On Functions of Several Split-Quaternionic Variables

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.

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 . 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 supersymmetry was shown in [2, 3] to be described by a space-time with a pseudo-Kähler metric of signature , 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 , where is a signature metric and are parallel endomorphisms of the tangent bundle with respect to the Levi-Civita connection of , such that 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 -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 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 , where 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 , 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 .

2. The Split Quaternions and Functions on

2.1. The Algebra of the Split Quaternions

The split quaternions are spanned over by the basis with algebraic relations , , . The inner product is defined by and (hence the signature). The conjugate of is , and then .

Another realization of in given by the following embedding : whose usefulness becomes obvious when we notice that it preserves the norm; that is,

The embedding above is extended to vectors in by the map where

so the embedding looks like

Similarly for an split-quaternionic matrix, we define its adjoint matrix as the image of the extension of to given by where and .

A related homomorphism is that of the Study map, which we denote by , which can be generalized for a matrix , where :

The Study map also has an associated determinant, where is the standard determinant of complex matrices, given by

For , we have 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.(2)For , , where .(3)For , , where . Furthermore, if is unitary, then ; that is, is symplectic, and .

Proof. Let . Then and by the property above we have HenceFor the second claim, if the adjoint of is unitary as matrix, then we show that is symplectic; that is, but by (12) we have that and combined with (14): we get thatand by multiplying the above equation by on the right we obtain as needed.

Following Wan and Wang [13] we define on the corresponding first-order differential operators that act as partial derivatives with respect to the variable in the th entry, , and or , in (6),and use them to define the analogs of and in complex analysis for the purposes of our study of functions on . We will denote by the partial derivatives and for notational simplicity.

Definition 2 (the Baston operator). Let be a domain. The operators , , and are defined as

Due to the particular embedding we chose, we can simplify notations and calculations, summarized in the following lemma.

Lemma 3. Properties of , , , , and are as follows: (1) where (2) for the same pair of indices and .(3)Let such that (clearly) for all . Let ; then without loss of generality we can suppose . Then if we consider the ordered pairs and , we have

Proof. We compute as Since both indices and run from to , the “symmetric terms” and both appear, and by the skew-symmetry of the wedge product (), and since any , the only remaining terms are the terms with , so we can cancel the and write (19) as where is the coefficient of term in .
To demonstrate , 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 , , but , , and similarly , , but , (in the other case if was odd we would have the completely symmetric situation, with ’s becoming ’s, etc.). So we calculate

Definition 4 (mixed Baston product). For we define the mixed Baston product of as where and is defined to be the sign of the permutation from to if and otherwise.
In particular, for , 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 “split-quaternionic Hessian” of , 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 in (just as in ) 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, 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 for all , and . Then we define the Moore determinant of a hyperhermitian matrix [denoted by ] 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 , we have and for for , if and if , and the hyperhermitian 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 , 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) aswhich 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 -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 () can be defined (again see [15]). They are defined on skew-symmetric matrices, so for a hyperhermitian matrix and matrix (and endomorphism) defined in Proposition 1, we define the map by From this follows the well-known equalities proved by Dyson [15]: This allows us to prove 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 showsand the corollary is proved.

Definition 7 (mixed discriminant of hyperhermitian matrices). Let be hyperhermitian matrices. The mixed discriminant [denoted by ] of is defined to be the coefficient of the monomial divided by in the polynomial given by , where det is again the Moore determinant. Note also that .

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

Let be a function of split-quaternionic variables, where . We define the partial derivative with respect to as and for , its conjugate is , we similarly define the partial derivative with respect to a conjugate variable as This immediately implies that the mixed partials calculation is since which extends from the fact that for and , . This also shows that the matrix is hyperhermitian; that is,

Definition 8 (the “split-quaternionic Hessian” of a function ). The “split-quaternionic Hessian” (denoted ) of a function defined on a domain in is defined analogously to the complex Hessian of a function, only with respect to split-quaternionic variables. For and , 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 function is defined as Generalized further, for functions , 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 , the mixed Monge-Amperè operator is equal to the regular Monge-Amperè operator:

Lemma 9. For any hyperhermitian matrix and any real diagonal real matrix of the form we have thatwhere is the matrix obtained by deleting rows and columns with indexes from a nonempty subset (see [4], pg. 10).

This lemma is proved as proposition in Alesker [4]; we will just be using a simple corollary.

