂2j¯-∂2j+1¯·s.This immediately implies that the mixed partials calculation is (40)∂2∂qi∂q¯j≔∂∂qi∂∂qj≔∂2i+∂2i+1¯·s∂2j¯-∂2j+1¯·s=∂2i∂2j¯-∂2i∂2j+1¯·s+∂2i+1¯·s·∂2j¯-∂2i+1¯·s·∂2j+1¯·s=∂2i∂2j¯-∂2i+1¯∂2j+1+∂2i+1¯∂2j-∂2i∂2j+1¯·s=∂2i2j¯2-∂2i+1¯2j+12+∂2i+1¯2j2-∂2i2j+1¯2·s,since s·∂k=∂k¯·s which extends from the fact that for z∈C and s∈Hs, s·z=z¯·s. This also shows that the matrix (∂2/∂qi∂q¯j)i,j=0,…,n-1 is hyperhermitian; that is, (41)∂2∂qi∂q¯j=∂2∂qj∂q¯i¯.Definition 8 (the “split-quaternionic Hessian” of a functionf).
The “split-quaternionic Hessian” (denotedHs) of a C2 function defined on a domain D in Hsn≅C2n is defined analogously to the complex Hessian of a function, only with respect to split-quaternionic variables. For i,j∈{0,1,…,n-1}(42)∂ijq≔∂2∂qi∂q¯j:f⟼∂ijqf≔∂2∂qi∂q¯jf≔∂2f∂qi∂q¯j,and Hs(f), the split-quaternionic Hessian of f, is defined as (43)Hs:C2D⟶MnHs,f⟼Hsf≔∂ijqfi,j=0,1,…,2n-1.
We now turn to the Monge-Amperè operator. As Alesker in [4] defined themixed Monge-Amperè operator in quaternionic space of a C2 function f is defined as (44)MAf:f⟼det⁡Hsf.Generalized further, for C2 functions f1,…,fn, themixed 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 (45)MAf1,…,fn≔det⁡f1,…,fn≔det⁡∂2f1∂qi∂q¯j,…,∂2fn∂qi∂q¯j.
Note also that, forf1=f2=⋯=fn=f, the mixed Monge-Amperè operator is equal to the regular Monge-Amperè operator: (46)det⁡f≔det⁡f,…,f=det⁡Hsf.Lemma 9.
For any hyperhermitiann×n matrix A and any real diagonal real matrix of the form T=t10⋱0tn we have that(47)det⁡A+T=∑I⊂1,…,n∏i∈Iti·det⁡MIA,where MIA is the matrix obtained by deleting rows and columns with indexes from a nonempty subset I⊂{1,2,…,n} (see [4], pg. 10).This lemma is proved as proposition1.1.11 in Alesker [4]; we will just be using a simple corollary.Corollary 10.
Ift2=⋯=tn=0 in Lemma 9 above, then (48)det⁡A+T=det⁡A+t1·det⁡M1A,where (49)A=a11∗∗M1Aand M{1}A is still a hyperhermitian (n-1)×(n-1) matrix, a11∈R.


### 2.4. 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 τR to be the following embedding (also a homomorphism like τ, see Proposition 1) for a split-quaternionic vector q=(qk)=(x4k+x4k+1i+x4k+2s+x4k+3t)∈Hsn and matrix A=A0+A1i+A2s+A3t, where Ai∈Mn(R): (50)Hsn∋q⟼τRq≔x4k,x4k+1,x4k+2,x4k+3k=0,1,…,n-1t∈R4n,GLnHs∋A⟼τRA≔A0-A1A2A3A1A0A3-A2A2A3A0-A1A3-A2A1A0∈M4nR.Using this real embedding the corresponding matricesIR,SR, and TR are (51)IR≔0-1001000000100-10,SR≔0010000110000100,TR≔000100-100-1001000and satisfy the split-quaternionic relations (1). Moreover, if we define (52)Iα≔diag⁡Iα,α=1,2,3for I1=IR, I2=SR, I3=TR, then one can calculate that τR(A) commutes with Iα; that is (53)Iα·τRA=τRA·Iα,α=1,2,3.For aC1 function f:Hsn→Hs, f=f0+f1i+f2s+f3t, and the real representation denoted by fR=(f0,f1,f2,f3)t, the partial derivative of f with respect to qj can be written as the (4×4) differential operator(54)∂qjfR≔∂f∂qjR≔∂x4j-∂x4j+1∂x4j+2∂x4j+3∂x4j+1∂x4j∂x4j+3-∂x4j+2∂x4j+2∂x4j+3∂x4j-∂x4j+1∂x4j+3-∂x4j+2∂x4j+1∂x4jfRand the derivative in the direction q∈Hsn as (55)∂qf≔DxfR≔∂q0fR⋮∂qn-1fR.For a linear transformation A∈GLn(Hs); A=(aij):Hsn→Hsn; q↦q′≔Aq and a C1 function f′:Hsn→Hs we define the pullback via A of f′ as f(q)≔f′(Aq)[=f′(q′)] and their corresponding real representation denoted as f'R and fR. Let x=τR(q) and y=τR(q′), so that y=τR(A)x, and f(q)=f′(Aq), that is, fR(x)=f'R(τR(A)x)Proposition 11.
With the same setup as above, we have(57)DxfRx=τRA∗Dyf'RτRAx;that is, (58)∂qjfq=∑k=0n-1a¯kj∂qk′f′Aq.Proof.
Denote by∂qjα the αth column of the functional operator ∂qj:f↦∂qjfR, for α=1,2,3,4, and then by the definitions of ∂qjα and Iα it follows directly that (59)∂qjα=Iα-1∂x4j∂x4j+1∂x4j+2∂x4j+3,α=1,2,3,4,with the understanding that I0=Id4×4 for the α=1th column. Hence we have (60)Dxα=Iα-1∂x=Iα-1∂x4j∂x4j+1∂x4j+2∂x4j+3j=0,…,n-1,where Dxα is the αth column of the functional operator Dx:f↦DxfR. Since y=τR(A)x, then by definition we have yk=∑m=04n-1‍τR(A)kmxm so that by the chain rule for functions of several variables (61)∂xjfRx=∑k=04n-1τRAkj∂ykf'RτRAx⟹DxfR=τRAt·∂yf'RτRAxdirectly in the first column, that is, Dxf1(x)=[τR(A)]tDyf′1(τR(A)x). Since f=f0+f1i+f2s+f3t and Dx is a linear operator, we use the commutation relations (53) to calculate (62)Dxfαx=Iα-1∂xfRx=Iα-1τRAt∂yf'RτRAx=τRAtIα-1∂yf'RτRAx=τRAtDyf′ατRAx=τRAtDyf′αy,forα=2,3,4.And hence (63)A∗∂q′f′RAq=τRAtDyf′Ry=τRAtDyf′RτRAx=DxfRx=∂qfRqand (57) follows.Corollary 12 (change of variables under split-quaternionic linear transformations).
Iff is a real-valued C2 function, then (64)∂2f∂qj∂q-kq=A∗∂2f′∂qj′∂q-k′AqA.Proof.
Ifu is real-valued, then ∂q¯ku=∂qku¯. Then by (58) (65)∂qiujq=∑k=0n-1a¯ki∂qk′uj′Aqand taking the conjugate of both sides gives us (66)∂q¯iujq=∂qiujq¯=∑k=0n-1a¯ki∂qk′uj′Aq¯=∑k=0n-1∂qk′uj′Aq¯aki=∑k=0n-1∂q-k′uj′Aqaki.Then applying (58) to the LHS of the line above (67)∑j,k=0n-1a¯ji∂2u∂qj∂q¯kAqaki=∑k=0n-1∂fk∂qj′qaki=∂∂qj∑k=0n-1fkqakj=∂2u′∂qj∂q¯kq,where fk(q)≔∂u/∂qk¯(Aq)=∂q¯ku(Aq), and hence (64) follows and the corollary is proved.


