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Abstract

In the present self-contained paper, we want, first, to construct a fundamental diagram, called (S.C), in homage to Carl Siegel and I. Satake that connects the following groups: \(\mathrm {SU}(m,m)\), \(\mathrm {SO}^{*}(2m)\), \(\mathrm {Sp}(2m,\mathbb {R}),\) \(\mathrm {Sp}(4m,\mathbb {R})\), \(\mathrm {SO}^{*}(4m)\). Then, we define and study three Clifford algebras related to that diagram. First, we consider the morphism from \(\mathrm {Sp}(2m,\mathbb {R})\) into \(\mathrm {SU}(m,m)\), shown in the construction of the diagram (S.C.). Then, we define a Clifford algebra \(Cl^{m,m}\), naturally associated with the group \(\mathrm {U}(m,m).\) Let (Eb) be an m-dimensional skew-hermitian space over \(\mathbb {H}\). For any \(x,y\in E,\) write \(b(x,y)=h(x,y)+ja(x,y).\) It is well known that h is a skew-hermitian complex form on \(\mathbb {E}_{2m}\), the complex 2m-dimensional vector space underlying E,  and a is a symmetric bilinear complex form on \(\mathbb {E}_{2m}\). We proved previously in [4] that the special unitary group \(\mathrm {SU}(E,b)\) of a skew-hermitian \(\mathbb {H}\)-right vector space (Eb),  m-dimensional over \(\mathbb {H}\), can be identified with the group \(\mathrm {SO}^{*}(2m)\) defined by E. Cartan. We define a real Clifford algebra, namely \(Cl_{\mathbb {R}}^{*}(2m),\) whose complexified algebra is \(C_{2m}^{+}(\mathbb {E}_{2m},a)\), the even complex Clifford algebra associated with a. Both algebras are associated with the geometry of the skew-hermitian \(\mathbb {H}\)-space (Eb). Let \(V=(\mathbb {R}^{2m},\mathrm {Sp}(2m,\mathbb {R}))\) be the standard model of a real symplectic space. We present some connections between the geometry of V and the algebras \(Cl^{m,m}\), \(C_{2m}^{+}(\mathbb {E}_{2m},a)\), \(Cl_{\mathbb {R}}^{*}(2m)\). The last section wants to give a sketch of the prospects offered by these algebras for the study of the real conformal symplectic geometry. An appendix gives some indispensable recalls and some complements.

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Data Availability Statement

The author confirms that the data supporting the findings of this study are available within the article and its supplementary materials.

Notes

  1. We recall that E. Cartan defined \(\mathrm {SO}^{*}(2m)\), (cf.the book written by S. Helgason given in the references, p. 340, the book written by B.G. Wybourne, p. 13, given in the references), as the subgroup of \(\mathrm {SO}(2m,\mathbb {C})\) constituted by matrices which leave invariant the skew-hermitian form:

    $$\begin{aligned} -z_{1}\overline{z_{m+1}}+z_{m+1}\overline{z_{1}}-z_{2}\overline{z_{m+2}}+z_{m+2}\overline{z_{2}}- \cdots -z_{m}\overline{z_{2m}}+z_{2m}\overline{z_{m}}. \end{aligned}$$
  2. In [18] Jean Dieudonné defined the notion of an orthogonal basis, an isotropic vector, an isotropic subspace and the index, say r, of b,  He showed that there exists an orthogonal basis \((\varepsilon _{l}\))\(_{1\le l\le m}\) such that for any l\(1\le l\le m,\) \(b(\varepsilon _{l},\varepsilon _{l})=j\), the quaternion unity j with \(j^{2}=-1.\) Then, J. Dieudonné proved that all sesquilinear skew-hermitian non-degenerate forms over \(\mathbb {H}\) defined on E are equivalent of maximal index \(\left[ \frac{m}{2}\right] \) (the integer part of \(\frac{m}{2}).\) Let \(x=\sum _{l=1}^{l=m}\varepsilon _{l}x^{l}\), \(y=\sum _{l=1}^{l=m}\varepsilon _{l}y^{l}.\) Write also the quaternions \(x^{l}\) and \(y^{l}\) as \(x^{l}=\xi ^{l}+j\xi ^{m+l}\), \(y^{l}=\eta ^{l}+j\eta ^{m+l}\), where, for any l\(1\le l\le m,\) \(\xi ^{l}\), \(\eta ^{l}\in \mathbb {C}\). Write also \(\xi ^{l}=a^{l}+ib^{l,},\) \(\eta ^{l}=c^{l}+id^{l},\) where, for any l\(1\le l\le m,\) \(a^{l}\), \(b^{l}\), \(c^{l}\), \(d^{l}\in \mathbb {R}\). An easy computation shows that for any \(x,y\in E,\) \(b(x,y)=\sum _{l=1}^{l=m}(-\overline{\xi ^{l}}\eta ^{m+l}+\overline{\xi ^{m+l}}\eta ^{l})+j(\sum _{l=1}^{l=2m}\xi ^{l}\eta ^{l}).\) An easy continuation of the development of the value of h(xy) gives the inclusion of \(\mathrm {SO}^{*}(2m)\) into \(\mathrm {Sp}(4m,\mathbb {R})\).