Corollary 10. If in Lemma 9 above, then where and is still a hyperhermitian matrix,

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 matrices, this endeavour is done easier via a real representation of the matrix algebra. In this light we define to be the following embedding (also a homomorphism like , see Proposition 1) for a split-quaternionic vector and matrix , where :

Using this real embedding the corresponding matrices , and are and satisfy the split-quaternionic relations (1). Moreover, if we define for , , , then one can calculate that commutes with ; that is

For a function , , and the real representation denoted by , the partial derivative of with respect to can be written as the differential operatorand the derivative in the direction as For a linear transformation ; ; and a function we define the pullback via of as and their corresponding real representation denoted as and Let and , so that , and , that is,

Proposition 11. With the same setup as above, we have that is,

Proof. Denote by the th column of the functional operator , for , and then by the definitions of and it follows directly that with the understanding that for the th column. Hence we have where is the th column of the functional operator . Since , then by definition we have so that by the chain rule for functions of several variables directly in the first column, that is, . Since and is a linear operator, we use the commutation relations (53) to calculate And hence and (57) follows.

Corollary 12 (change of variables under split-quaternionic linear transformations). If is a real-valued 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 , and hence (64) follows and the corollary is proved.

3. Statement of the Theorem

3.1. The Main Result of This Paper

Theorem 13. Let be a domain and . Then where is a multi-index and is the holomorphic volume form in .

3.2. Proof of Theorem 13 for the Case

First we prove Theorem 13 for the case (base case) and then proceed by induction.

Proof. Let 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 to itself yields where again and is the holomorphic volume form in .
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 givesThe third row of the above then also implies that so that is actually since [see (41)]. This concludes the proof of Theorem 13 for the case .

3.3. Proof of Induction

We now assume that Theorem 13 is true for some ; we want to prove by induction that it holds for . We consider a function 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 functions.

Lemma 14. Linear combinations of delta functions is dense in the space of generalized functions , where is the Schwartz space of rapidly decreasing functions on .

Proof. Consider the Fréchet space with the Fréchet topology and its dual space . We wish to show that the Schwartz space is dense in , 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 we have a copy of .
But since is a nuclear Fréchet space which is also barreled (see [16], pg. 107, 147) then is (semi)reflexive; that is, as vector spaces and hence the Schwartz space is trivially dense in the dual of . Consider the subspace and its closure inside , and suppose . By the Hahn-Banach theorem, there is a linear functional such that and . Thus there exists a nonzero Schwartz function such that the functional But then for any (all) , since the Radon transform (defined on hyperplanes) (for nonzero ) is injective (proved in [11, 17]), which is a contradiction since was assumed to be nonzero.
Hence ; that is, the span of delta functions is dense in .

Proof of Theorem 13 for . By Lemma 14 and the properties of the mixed Baston product and mixed discriminant, it suffices to provefor in the case , where is a split-quaternionic hyperplane, which implies Theorem 13. We proceed by finding a unitary linear transformation such that . We can use the pullback functions , and by Corollary 12 we have that where and . Then since is unitary; that is, . Hence it follows by the definition of the mixed discriminant and using the simpler notation where again . From these considerations, it then suffices to prove (80) in the case where and . We can compute the split-quaternionic Hessian of , where the derivatives are now weak derivatives of distributions. For any test function we havewhere is the volume measure on and is the volume measure on . Translated into the complex representations, and so that, for , But lettingwhich means that and similarly are exact forms and hence by Stokes’ theorembecause of the compact support of , and . If then since and do not appear in and , respectively, the integral and hence may not necessarily be zero. Applying the partial derivative with respect to to (84) with we get the first entry in the split-quaternionic Hessian matrix for , and combining with (87) we obtain since if , and the second equality is by definition for . Using Corollary 10 we havewhere are the corresponding hyperhermitian minors of the original matrices. Then from (89) it follows that for any test function we haveSince the domain of integration is , the integral only depends on no more than 2nd-order derivatives of in the direction of Thus we can assume that there exists polynomials of order no more than such that for .
Let denote , and then it follows that Thus we obtain 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.

4. Split Quaternions and Structures on Manifolds

The operator 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, , is integrable almost complex structure. Suppose also that there is with and . If the eigen-bundles of are involutive, is called integrable. When is integrable, again has 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 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 , however we can define an analog of the Baston operator . Denote by and the standard operators for the structure . Then is a globally defined 2-form on , which is of type 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 -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 . The condition is also equivalent to existence of a connection on for which 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 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 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 which has values in split-complex numbers . Then for real nondegenerate -forms and . Moreover, when the structure is hypersymplectic, forms and are closed. A split special Lagrangian manifold (of phase zero) then is defined as a submanifold of of real dimension , for which the form vanishes on and 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research of Gueo Grantcharov is partially supported by grant from Simons Foundation (no. 246184).