## 2.1. The Algebra of the Split Quaternions

The split quaternionsHs are spanned over R by the basis {1,i,s,t} with algebraic relations 12=s2=t2=1, i2=-1, is=-si=t. The inner product ·,· is defined by 1,1=i,i=1 and s,s=t,t=-1 (hence the (2,2) signature). The conjugate of q=x0+x1i+x2s+x3t is q-≔x0-x1i-x2s-x3t, and then q,q=qq-=x02+x12-x22-x32.Another realization ofHs in HC given by the following embedding τ: (2)Hs∋q=z+w·s↪τq≔z¯w¯wz∈EndC2whose usefulness becomes obvious when we notice that it preserves the norm; that is, (3)q2=q,q=zz--ww-=det⁡z¯w¯wz=det⁡τq.The embeddingτ above is extended to vectors in Hsn by the map (4)τ:Hsn↪EndRC2n,where(5)q⟼τq≔τqkk=0,…,n-1=τq0⋮τqk⋮τqn-1,qk≔z2k+z2k+1·s,τqk=τz2k+z2k+1·s≔z¯2kz¯2k+1z2k+1z2kso the embedding looks like(6)Hsn∋q=q0⋮qn-1⟼τq≔τq0⋮τqn-1=z¯0w¯0w0z0⋮z¯n-1w¯n-1wn-1zn-1.Similarly for ann×n split-quaternionic matrix, we define its adjoint matrix as the image of the extension of τ to M2n(C) given by (7)A=Aij0≤i,j≤n-1⟼τA≔τAij0≤i,j≤n-1,where Aij=aij+bij·s and aij,bij∈C.A related homomorphism is that of the Study map, which we denote byμ, which can be generalized for a matrix Mn(Hs)∋M=A+B·s, where A,B∈Mn(C): (8)M=A+B·s⟼μM≔τA¯τB¯τBτA.The Study map also has an associated determinant, wheredetC is the standard determinant of complex matrices, given by (9)Sdet⁡M≔detCμM.Forqk=z2k+z2k+1s, we have τ(q0·q1)=τ(q0)·τ(q1) and we can check that (10)τq0·q1=z0z1+w0w¯1¯z0w1+z¯1w0¯z0w1+z¯1w0z0z1+w0w¯1=z¯0z¯1+w¯0w1z¯0w¯1+z1w¯0z0w1+z¯1w0z0z1+w0w¯1=z¯0w¯0w0z0·z¯1w¯1w1z1=τq0·τq1.So it is natural to expect that this representation extends linearly to matrix groups, that is, the following.Proposition 1.
For any split-quaternionic matricesA and B, we have the following:(1)
τ(AB)=τ(A)τ(B) for A=Ap×m and B=Bm×l, where the superscripts are the respective dimensions.(2)
ForA∈Mn(Hs), τ(A∗)=Tτ(A)∗T, where T≔iτ(i)=idiag-i00i=diag100-1.(3)
ForA∈Mn(Hs), Sτ(A)=τ(A)¯S, where S≔τ(s)=diag0110. Furthermore, if A∈Un(Hs) is unitary, then τ(A)Iτ(A)t=I; that is, A is symplectic, and I≔iτ(-t)=idiag0-ii0=diag01-10.Proof.
LetC=AB. Then Cij=∑k=0p-1‍AikBkj and by the property above we have (11)τCij=τ∑k=0p-1AikBkj=∑k=0p-1τAikBkj=∑k=0p-1τAikτBkj=τAi0,…,τAip-1τB0j⋮τBp-1j=τAikk=0,…,p-1τBkjk=0,…,p-1t=∑k=0p-1τAikτBkjij.Hence(12)τC=τCij=∑k=0p-1τAikτBkjij=τA·τB,τA∗=τAt¯=τA¯ji=τAji¯=τa¯ji0+s·-aji1¯=aji0-aji1¯-aji1aji0¯=Taji0aji1¯aji1aji0¯T=Taij0¯aij1¯aij1aij0t¯T=TτAt¯T=TτA∗T,SτA¯=diag⁡0110aij0¯aij1¯aij1aij0¯=diag⁡0110aij0aij1aij1¯aij0¯=aij1¯aij0¯aij0aij1=aij0¯aij1¯aij1aij0diag⁡0110=τAS.
For the second claim, if the adjoint ofU∈U(n,Hs) is unitary as (2n×2n) matrix, then we show that τ(U) is symplectic; that is, τ(U)Iτ(U)t=I(13)Un,Hs∋U⟺UU∗=Idn⟺τUU∗=τUτU∗=Id2n,but by (12) we have that τ(U∗)=Tτ(U)∗T and combined with (14): (14)τASt=SτAt=SτA¯t=τAt¯S⟹SτAtS=τAt¯we get that(15)Id2n=τUτU∗=τUTτUt¯T=τUISSτUtSIS=τUIτUt-Iand by multiplying the above equation by I on the right we obtain (16)I=Id2nI=τUIτUt-II=τUIτUt-I2=τUIτUtas needed.Following Wan and Wang [13] we define on C2n∋z=(z0,…,z2n-1) the corresponding first-order differential operators that act as partial derivatives with respect to the variable in the (kα)th entry, k=0,…,2n-1, and α=0 or 1, in (6),(17)∇00∇01∇10∇11⋮⋮∇2n-20∇2n-21∇2n-10∇2n-11≔∂z¯0∂z¯1∂z1∂z0⋮⋮∂z¯2n-2∂z¯2n-1∂z2n-1∂z2n-2,and use them to define the analogs of ∂ and ∂¯ in complex analysis for the purposes of our study of functions on Hsn. We will denote by ∂i,∂j¯ the partial derivatives ∂zi and ∂z¯j for notational simplicity.Definition 2 (the Baston operator).
LetD⊂R4n be a domain. The operators d0, d1, and Δ are defined as (18)d0≔ω=∑kfkdzk⟼d0ω≔∑k∇k0ωdzkd1≔f⟼d1f≔∑k∇k1fdzkΔ≔d0d1:f⟼Δf≔d0d1f.Due to the particular embedding we chose, we can simplify notations and calculations, summarized in the following lemma.Lemma 3.
Properties of∇k0, ∇k1, d0, d1, and Δ are as follows: (1)
(19)Δf≔d0d1f=∑0≤i<j≤2n-1Δijfdzi∧dzj,where(20)Δij≔∇i0∇j1-∇i1∇j0=det⁡∇i0∇i1∇j0∇j1.(2)
Δij=-Δji for the same pair of indices i and j.(3)
LetI={(2k,2k+1)∣k=0,…,n-1}={(0,1),…,(2n-2,2n-1)} such that (clearly) 2k<2k+1 for all k. Let r≠s∈{0,1,…,n-1}; then without loss of generality we can suppose r<s. Then if we consider the ordered pairs (i0,i1)≔(2r,2r+1) and (j0,j1)≔(2s,2s+1), we have (21)Δi0j1=-Δi1j0¯.Proof.
We computeΔf as (22)Δf≔d0d1f=d0∑j=02n-1∇j1fdzj=∑i=02n-1∇i0dzi∧∑j=02n-1∇j1fdzj=∑i,j=02n-1∇i0∇j1fdzi∧dzj.
Since both indicesi and j run from 0 to 2n-1, the “symmetric terms” ∇i0∇j1fdzi∧dzj and ∇j0∇i1fdzj∧dzi both appear, and by the skew-symmetry of the wedge product (dzi∧dzj=-dzj∧dzi), (23)∇i0∇j1fdzi∧dzj+∇i1∇j0fdzj∧dzi=∇i0∇j1f-∇i1∇j0fdzi∧dzj⟹Δf=∑i,j=02n-1∇i0∇j1fdzi∧dzj=12∑i,j=02n-1∇i0∇j1f-∇i1∇j0ffdzi∧dzjand since any dzk∧dzk=0, the only remaining terms are the terms with 0≤i<j≤2n-1, so we can cancel the 1/2 and write (19) as (24)Δf=∑0≤i<j≤2n-1∇i0∇j1f-∇i1∇j0fdzi∧dzj=∑0≤i<j≤2n-1Δijfdzi∧dzj,where Δij is the coefficient of dzi∧dzjth term in Δf.
To demonstrate(2), we use commutation of ∇kα∇jβ=∇jβ∇kα:(25)-Δji≔-∇j0∇i1-∇j1∇i0=∇j1∇i0-∇j0∇i1=∇i0∇j1-∇i1∇j0=Δij.
Proving the final claim is done in two nearly identical calculations, depending on the parity ofr and s. We will prove one case, where without loss of generality we assume r is even and we still have r<s. This means that ∇i00=∇(2r)0=∂2r¯, ∇i10=∇(2r+1)0=∂2r+1, but ∇i01=∇(2r)1=∂2r+1¯, ∇i11=∇(2r+1)1=∂2r, and similarly ∇j00=∇(2s)0=∂2s¯, ∇j10=∇(2s+1)0=∂2s+1, but ∇j01=∇(2s)1=∂2s+1¯, ∇j11=∇(2s+1)1=∂2s (in the other case if s was odd we would have the completely symmetric situation, with 2r+1’s becoming 2r¯’s, etc.). So we calculate (26)-Δi1j0¯=-Δi1j0¯=Δj0i1¯=∇j00∇i11-∇j01∇i10¯=∂2s¯∂2r-∂2s+1¯∂2r+1¯=∂2s¯∂2r¯-∂2s+1¯∂2r+1¯=∂2r¯∂2s-∂2r+1¯∂2s+1=∇2r0∇2s+11-∇2r1∇2s+10≕Δ2r2s+1=Δi0j1.Definition 4 (mixed Baston product).
Forf1,…,fn∈C2 we define the mixed Baston product of f1,…,fn as (27)Δnf1,…,fn≔Δf1∧Δf2∧⋯∧Δfn≔∑i1,j1,…Δi1j1f1⋯Δinjnfndzi1∧dzj1∧⋯∧dzin∧dzjn=∑i1,j1,…δ1,2,3,4,…,2ni1j1i2j2,…,injnΔi1j1f1⋯ΔinjnfndzI,where I={1,2,…,2n} and δ1,2,3,4,…,2ni1j1i2j2,…,injn is defined to be the sign of the permutation from (i1,j1,…,in,jn) to (1,2,3,…,2n) if {ik,jk∣k=1,2,…,2n}=I and 0 otherwise.
In particular, forf1=f2=⋯=fn=f, the mixed Baston product coincides with the n-times wedged Baston of f; that is, (28)Δnf≔Δnf,…,f=Δfn≔:⋀nΔf≔Δf∧⋯∧Δf︸n-times.The results and definitions above will allow us to translate then-times wedged Baston of f in terms of the “split-quaternionic Hessian” of f, defined in terms of the Moore determinant, which are precisely the next sections.