  3. This assertion is also a consequence of the following fact. Let \(u\in \mathrm {SU}(E,h)\cap \mathrm {SO}(E,a)\). If \(u(x)=x\lambda \) for some non-zero element \(x\in E\) and some \(\lambda \in \mathbb {H}\), and if \(x'=x\kappa \) and \(\lambda '=\kappa ^{-1}\lambda \kappa \) for some invertible \(\kappa \in \mathbb {H}\), then \(u(x')=x'\lambda '.\) We have \(\lambda ^{'}={\overline{\lambda }}\) if \(\lambda \in \mathbb {C}\) and \(\kappa =j.\)

  4.  We give in the appendix Sect. 5.3 a summary of the results obtained by I. Satake. I. Satake considers three main cases where we have successively: \(F=\mathbb {R}\), \(F=\mathbb {C}\), \(F=\mathbb {H}\). When \(F=\mathbb {R}\), the fundamental group G is the symplectic classical group \(G=\mathrm {Sp}(2m,\mathbb {R}\)). When \(F=\mathbb {C}\), I. Satake considers (Vh),  a finite n-dimensional skew-hermitian complex vector space. For \(F=\mathbb {H},\) he considers (Vh) a finite n-dimensional skew-hermitian space over \(\mathbb {H}\). I. Satake denotes by the same letter either the form h or its matrix realization in a suitable orthogonal basis. If (pq) denotes the usual signature of the form h,  we recall that h is called positive (resp. negative) and write \(h\gg 0\) (resp. \(h\ll 0)\) if \(q=0\) (resp. \(p=0)\) and that h is called indefinite if \(p>0,\) \(q>0.\) Then, \(\mathcal {P}_{n}(F)\) is defined as the set of matrices \(h\in \mathcal {M}(n,F)\) such that \(^{t}{\overline{h}}=h\) and \(h\gg 0\), where \(\mathcal {M}(n,F)\) is the set of \((n)\times (n)\) matrices with coefficients in F. We recall in this footnote that the symmetric space \({\mathfrak {D}}(V,h)\) can be identified with the space of all complex structures J satisfying the relation \(hJ\in \mathcal {P}_{n}(F).\)

  5. We recall that in the appendix, Sect. 5.3, we present the work of I. Satake concerning the embeddings of symmetric domains.

  6. We verify easily that the form \(\eta _{\varOmega }\) is hermitian.Indeed, put: \(v=x_{1}+iy_{1},v'=x_{2}+iy_{2}\). Then, first we have: \(\eta _{\varOmega }(v,v')=if_{\varOmega }\)(\({\overline{v}}\),v’)\(=\) \(i(\) \(\varOmega (x_{1},x_{2})+i\varOmega (\text {x}_{1},y_{2})-i\varOmega (y_{1},x_{2})+\varOmega (y_{1},y_{2}))=\) \(\varOmega (y_{1},x_{2})-\varOmega (x_{1},y_{2})+i\varOmega (x_{1},x_{2})+i\varOmega (y_{1},y_{2})=A+iB.\)

    Then, on the other hand: \(\eta _{\varOmega }(v',v)=if_{\varOmega }(\overline{v'},v)=if_{\varOmega }(x_{2}-i\) \(y_{2},x_{1}+iy_{1})=\varOmega (y_{2},x_{1})-\varOmega (x_{2},y_{1})+i\varOmega (x_{2},x_{1})+i\varOmega (y_{2},y_{1})=-\varOmega (x_{1},y_{2})+\varOmega (y_{1},x_{2})-i\varOmega (x_{1},x_{2})-i\varOmega (y_{1}\),y\(_{2}\)) , since \(\varOmega \) is skew-symmetric. Now, put:\(\eta _{\varOmega }(v',v)=A_{1}+iB_{1}.\)the , we find immediately that \(\overline{A_{1}+iB_{1}}=A+iB.\) Therefore, \(\eta _{\varOmega }\) is well hermitian.