## 2.2. Split-Quaternionic Determinants

Due to the noncommutativity of the multiplication inHs (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 tohyperhermitian matrices; that is, A∈GLn(Hs) such that A=A∗; 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 n! terms in the sum over permutations of the symmetric group Sn.Definition 5 (the Moore determinant, see [14] or [15]).
For a permutationσ∈Sn, write σ 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 (29)σ=n11⋯n1r1n21⋯n2r2⋯nj1⋯njrj,where, for i, we have ni1>nik for all k>1, and n11>n21>⋯>nj1. Then we define the Moore determinant of a hyperhermitian matrix A=(aij) [denoted by det(A)] as (30)det⁡A≔∑σ∈Snσan11n12⋯an1r1n11an21n22⋯anjrjnj1.
Another equivalent definition of the Moore determinant is the inductive one (see [15]), defined as, for a hyperhermitian n×n matrix A=(aij), the inductive definition is given as follows: for n=1, we have det(A)≔a11 and for n>2(31)det⁡A=∑k=1nϵrkarkdet⁡Ar,kfor r∈I={1,2,…,n}, ϵrk=+1 if k=r and ϵrk=-1 if k≠r, and A(k,r) the hyperhermitian (n-1)×(n-1) matrix obtained by interchanging the kth and rth columns and then deleting both the kth row and column of the corresponding matrix. For any matrix A∈Mn(Hs), it can be easily checked that A·A∗=A∗·A is also hyperhermitian, which leads to the equalities (32)det⁡A∗·A=detCμA.The Moore determinant is related to the Study determinant from (8) as(33)det⁡A∗·A=detCτA=detCμA=Sdet⁡A,which is given by the middle equality which can be seen easily by noticing that μ(A) and τ(A) are similar matrices (having the same exact entries in different arrays, except that the former consists of 4 n-blocks and the latter consists of n22-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 asself-adjoint matrices, for which the Pfaffians (Pf) can be defined (again see [15]). They are defined on 2n×2n skew-symmetric matrices, so for a hyperhermitian matrix A and matrix (and endomorphism) T defined in Proposition 1, we define the map ρ by (34)A⟼ρA≔T·τA.From this follows the well-known equalities proved by Dyson [15]: (35)det⁡A=PfρA=Sdet⁡Ai.e.,det⁡A2=detCτA.This allows us to prove det is a homomorphism and, in particular, the following corollary.Corollary 6.
For any hyperhermitian matrixA and any split-quaternionic matrix C, the matrix C∗·A·C is also hyperhermitian and (36)det⁡C∗·A·C=det⁡C∗·C·det⁡A.Proof.
Using the identities (32), (33), and (35) above, Proposition 1, and the multiplicative properties of complex determinants and τ, a direct calculation shows(37)det⁡C∗·A·C2=detCτC∗·A·C=detCτC∗·τA·τC=detCτC∗·detCτA·detCτC=detCτC∗detCτCdetCτA=detCτC∗·C·detCτA=det⁡C∗·C2·det⁡A2and the corollary is proved.Definition 7 (mixed discriminant of hyperhermitian matrices).
LetA1,…,An be hyperhermitian n×n matrices. Themixed discriminant [denoted by det(A1,…,An)] of A1,…,An is defined to be the coefficient of the monomial x1,…,xn divided by n! in the polynomial given by det(x1A1+⋯+xnAn), where det is again the Moore determinant. Note also that det(A,…,A)=det(A).


## 2.3. The “Split-Quaternionic Hessian” of a Function

Letf be a function of n split-quaternionic variables, where qi≔z2i+z¯2i+1·s. We define the partial derivative with respect to qi as (38)∂∂qi≔∂2i+∂2i+1¯·s;and for qj≔z2j+z2j+1·s, its conjugate is q¯j≔z¯2j-z¯2j+1·s, we similarly define the partial derivative with respect to a conjugate variable as (39)∂∂q¯j≔∂∂qj¯≔∂2j¯-∂2j+1¯·s.This immediately implies that the mixed partials calculation is (40)∂2∂qi∂q¯j≔∂∂qi∂∂qj≔∂2i+∂2i+1¯·s∂2j¯-∂2j+1¯·s=∂2i∂2j¯-∂2i∂2j+1¯·s+∂2i+1¯·s·∂2j¯-∂2i+1¯·s·∂2j+1¯·s=∂2i∂2j¯-∂2i+1¯∂2j+1+∂2i+1¯∂2j-∂2i∂2j+1¯·s=∂2i2j¯2-∂2i+1¯2j+12+∂2i+1¯2j2-∂2i2j+1¯2·s,since s·∂k=∂k¯·s which extends from the fact that for z∈C and s∈Hs, s·z=z¯·s. This also shows that the matrix (∂2/∂qi∂q¯j)i,j=0,…,n-1 is hyperhermitian; that is, (41)∂2∂qi∂q¯j=∂2∂qj∂q¯i¯.Definition 8 (the “split-quaternionic Hessian” of a functionf).
The “split-quaternionic Hessian” (denotedHs) of a C2 function defined on a domain D in Hsn≅C2n is defined analogously to the complex Hessian of a function, only with respect to split-quaternionic variables. For i,j∈{0,1,…,n-1}(42)∂ijq≔∂2∂qi∂q¯j:f⟼∂ijqf≔∂2∂qi∂q¯jf≔∂2f∂qi∂q¯j,and Hs(f), the split-quaternionic Hessian of f, is defined as (43)Hs:C2D⟶MnHs,f⟼Hsf≔∂ijqfi,j=0,1,…,2n-1.
We now turn to the Monge-Amperè operator. As Alesker in [4] defined themixed Monge-Amperè operator in quaternionic space of a C2 function f is defined as (44)MAf:f⟼det⁡Hsf.Generalized further, for C2 functions f1,…,fn, themixed 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 (45)MAf1,…,fn≔det⁡f1,…,fn≔det⁡∂2f1∂qi∂q¯j,…,∂2fn∂qi∂q¯j.
Note also that, forf1=f2=⋯=fn=f, the mixed Monge-Amperè operator is equal to the regular Monge-Amperè operator: (46)det⁡f≔det⁡f,…,f=det⁡Hsf.Lemma 9.
For any hyperhermitiann×n matrix A and any real diagonal real matrix of the form T=t10⋱0tn we have that(47)det⁡A+T=∑I⊂1,…,n∏i∈Iti·det⁡MIA,where MIA is the matrix obtained by deleting rows and columns with indexes from a nonempty subset I⊂{1,2,…,n} (see [4], pg. 10).This lemma is proved as proposition1.1.11 in Alesker [4]; we will just be using a simple corollary.Corollary 10.
Ift2=⋯=tn=0 in Lemma 9 above, then (48)det⁡A+T=det⁡A+t1·det⁡M1A,where (49)A=a11∗∗M1Aand M{1}A is still a hyperhermitian (n-1)×(n-1) matrix, a11∈R.


## 2.4. 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 τR to be the following embedding (also a homomorphism like τ, see Proposition 1) for a split-quaternionic vector q=(qk)=(x4k+x4k+1i+x4k+2s+x4k+3t)∈Hsn and matrix A=A0+A1i+A2s+A3t, where Ai∈Mn(R): (50)Hsn∋q⟼τRq≔x4k,x4k+1,x4k+2,x4k+3k=0,1,…,n-1t∈R4n,GLnHs∋A⟼τRA≔A0-A1A2A3A1A0A3-A2A2A3A0-A1A3-A2A1A0∈M4nR.Using this real embedding the corresponding matricesIR,SR, and TR are (51)IR≔0-1001000000100-10,SR≔0010000110000100,TR≔000100-100-1001000and satisfy the split-quaternionic relations (1). Moreover, if we define (52)Iα≔diag⁡Iα,α=1,2,3for I1=IR, I2=SR, I3=TR, then one can calculate that τR(A) commutes with Iα; that is (53)Iα·τRA=τRA·Iα,α=1,2,3.For aC1 function f:Hsn→Hs, f=f0+f1i+f2s+f3t, and the real representation denoted by fR=(f0,f1,f2,f3)t, the partial derivative of f with respect to qj can be written as the (4×4) differential operator(54)∂qjfR≔∂f∂qjR≔∂x4j-∂x4j+1∂x4j+2∂x4j+3∂x4j+1∂x4j∂x4j+3-∂x4j+2∂x4j+2∂x4j+3∂x4j-∂x4j+1∂x4j+3-∂x4j+2∂x4j+1∂x4jfRand the derivative in the direction q∈Hsn as (55)∂qf≔DxfR≔∂q0fR⋮∂qn-1fR.For a linear transformation A∈GLn(Hs); A=(aij):Hsn→Hsn; q↦q′≔Aq and a C1 function f′:Hsn→Hs we define the pullback via A of f′ as f(q)≔f′(Aq)[=f′(q′)] and their corresponding real representation denoted as f'R and fR. Let x=τR(q) and y=τR(q′), so that y=τR(A)x, and f(q)=f′(Aq), that is, fR(x)=f'R(τR(A)x)Proposition 11.
With the same setup as above, we have(57)DxfRx=τRA∗Dyf'RτRAx;that is, (58)∂qjfq=∑k=0n-1a¯kj∂qk′f′Aq.Proof.
Denote by∂qjα the αth column of the functional operator ∂qj:f↦∂qjfR, for α=1,2,3,4, and then by the definitions of ∂qjα and Iα it follows directly that (59)∂qjα=Iα-1∂x4j∂x4j+1∂x4j+2∂x4j+3,α=1,2,3,4,with the understanding that I0=Id4×4 for the α=1th column. Hence we have (60)Dxα=Iα-1∂x=Iα-1∂x4j∂x4j+1∂x4j+2∂x4j+3j=0,…,n-1,where Dxα is the αth column of the functional operator Dx:f↦DxfR. Since y=τR(A)x, then by definition we have yk=∑m=04n-1‍τR(A)kmxm so that by the chain rule for functions of several variables (61)∂xjfRx=∑k=04n-1τRAkj∂ykf'RτRAx⟹DxfR=τRAt·∂yf'RτRAxdirectly in the first column, that is, Dxf1(x)=[τR(A)]tDyf′1(τR(A)x). Since f=f0+f1i+f2s+f3t and Dx is a linear operator, we use the commutation relations (53) to calculate (62)Dxfαx=Iα-1∂xfRx=Iα-1τRAt∂yf'RτRAx=τRAtIα-1∂yf'RτRAx=τRAtDyf′ατRAx=τRAtDyf′αy,forα=2,3,4.And hence (63)A∗∂q′f′RAq=τRAtDyf′Ry=τRAtDyf′RτRAx=DxfRx=∂qfRqand (57) follows.Corollary 12 (change of variables under split-quaternionic linear transformations).
Iff is a real-valued C2 function, then (64)∂2f∂qj∂q-kq=A∗∂2f′∂qj′∂q-k′AqA.Proof.
Ifu is real-valued, then ∂q¯ku=∂qku¯. Then by (58) (65)∂qiujq=∑k=0n-1a¯ki∂qk′uj′Aqand taking the conjugate of both sides gives us (66)∂q¯iujq=∂qiujq¯=∑k=0n-1a¯ki∂qk′uj′Aq¯=∑k=0n-1∂qk′uj′Aq¯aki=∑k=0n-1∂q-k′uj′Aqaki.Then applying (58) to the LHS of the line above (67)∑j,k=0n-1a¯ji∂2u∂qj∂q¯kAqaki=∑k=0n-1∂fk∂qj′qaki=∂∂qj∑k=0n-1fkqakj=∂2u′∂qj∂q¯kq,where fk(q)≔∂u/∂qk¯(Aq)=∂q¯ku(Aq), and hence (64) follows and the corollary is proved.