  7. \(E_{2m,2m}\) is the regular pseudo-Euclidean space \(\mathbb {R}^{4m}\) endowed with the regular quadratic form of signature (2m, 2m) (cf. Sect. 2.1.1).

  8. We recall that the Clifford group is the group of all invertible elements of the Clifford algebra such that for any g in G(2m, 2m),  for any x in \(E_{2m,2m}\), we have \(\varphi (g)x=gxg^{-1}\in E_{2m,2m}\). Recall also the two following exact sequences:

    $$\begin{aligned} 1\longrightarrow \mathbb {R}^{*}\longrightarrow \mathrm {G}(2m,2m)\longrightarrow \mathrm {O}(2m,2m)\longrightarrow 1 \end{aligned}$$

    and

    $$\begin{aligned} 1\longrightarrow \mathbb {R}^{*}\longrightarrow \mathrm {G^{+}}(2m,2m)\longrightarrow \mathrm {SO}(2m,2m)\longrightarrow 1. \end{aligned}$$
  9. This is a purely algebraic argument. Consider the transformation \(\zeta :x\longrightarrow xi\) (where \(i^{2}=-1).\) \(\zeta \) belongs to \(\mathrm {U}(\eta ).\) Then, there exists an element \(\delta \) in \(C_{2m,2m}^{+}\) such that for any w in \(E_{2m,2m}\) we have \(\zeta (w)=\delta w\delta ^{-1}\). Whence, \(\widetilde{J_{2}}(w)=\delta w\delta ^{-1}\). Denote by \(e_{l}\) \((1\le l\le 4m)\) the elements of an orthonormal basis of \(E_{2m,2m}\). We can choose for \(\delta \) (up to a non-zero real factor) the element

    $$\begin{aligned} \delta _{0}=\prod _{1\le l\le 4m}(1+(e_{l}i)e_{l}^{-1}). \end{aligned}$$

    Now, assume that \(\lambda =-1.\) Then, \(\widetilde{J_{2}}(b_{u})=-b_{u}\) means that \(b_{u}\delta _{0}b_{u}^{-1}=-\delta _{0}\). But, this is contradictory because the automorphism \(y\longrightarrow b_{u} yb_{u}^{-1}\) operates trivially on the center of \(C_{2m,2m}^{+}\) and \(\delta _{0}\) has a non-zero component in \(C_{2m,2m}\), the non trivial line in this center.

  10. The following well-known result is used by J. Dieudonné, “On the automorphisms of the Classical groups”, Memoirs of the American Mathematical Society, 2, pp. 1–95, page 3, 1951, as a lemma.

    “Let E be an n-dimensional vector space over a field K and let u be an involution in \(\mathrm {GL}(n,K).\) If the characteristic of K is not 2,  E is the direct sum of two subspaces V and W,  (one of which may possibly be 0),  such that \(u(x)=x\) on V and \(u(x)=-x\) on W. V and W are called, respectively, the plus-subspace and the minus-subspace. They determine u completely.”

    The plus-subspace, respectively, the minus-subspace, will be denoted by \((+)\)-subspace, respectively, by \((-)\)-subspace in the sequel.

  11. We denote by \(C_{n}^{j}\) the classical binomial coefficient.

  12. We recall simply the proof of this result. Writing

    $$\begin{aligned} b(xj,y)=-jb(x,y)=-jh(x,y)+a(x,y), \end{aligned}$$

    we find that \(h(xj,y)=a(x,y)\) and, moreover, \(a(xj,y)=-h(x,y)\). That is, \(h(Tx,y)=a(x,y)\) and \(a(Tx,y)=-h(x,y)\).

  13. We recall this theorem: “If the quadratic form is different from zero, the even Clifford \(C^{+}(E,Q)\) can be realized as the Clifford algebra \(C(E_{1},Q_{1})\) where \(E_{1}=(e_{1})^{\perp }\) with \(e_{1}\in E, Q(e_{1})\ne 0\), and \(Q_{1}=-Q(e_{1})Q.\)” (cf., for example, [13,14,15, 27, 28, 30]).