## 3. Statement of the Theorem

### 3.1. The Main Result of This Paper

Theorem 13.
LetD⊂R4n≅Hsn be a domain and f:D→Hs. Then (68)Δnf≔Δfn≔Δf∧Δf∧⋯∧Δf︸n-times=n!·det⁡Hsf·dzI,where I={0,1,…,2n-1} is a multi-index and dzI≔dz0∧dz1∧⋯∧dz2n-1 is the holomorphic volume form in C2n.


### 3.2. Proof of Theorem13 for the Case n=2

First we prove Theorem13 for the case n=2 (base case) and then proceed by induction.Proof.
Letn=2 and consider the embedding τ from Section 2.1: (69)Hs2∋q=q0q1=z0+z1·sz2+z3·s⟼︸ττq=τ0q0τ1q1=z¯0z¯1z1z0z¯2z¯3z3z2.The correspoding operators are (70)∇00∇01∇10∇11∇20∇21∇30∇31≔∂z¯0∂z¯1∂z1∂z0∂z¯2∂z¯3∂z3∂z2and we use Lemma 3 to complete the proof:(71)Δf=∑0≤i<j≤3Δijfdzi∧dzjfrom which it follows that wedging Δf to itself yields (72)Δf2≔Δf∧Δf,where again I={0,1,…,2(2)-1} and dzI≔dz0∧dz1∧dz2∧dz3 is the holomorphic volume form in Λ4∗C2(2).
On the other hand (40) tells us how to compute entries in the split-quaternionic Hessian, which works out beautifully: (73)∂00q=∂00¯2-∂1¯12=Δ01,∂11q=∂22¯2-∂3¯32=Δ23,∂10q=∂20¯2-∂3¯12+∂3¯02-∂21¯2·s=-Δ12+Δ02·s
and when combined with Lemma3, that gives(74)Δ12¯=-Δ03,Δ02¯=-Δ13.
The third row of the above then also implies that(75)∂10q2=∂10q·∂10q¯=-Δ12-Δ12¯-Δ02Δ02¯=Δ12Δ12¯-Δ02Δ02¯=Δ02Δ13-Δ03Δ12,so that (Δf)2 is actually (76)Δf2=2Δ01fΔ23f-Δ02fΔ13f-Δ03fΔ12f·dzI=2∂00qf∂11qf-∂10qf∂10q¯f·dzI=2·det⁡∂00qf∂01qf∂10qf∂11qf·dzI=2·det⁡Hsf·dzIsince ∂ijq=∂jiq¯ [see (41)]. This concludes the proof of Theorem 13 for the case n=2.


### 3.3. Proof of Induction

We now assume that Theorem13 is true for some n-1∈N; we want to prove by induction that it holds for n. We consider a C2 function f(q0,…,qn-1) of n 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 C2 functions.Lemma 14.
Linear combinations of delta functions is dense in the space of generalized functionsD(Hsn)=(S(Hsn))∗, where S(Hsn) is the Schwartz space of rapidly decreasing functions on Hsn.Proof.
Consider the Fréchet spaceS(Hsn) with the Fréchet topology and its dual space D(Hsn). We wish to show that the Schwartz space S(Hsn) is dense in (D(Hsn))∗, the dual of the distribution space. It is well known that the evaluation map is an injection from a topological vector space X into its double dual X∗∗; hence for X=S(Hsn) we have a copy of S(Hsn)⊂(D(Hsn))∗=(S(Hsn))∗∗.
But sinceS(Hsn) is a nuclear Fréchet space which is also barreled (see [16], pg. 107, 147) then S(Hsn) is (semi)reflexive; that is, S(Hsn)≅(SHsn)∗∗=(D(Hsn))∗ as vector spaces and hence the Schwartz space is trivially dense in the dual of D(Hsn). Consider the subspace (77)S=spanδE∣E⊂Hsn-split-quaternionic  hyperplaneand its closure S¯ inside (D(Hsn))∗, and suppose S¯≠(D(Hsn))∗. By the Hahn-Banach theorem, there is a linear functional l such that lS≡0 and l(D(Hsn))∗-S¯>0. Thus there exists a nonzero Schwartz function g such that the functional (78)l:f⟼lf≔∫Hsng·f≠0.But then for any (all) δE∈S, (79)lδE=∫Hsng·δE=∫Eg≕RgE=0⟺g≡0since the Radon transform (defined on hyperplanes) Rg:E↦∫E‍g (for nonzero g) is injective (proved in [11, 17]), which is a contradiction since g was assumed to be nonzero.
HenceS¯=(D(Hsn))∗; that is, the span of delta functions S is dense in (D(Hsn))∗.Proof of Theorem13 for n>2.
By Lemma14 and the properties of the mixed Baston product and mixed discriminant, it suffices to prove(80)Δnf1,…,fn=n!·det⁡f1,…,fndzIfor n>2 in the case f1(q)=δE, where E={q∣∑i‍λiqi=0} is a split-quaternionic hyperplane, which implies Theorem 13. We proceed by finding a unitary linear transformation A such that E={q∣q1=0}. We can use the pullback functions fk′(q′)=fk(Aq′), and by Corollary 12 we have that (81)Fk′≔∂2fk′∂qi′∂q-j′q′=A∂2fk∂qi∂q-jAq′A∗≕A·Fk·A∗,where q≔Aq′ and k=1,…,n. Then (82)∑kxkFk′=∑kxkA·Fk·A∗=A·∑kxkFk·A∗⟹det⁡∑kxkFk′=det⁡A·∑kxkFk·A∗=det⁡A·A∗·det⁡∑kxkFk=det⁡∑kxkFksince A is unitary; that is, A·A∗=Id. Hence it follows by the definition of the mixed discriminant and using the simpler notation (83)det⁡δ∑kλkqk=0,f2,…,fnq=det⁡δ∑kλkqk=0,f2,…,fnAq′=detδq1′=0,f2′,…,fn′q′,where again q=Aq′. From these considerations, it then suffices to prove (80) in the case where f1=δE and E={q1=0}. We can compute the split-quaternionic Hessian of δE, where the derivatives are now weak derivatives of distributions. For any test function ϕ∈C0∞(Hsn) we have(84)∂δE∂q-jϕ≔∫Hsnϕ·∂δE∂q-jdq=-∫Hsn∂ϕ∂q-j·δE·dq=∫E∂ϕ∂q-jdqE,where dq=⋀k=1ndqk is the volume measure on Hs and dqE=dq2∧⋯∧dqn is the volume measure on E. Translated into the complex representations, dq=⋀k=12ndzk∧dz¯k≕dZ and dqE=⋀k=32ndzk∧dz¯k≕dZ1 so that, for j=1,2,…,n, (85)∂ϕ∂q-jdqE=∂2j-1¯ϕ-∂2j¯ϕ·sdZ1=∂2j-1¯ϕ·dZ1-∂2j¯ϕ·dZ1·s.
But letting(86)dZi^≔dz1∧dz¯1∧⋯∧dzi^∧dz¯2j∧⋯∧dz2n∧dz¯2n,dZi¯^≔dz1∧dz¯1∧⋯∧dzi∧dz¯i^∧⋯∧dz2n∧dz¯2n,∂2j-1¯ϕ·dZ=∑k=12nδk2j-1∂ϕ∂z¯kdZ2j-1¯^∧dz¯k=∑k=12n∂∂z¯kϕ·dZ2j-1¯^∧dz¯k+∑k=12n∂∂zkϕ·dZ2j-1¯^∧dzk︸=0=dϕ·dZ2j-1¯^,which means that ∂2j-1¯ϕ·dZ=dϕ·dZ2j-1¯^ and similarly ∂2j¯ϕ·dZ=dϕ·dZ2j¯^ are exact forms and hence by Stokes’ theorem(87)∫E∂ϕ∂q-jdqE=∫E∂2j-1¯ϕ·dZ1-∂2j¯ϕ·dZ1·s=∫E∂2j-1¯ϕ·dZ1-∫E∂2j¯ϕ·dZ1·s=∫Edϕ·dZ2j-1¯^-∫Edϕ·dZ2j¯^=∫∂Eϕ·dZ2j-1¯^-∫∂Eϕ·dZ2j¯^=limb→∞⁡∫∂Ebϕ·dZ2j-1¯^-∫∂Ebϕ·dZ2j¯^=0-0,ifj≠1because of the compact support of ϕ, and ∂Eb={zi≤b:i=3,4,…,2n}. If j=1 then since dz¯1 and dz¯2 do not appear in dZ1¯^ and dZ2¯^, respectively, the integral and hence ∂δE/∂q-1(ϕ) may not necessarily be zero. Applying the partial derivative with respect to q1 to (84) with j=1 we get the first entry in the split-quaternionic Hessian matrix for δE, and combining with (87) we obtain (88)∂2δE∂qi∂q-j=∂2δE∂q1∂q-10⋱00=Δq1δE0⋱00since ∂2δE/∂qi∂q-j=0 if i≠1≠j, and the second equality is by definition for n=1. Using Corollary 10 we have(89)n·det⁡δE,f2,…,fn=Δq1δE·det⁡M1F2,…,M1Fn,where M{1}Fk=∂2fk/∂qi∂q-ji,j=2,…,n are the corresponding hyperhermitian minors of the original matrices. Then from (89) it follows that for any test function ϕ∈C0∞(Hsn) we have(90)∫Hsnϕ·detδE,f2,…,fn=∫Hsn1n·ϕ·Δq1δE·detM1F2,…,M1Fn=∫E1n·Δq1ϕ·det⁡M1F2,…,M1Fnq1=0.Since the domain of integration is E={q1=0}, the integral only depends on no more than 2nd-order derivatives of f2,…,fn in the direction of q1. Thus we can assume that there exists polynomials pk of order no more than 2 such that fk(q1,…,qn)=pk(q1)f^k(q2,…,qn) for k=2,3,…,n.
LetM{1}F^k denote ∂2f^k/∂qi∂q-ji,j=2,…,n, and then it follows that (91)det⁡M1F2,…,M1Fn=p2q1⋯pnq1·det⁡M1F^2,…,M1F^n=p2q1⋯pnq1·det⁡f^2,…,f^n.Thus we obtain (92)∫Hsnϕ·det⁡δE,f2,…,fn=∫E1n·Δq1ϕ·det⁡M1F2,…,M1Fnq1=0=∫E1n!·Δq1ϕ·p2q1⋯pnq1q1=0·Δn-1f^2,…,f^n.On the other hand, using properties of the mixed Baston product and our inductive hypothesis used in the last equality we have (93)∫Hsnϕ·ΔnδE,f2,…,fn=∫Hsnϕ·∑i1,j1,…δ1,2,3,4,…,2ni1j1i2j2,…,injnΔi1j1δE⋯Δinjnfn=∫Hsn2ϕ·∑i1,j1,…δ1,2,3,4,…,2n12i2j2,…,injnΔ12δE⋯Δinjnfn=∫Hsnϕ·Δq1δE·p2q1⋯pnq1∑i2,j2,…δ1,2,3,4,…,2n12i2j2,…,injnΔi2j2f^2⋯Δinjnf^n=∫HsnΔq1ϕ·p2q1⋯pnq1q1=0∑i2,j2,…δ1,2,3,4,…,2n12i2j2,…,injnΔi2j2f^2⋯Δinjnf^n=∫HsnΔq1ϕ·p2q1⋯pnq1q1=0·Δn-1f^2,…,f^n=∫E1n!·Δq1ϕ·p2q1⋯pnq1q1=0·Δn-1f^2,…,f^n.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.


## 3.1. The Main Result of This Paper

Theorem 13.
LetD⊂R4n≅Hsn be a domain and f:D→Hs. Then (68)Δnf≔Δfn≔Δf∧Δf∧⋯∧Δf︸n-times=n!·det⁡Hsf·dzI,where I={0,1,…,2n-1} is a multi-index and dzI≔dz0∧dz1∧⋯∧dz2n-1 is the holomorphic volume form in C2n.