  14. This is a well known theorem shown for example by N. Bourbaki [13, pp. 150–151]. As here, u is a rotation of \(\mathbb {E}_{2m}\), \(\Phi _{u}\) is an inner automorphism of \((\mathbb {E}_{2m},a)\) and \(b_{u}\) is the product of an even number of non isotropic vectors in \(\mathbb {E}_{2m}.\)

  15. We recall that, by definition, the ordinary Clifford group G(EQ) of a regular quadratic space (EQ) is the multiplicative group constituted by the invertible elements g of the corresponding Clifford algebra Cl(EQ) of (EQ) that satisfy the condition \(\forall x,x\in E,\) \(y=gxg^{-1}\in E.\)

  16. In 1935 E. Cartan was concerned with the study of which homogeneous and bounded symmetric domains in \(\mathbb {C}^{n}\) are symmetric spaces. The statement is true for \(n=1,2,3.\) In 1959, Piatecki-Szapiro gave counter-examples for \(n=4,5.\) Bounded symmetric domains in \(\mathbb {C}^{n}\) appear as very particular Hermitian spaces of non compact type. The fact that these are all such spaces, partially shown by E. Cartan, is essentially due to Harish Chandra.

    We want to recall simply some notions that the reader will find, for example, in references [24, 32] or in the paper written by F. Viviani: “A tour on Hermitian symmetric domains”, Lectures notes in Mathematics, Springer-Verlag, October 2013.

    Irreducible Hermitian symmetric manifolds of compact and non compact type can be classified using Lie Theory. They are diffeomorphic to a quotient G/K where G is a simple Lie group, compact in the compact type case and non compact in the non compact type case, and \(K\subset G\) is a maximal compact proper subgroup whose center is equal to \(S^{1}\). Using either Lie Theory or Jordan theory it is possible to give a classification of irreducible Hermitian symmetric manifolds of non compact type and of their compact duals.We recall the following theorem found in 1956 by Harish Chandra (cf. Krzysztof Maurin,“Riemannian ideas in mathematical physics: The Riemann Legacy”, Springer, 2010, p. 38).

    Theorem. (i) Every bounded symmetric domain \({\mathfrak {D}}\subset \mathbb {C}^{n}\) is a Hermitian symmetric space of non compact type, that is \({\mathfrak {D}}=G/H\) where G is a non compact Lie group. In particular, \({\mathfrak {D}}\) is simply connected. (ii) Let M be a Hermitian symmetric space of non compact type. Then, there exists a bounded symmetric space \({\mathfrak {D}}\subset \mathbb {C}^{n}\) and a holomorphic diffeomorphism from M onto \({\mathfrak {D}}\).

  17. We give this quote from H. Freudenthal [24]: “The first indication of the magic square is to be found in Tit’s “Thèse d’agrégé”, though its arithmetical properties are not mentioned. It was probably found and used as a heuristic tool by Freudenthal and Rozenfeld. The problem arose to fill the fourth line with a quadruple of geometries over the four Hurwitz algebras. In 1956 Rozenfeld proved a unified solution for the whole magic square. (...) In 1958 (or perhaps 1957) Tits and Freudenthal found independent explanations”. Hans Freudenthal shows in [24] that the first row of the magic square is relative to 2-dimensional elliptic geometry; the second one to 2-dimensional projective geometry; the third one to 5-dimensional symplectic geometry and the fourth one to a geometry called by Freudenthal metasymplectic.

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Acknowledgements

The author wants to thank the referees for their thorough inspection of the manuscript and their useful and constructive remarks. He wants also to present Rafał Abłamowicz, former Editor-in-Chief of Advances in Applied Clifford Algebras, with all his grateful thanks for his careful reading of the manuscript and his generous help in the preparation of the files. Thanks also are due to Jean-Claude Yakoubsohn, Institut de Mathématiques de Toulouse, for his efficient and friendly help in formatting the text.

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Appendix

Appendix

1.1 A Few Complements

1.1.1 Transvections [9, 23, 39]

Definition 5

Let V be a right vector space n-dimensional over a skew-field K. We call a transvection of V any linear mapping f from V into V with the properties:

$$\begin{aligned} \mathrm{rk}(f-E)=1 \quad \text {and} \quad \mathrm{Im}(f-E)\subseteq \mathrm{Ker}(f-E) \end{aligned}$$

where E is the identity linear transformation.

This is a well known proposition shown for example in [15, 17].

Proposition 1

The transvections of a vector space V generate the special linear or unimodular group \(\mathrm {SL}(V)\).