## 3.2. Proof of Theorem13 for the Case n=2

First we prove Theorem13 for the case n=2 (base case) and then proceed by induction.Proof.
Letn=2 and consider the embedding τ from Section 2.1: (69)Hs2∋q=q0q1=z0+z1·sz2+z3·s⟼︸ττq=τ0q0τ1q1=z¯0z¯1z1z0z¯2z¯3z3z2.The correspoding operators are (70)∇00∇01∇10∇11∇20∇21∇30∇31≔∂z¯0∂z¯1∂z1∂z0∂z¯2∂z¯3∂z3∂z2and we use Lemma 3 to complete the proof:(71)Δf=∑0≤i<j≤3Δijfdzi∧dzjfrom which it follows that wedging Δf to itself yields (72)Δf2≔Δf∧Δf,where again I={0,1,…,2(2)-1} and dzI≔dz0∧dz1∧dz2∧dz3 is the holomorphic volume form in Λ4∗C2(2).
On the other hand (40) tells us how to compute entries in the split-quaternionic Hessian, which works out beautifully: (73)∂00q=∂00¯2-∂1¯12=Δ01,∂11q=∂22¯2-∂3¯32=Δ23,∂10q=∂20¯2-∂3¯12+∂3¯02-∂21¯2·s=-Δ12+Δ02·s
and when combined with Lemma3, that gives(74)Δ12¯=-Δ03,Δ02¯=-Δ13.
The third row of the above then also implies that(75)∂10q2=∂10q·∂10q¯=-Δ12-Δ12¯-Δ02Δ02¯=Δ12Δ12¯-Δ02Δ02¯=Δ02Δ13-Δ03Δ12,so that (Δf)2 is actually (76)Δf2=2Δ01fΔ23f-Δ02fΔ13f-Δ03fΔ12f·dzI=2∂00qf∂11qf-∂10qf∂10q¯f·dzI=2·det⁡∂00qf∂01qf∂10qf∂11qf·dzI=2·det⁡Hsf·dzIsince ∂ijq=∂jiq¯ [see (41)]. This concludes the proof of Theorem 13 for the case n=2.


## 3.3. Proof of Induction

We now assume that Theorem13 is true for some n-1∈N; we want to prove by induction that it holds for n. We consider a C2 function f(q0,…,qn-1) of n 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 C2 functions.Lemma 14.
Linear combinations of delta functions is dense in the space of generalized functionsD(Hsn)=(S(Hsn))∗, where S(Hsn) is the Schwartz space of rapidly decreasing functions on Hsn.Proof.
Consider the Fréchet spaceS(Hsn) with the Fréchet topology and its dual space D(Hsn). We wish to show that the Schwartz space S(Hsn) is dense in (D(Hsn))∗, the dual of the distribution space. It is well known that the evaluation map is an injection from a topological vector space X into its double dual X∗∗; hence for X=S(Hsn) we have a copy of S(Hsn)⊂(D(Hsn))∗=(S(Hsn))∗∗.
But sinceS(Hsn) is a nuclear Fréchet space which is also barreled (see [16], pg. 107, 147) then S(Hsn) is (semi)reflexive; that is, S(Hsn)≅(SHsn)∗∗=(D(Hsn))∗ as vector spaces and hence the Schwartz space is trivially dense in the dual of D(Hsn). Consider the subspace (77)S=spanδE∣E⊂Hsn-split-quaternionic  hyperplaneand its closure S¯ inside (D(Hsn))∗, and suppose S¯≠(D(Hsn))∗. By the Hahn-Banach theorem, there is a linear functional l such that lS≡0 and l(D(Hsn))∗-S¯>0. Thus there exists a nonzero Schwartz function g such that the functional (78)l:f⟼lf≔∫Hsng·f≠0.But then for any (all) δE∈S, (79)lδE=∫Hsng·δE=∫Eg≕RgE=0⟺g≡0since the Radon transform (defined on hyperplanes) Rg:E↦∫E‍g (for nonzero g) is injective (proved in [11, 17]), which is a contradiction since g was assumed to be nonzero.
HenceS¯=(D(Hsn))∗; that is, the span of delta functions S is dense in (D(Hsn))∗.Proof of Theorem13 for n>2.
By Lemma14 and the properties of the mixed Baston product and mixed discriminant, it suffices to prove(80)Δnf1,…,fn=n!·det⁡f1,…,fndzIfor n>2 in the case f1(q)=δE, where E={q∣∑i‍λiqi=0} is a split-quaternionic hyperplane, which implies Theorem 13. We proceed by finding a unitary linear transformation A such that E={q∣q1=0}. We can use the pullback functions fk′(q′)=fk(Aq′), and by Corollary 12 we have that (81)Fk′≔∂2fk′∂qi′∂q-j′q′=A∂2fk∂qi∂q-jAq′A∗≕A·Fk·A∗,where q≔Aq′ and k=1,…,n. Then (82)∑kxkFk′=∑kxkA·Fk·A∗=A·∑kxkFk·A∗⟹det⁡∑kxkFk′=det⁡A·∑kxkFk·A∗=det⁡A·A∗·det⁡∑kxkFk=det⁡∑kxkFksince A is unitary; that is, A·A∗=Id. Hence it follows by the definition of the mixed discriminant and using the simpler notation (83)det⁡δ∑kλkqk=0,f2,…,fnq=det⁡δ∑kλkqk=0,f2,…,fnAq′=detδq1′=0,f2′,…,fn′q′,where again q=Aq′. From these considerations, it then suffices to prove (80) in the case where f1=δE and E={q1=0}. We can compute the split-quaternionic Hessian of δE, where the derivatives are now weak derivatives of distributions. For any test function ϕ∈C0∞(Hsn) we have(84)∂δE∂q-jϕ≔∫Hsnϕ·∂δE∂q-jdq=-∫Hsn∂ϕ∂q-j·δE·dq=∫E∂ϕ∂q-jdqE,where dq=⋀k=1ndqk is the volume measure on Hs and dqE=dq2∧⋯∧dqn is the volume measure on E. Translated into the complex representations, dq=⋀k=12ndzk∧dz¯k≕dZ and dqE=⋀k=32ndzk∧dz¯k≕dZ1 so that, for j=1,2,…,n, (85)∂ϕ∂q-jdqE=∂2j-1¯ϕ-∂2j¯ϕ·sdZ1=∂2j-1¯ϕ·dZ1-∂2j¯ϕ·dZ1·s.
But letting(86)dZi^≔dz1∧dz¯1∧⋯∧dzi^∧dz¯2j∧⋯∧dz2n∧dz¯2n,dZi¯^≔dz1∧dz¯1∧⋯∧dzi∧dz¯i^∧⋯∧dz2n∧dz¯2n,∂2j-1¯ϕ·dZ=∑k=12nδk2j-1∂ϕ∂z¯kdZ2j-1¯^∧dz¯k=∑k=12n∂∂z¯kϕ·dZ2j-1¯^∧dz¯k+∑k=12n∂∂zkϕ·dZ2j-1¯^∧dzk︸=0=dϕ·dZ2j-1¯^,which means that ∂2j-1¯ϕ·dZ=dϕ·dZ2j-1¯^ and similarly ∂2j¯ϕ·dZ=dϕ·dZ2j¯^ are exact forms and hence by Stokes’ theorem(87)∫E∂ϕ∂q-jdqE=∫E∂2j-1¯ϕ·dZ1-∂2j¯ϕ·dZ1·s=∫E∂2j-1¯ϕ·dZ1-∫E∂2j¯ϕ·dZ1·s=∫Edϕ·dZ2j-1¯^-∫Edϕ·dZ2j¯^=∫∂Eϕ·dZ2j-1¯^-∫∂Eϕ·dZ2j¯^=limb→∞⁡∫∂Ebϕ·dZ2j-1¯^-∫∂Ebϕ·dZ2j¯^=0-0,ifj≠1because of the compact support of ϕ, and ∂Eb={zi≤b:i=3,4,…,2n}. If j=1 then since dz¯1 and dz¯2 do not appear in dZ1¯^ and dZ2¯^, respectively, the integral and hence ∂δE/∂q-1(ϕ) may not necessarily be zero. Applying the partial derivative with respect to q1 to (84) with j=1 we get the first entry in the split-quaternionic Hessian matrix for δE, and combining with (87) we obtain (88)∂2δE∂qi∂q-j=∂2δE∂q1∂q-10⋱00=Δq1δE0⋱00since ∂2δE/∂qi∂q-j=0 if i≠1≠j, and the second equality is by definition for n=1. Using Corollary 10 we have(89)n·det⁡δE,f2,…,fn=Δq1δE·det⁡M1F2,…,M1Fn,where M{1}Fk=∂2fk/∂qi∂q-ji,j=2,…,n are the corresponding hyperhermitian minors of the original matrices. Then from (89) it follows that for any test function ϕ∈C0∞(Hsn) we have(90)∫Hsnϕ·detδE,f2,…,fn=∫Hsn1n·ϕ·Δq1δE·detM1F2,…,M1Fn=∫E1n·Δq1ϕ·det⁡M1F2,…,M1Fnq1=0.Since the domain of integration is E={q1=0}, the integral only depends on no more than 2nd-order derivatives of f2,…,fn in the direction of q1. Thus we can assume that there exists polynomials pk of order no more than 2 such that fk(q1,…,qn)=pk(q1)f^k(q2,…,qn) for k=2,3,…,n.
LetM{1}F^k denote ∂2f^k/∂qi∂q-ji,j=2,…,n, and then it follows that (91)det⁡M1F2,…,M1Fn=p2q1⋯pnq1·det⁡M1F^2,…,M1F^n=p2q1⋯pnq1·det⁡f^2,…,f^n.Thus we obtain (92)∫Hsnϕ·det⁡δE,f2,…,fn=∫E1n·Δq1ϕ·det⁡M1F2,…,M1Fnq1=0=∫E1n!·Δq1ϕ·p2q1⋯pnq1q1=0·Δn-1f^2,…,f^n.On the other hand, using properties of the mixed Baston product and our inductive hypothesis used in the last equality we have (93)∫Hsnϕ·ΔnδE,f2,…,fn=∫Hsnϕ·∑i1,j1,…δ1,2,3,4,…,2ni1j1i2j2,…,injnΔi1j1δE⋯Δinjnfn=∫Hsn2ϕ·∑i1,j1,…δ1,2,3,4,…,2n12i2j2,…,injnΔ12δE⋯Δinjnfn=∫Hsnϕ·Δq1δE·p2q1⋯pnq1∑i2,j2,…δ1,2,3,4,…,2n12i2j2,…,injnΔi2j2f^2⋯Δinjnf^n=∫HsnΔq1ϕ·p2q1⋯pnq1q1=0∑i2,j2,…δ1,2,3,4,…,2n12i2j2,…,injnΔi2j2f^2⋯Δinjnf^n=∫HsnΔq1ϕ·p2q1⋯pnq1q1=0·Δn-1f^2,…,f^n=∫E1n!·Δq1ϕ·p2q1⋯pnq1q1=0·Δn-1f^2,…,f^n.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.