1.2 Basics on Hermitian Symmetric Domains [24, 26, 32, 43, 52]

1.2.1 Fundamental Concepts

We recall some fundamental concepts. A Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the hermitian structure. First studied by E. Cartan, they form a natural generalization of the notion of Riemannian symmetric spaces from real manifolds to complex manifolds.Footnote 16

Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non compact space that, as A. Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non compact space can be realized as a bounded symmetric domain in a complex vector space.

The irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact Lie groups by maximal closed connected subgroups which contain a maximal torus and have center isomorphic to T, that denotes the multiplicative group of complex numbers of modulus equal to 1. There is a complete list of their compact duals, a classification of the irreducible spaces, with four classical series, studied by E. Cartan, and two exceptional cases. The classification can be deduced from Borel-de Siebenthal theory, which classifies closed connected subgroups containing a maximal torus.

1.2.2 A Table

This is an extract of the table of irreducible Riemannian globally symmetric spaces of type I and type III established by E. Cartan and that concerns the subject of the present paper, cf. for example, [26, pp. 339–355], [43]. In the table, \(\mathrm {S}(\mathrm {U}_{p}\times \mathrm {U}_{q})\) denotes \(\mathrm {SU}(p,q)\cap \mathrm {U}(p+q))\) and K denotes the maximal compact subgroup of G.

Type

G

K

E. Cartan’s class

\(I_{p,q}\)

\(\mathrm {SU}(p,q)\)

\(\mathrm {S}(\mathrm {U}_{p}\times \mathrm {U}_{q}\))

AIII

\(II_{m}\)

\(\mathrm {SO^{*}}(2m)\)

\(\mathrm {U}(m)\)

DIII

\(III_{m}\)

\(\mathrm {Sp}(2m,\mathbb {R})\)

\(\mathrm {U}(m)\)

CI

1.3 Diagrams

We use notation initiated by Carl Siegel [45] and revisited notably by I. Satake [41,42,43]. Then, the fundamental diagram given in the first section can be written equivalently as the following one:

figure a

This diagram can be also rewritten equivalently as follows where b, resp. \(b_{1}\), is the skew-symmetric \(\mathbb {H}\)-sesquilinear form defined on \(\mathbb {H}^{m}\), resp. \(\mathbb {H}^{2m}\).

figure b

1.4 The Work of I. Satake

I. Satake presented in [41,42,43] a thorough study of the algebraic structures of symmetric domains. His work is beyond the goals of the present paper, but, nevertheless, it is useful. We want to give a summary of the results useful for the present paper. Satake studied the unbounded realizations of symmetric domains via the notion of Siegel domains and, more precisely, the equivariant holomorphic mappings of a symmetric domain into a Siegel space.

1.4.1 A Few Definitions

1-The Notion of Siegel Domain Let U be a real vector space, \(\dim _{\mathbb {R}}U=m>0;\) let V be a complex vector space, \(\dim _{\mathbb {C}}V=n,\) (viewed also as a real 2n-dimensional vector space), relative to a fixed complex structure \(J_{0}\); let \(\Omega \) be a (non-degenerate) open convex cone in U with vertex at the origin, and let H be a hermitian mapping from \(V\times V\) into \(U_{\mathbb {C}}\) (the complexification of U) which is \(\mathbb {C}\)-linear in the second variable and \(\mathbb {C}\)-anti linear in the first variable. Assume that H is \(\Omega \)-positive which means that we have: for any \(v\in V,\) \(v\ne 0,\) \(H(v,v)\in {\overline{\Omega }}\setminus {0}.\) By definition, the Siegel domain is

$$\begin{aligned} \mathcal {D=D}(U,V,\Omega ,H)=\{(u,v)\in U_{\mathbb {C}}\times V\mid \mathrm{Im}\, u - H(v,v)\in \Omega \}. \end{aligned}$$

2-Classical Domains

Notation and Definitions Let F be either \(\mathbb {R}\), \(\mathbb {C}\), or \(\mathbb {H}\). We denote by \(\iota _{0}\) the standard involution of F,  that is, the identity if \(F=\mathbb {R}\); the complex conjugation if \(F=\mathbb {C}\); the standard involution \(\star \) if \(F=\mathbb {H}\), such that for any \(q=u+jv\in \mathbb {H},(u,v\in \mathbb {C}),\) \(q\longrightarrow q^{\star }={\overline{u}}-jv.\)

Let V be a n-dimensional vector space over \(\mathbb {R}\) endowed with a right F-module structure. We use the definitions and the notation recalled in footnote 4. Satake defined the following three sets:

$$\begin{aligned}&\mathcal {H}_{n}(F)=\{h\in \mathcal {M}(n,F)\mid ^{t}{\overline{h}}=h\}; \quad \mathcal {P}_{n}(F)=\{h\in \mathcal {H}_{n}(F)\mid h\gg 0\};\\&\qquad \mathcal {P}_{n}^{(1)}(F)=\{h\in \mathcal {P}_{n}(F)\mid N(h)=1\}, \end{aligned}$$

where N denotes the usual reduced norm in the algebra \(\mathcal {A}\) of F-linear transformations of V defined for example in [2, pp. 121–124]. An involution \(\alpha \) of \(\mathcal {M}(n,F)\) is called positive if the quadratic form \(\mathrm{tr}(\alpha (x)x),\) (often denoted by A. Weil as \(\mathrm{tr}(x^{\alpha }x)\) (cf. [51])), on \(\mathcal {M}(n,F)\) is positive definite.

Satake proved the following theorem.

Theorem 10

   

  1. (i)

    The symmetric space associated with \(\mathrm {SL}(n,F)\) is given by \(\mathcal {P}_{n}^{(1)}(F)\).

  2. (ii)

    It can be identified with the space of all positive involutions of \(\mathcal {M}(n,F).\)

  3. (iii)

    The symmetric space \(\mathcal {D=D}(V,h)\) associated with the group \(G=\mathrm {SU}(V,h)\) can be identified with the space of all complex structures J satisfying the condition \(hJ\in \mathcal {P}_{n}(F).\)

Satake considers in [43],pp.273-279, the following symmetric domains \(\mathcal {D}\) of type \(( III \) \(\,_{\frac{n}{2}}\)),(\(I_{p,q}\)),(\( II_{n} \)), in Siegel’s notations,(cf.[45]), according as F is either \(\mathbb {R}\) or \(\mathbb {C}\) or \(\mathbb {H}\). He studies very deeply the holomorphic embedding of a symmetric domain \( D \) into a symmetric domain \( D' ,\) that is beyond the goals of the present paper.

1.5 Magic Square

There is a link between the diagram given above, called (S.C.) and the Tits-Freudenthal magic square.Footnote 17

In contemporary notation, the magic square can be set up as follows:

\(A\setminus \)B

\(\mathbb {R}\)

\(\mathbb {C}\)

\(\mathbb {H}\)

O

\(\mathbb {R}\)

A\(_{1}\)

A\(_{2}\)

C\(_{3}\)

F\(_{_{4}}\)

\(\mathbb {C}\)

A\(_{2}\)

A\(_{1}\)+A\(_{2}\)

A\(_{5}\)

E\(_{6}\)

\(\mathbb {H}\)

C\(_{3}\)

A\(_{5}\)

D\(_{6}\)

E\(_{7}\)

O

F\(_{4}\)

E\(_{6}\)

E\(_{7}\)

E\(_{8}\)

This magic square associates a Lie algebra to a pair of division algebras AB. The construction is symmetric in A and B although originally it was not symmetric. Therefore, we are led to the following square.

A\(\setminus \)B

\(\mathbb {R}\)

\(\mathbb {C}\)

\(\mathbb {H}\)

\(\mathbb {R}\)

\(\mathfrak {sl(n,\mathbb {R})}\)

\(\mathfrak {sl(n,\mathbb {C}}\))

\(\mathfrak {sl(n,\mathbb {H}}\))

\(\mathbb {C}\)

\(\mathfrak {sp(2n,\mathbb {R})}\)

\(\mathfrak {su(n,n)}\)

\(\mathfrak {so^{*}(4n)}\)

\(\mathbb {H}\)

\(\mathfrak {sp(2n,\mathbb {C})}\)

\(\mathfrak {sl(2n,\mathbb {C}}\))

\(s\mathfrak {o(4n,\mathbb {C}}\))

The second row of that square is associated with the embeddings of groups: \(\mathrm {Sp}(2n,\mathbb {R})\longrightarrow \mathrm {SU}(n,n)\longrightarrow \mathrm {SO}^{*}(4n),\) found previously.

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Anglès, P. Siegel-Satake Cross and Associated Clifford Algebras. Adv. Appl. Clifford Algebras 32, 53 (2022). https://doi.org/10.1007/s00006-022-01223-1

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