## 4. Split Quaternions and Structures on Manifolds

The operatorΔ above can be generalized for any manifold with a special structure which we callsplit-hypercomplex (other known names are parahypercomplex and neutral hypercomplex). Let M be a manifold and let I be a complex structure on it; that is, I:TM→TM, I2=-Id is integrable almost complex structure. Suppose also that there is S:TM→TM with S2=Id and IS=-SI. If the ±1 eigen-bundles of S are involutive, S is called integrable. When S is integrable, T=IS again has T2=Id and it is known that it is integrable. We call such (M,I,S,T) with integrable I,S,T split-hypercomplex manifold and (I,S,T)-split-hypercomplex structure. Clearly the left multiplication by i,s,t in Hsn 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 f:M→R, however we can define an analog of the Baston operator Δ. Denote by ∂ and ∂¯ the standard operators for the structure I. Then Δf=∂S∂¯f is a globally defined 2-form on M, which is of type (2,0) with respect to I.It is known that when∂∘S∘∂¯f is nondegenerate it defines a pseudo-Riemannian metric g on M of split signature, such that I is an isometry and S,T are anti-isometries of g, called split-hyperhermitian. Any split-hyperhermitian structure defines 3 nondegenerate 2-forms by ωI(X,Y)=g(IX,Y), ωS(X,Y)=g(SX,Y), ωT(X,Y)=g(TX,Y), for which ωS+iωT is nondegenerate (2,0)-form with respect to I. In particular such metric is necessary of split signature and M has dimension divisible by four. The relation with a function f as above is ∂∘S∘∂¯f=ωS+iωT and conversely, from nondegenerate form ∂∘S∘∂¯f on a split-hypercomplex manifold, one recovers g.However not every hyperhermitian metric arises in such a way. There is an additional integrability condition ong which is obtained as follows: If ωS+iωT=∂S∂¯f for some f, then ∂(ωS+iωT)=0. The condition is also equivalent to existence of a connection ∇ on M for which ∇g=∇I=∇S=∇T=0 and g(T∇(X,Y),Z) is totally skew-symmetric, where T∇ is the torsion of ∇ [10]. On a split-hyperhermitian manifold M admitting such connection with skew-torsion, such function f locally always exists [10] but may not exist globally.The main result of Section4 then gives that on Hsn≅C2n(94)∂∘S∘∂¯fn=det⁡Hsfdz1∧⋯∧dz2n,where Hs is the split-quaternionic Hessian of f. In the quaternionic case, this gives rise to the so-calledquaternionic Monge-Amperè equation, which arises if we want to find f 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 Hsn 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=ωI-sωT which has values insplit-complex numbers D={x+sy∣s2=1}. Then Ωn=Ω1+sΩ2 for real nondegenerate 2n-forms Ω1 and Ω2. Moreover, when the structure is hypersymplectic, forms Ω1 and Ω2 are closed. A split special Lagrangian manifold (of phase zero) then is defined as a submanifold M of Hsn of real dimension 2n, for which the form Ω2 vanishes on TM and Ω1 is nondegenerate. Such manifold is necessarily complex, since its tangent bundle is preserved by I. This is a partial case of split special Lagrangian manifolds, which are analogs of the holomorphic Lagrangian submanifolds in hyperkähler manifold.



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