Boris Rosenfeld

Algebras and Lie Groups

In 0.1.7 we have defined n-algebras, that is, rings that are linear spaces L n satisfying the axiom A.1° (0.13). In 0.1.7 we mentioned that the field ℂ of complex numbers is a commutative associative 2-algebra with basis e 0 = 1, e l = i (i 2 = −1).

Geometries of Exceptional Lie Groups. Metasymplectic Geometries

There are two locally non-isomorphic simple real Lie groups G 2: a compact group with character δ = −14 (the Dynkin diagram is given in Fig. 1.3) and a split group with character δ = 2 (the Satake diagram is given in Fig. 1.9).

Affine and Projective Geometries

In 0.1.6 we have defined the real linear spaces L n. If in this definition we replace the field ℝ of real numbers by the field ℂ of complex numbers or by the skew field ℍ of quaternions, we obtain the complex linear space ℂL n or quaternionic linear space ℍL n. We have mentioned the space ℂL n in 1.6.1. Since ℍ is noncommutative, in the products (0.7) of a vector a and a quaternionic scalar λ, the scalar must always be at the right of the vector.

Trigonometry in Islamic Mathematics

Trigonometry is the connecting link between mathematics and astronomy, between the way calendars are calculated, the gnomon, and the sundial. In the Islamic world, the calculation of spherical triangles was necessary to carry out ritual customs. The qibla, the direction to Mecca, was indicated next to the hour lines on all public sundials. The first trigonometric problems appeared in the field of spherical astronomy. Around the year 773, one of the Indian siddhāntas (astronomy books) was made known in Baghdad. The Indian astronomers Varāhamihira (fifth century) and Brahmagupta (sixth century) solved different problems in spherical astronomy by means of rules equivalent to a general sine theorem for a spherical triangle ABC with sides a, b, c and angles A, B, C (where angle A is opposite to side a, etc.), namely, sin A= sin a ð Þ 1⁄4 sin B= sin b ð Þ 1⁄4 sin C= sin c ð Þ and to the cosine theorem for the same triangle cos a 1⁄4 cos b cos cþ sin b sin c cos A: In the ninth century, Ptolemy’s Almagest and Menelaus’ Spherics were also translated, and commentaries were written to these works. Many trigonometric problems were solved in Ptolemy’s Almagest, in which Menelaus’ theorem on the spherical complete quadrilateral was used. The cases of this theorem used by Ptolemy are equivalent to the sine and tangent theorems for a right‐angled spherical triangle. The Almagest, the Spherics, and the Indian siddhāntas formed the basis on which Arab mathematicians built their trigonometry. The ancient Greek astronomers only used one trigonometric function, the chord of an arc. The “theorem of Ptolemy,” which is equivalent to the formula for the sine of the sum of the angles, forms, together with the formula for the chord of the half arc, the basis for the chord table in the Almagest. The Indian people replaced the chord with the sine, introduced the cosine and the versed sine, and compiled a small table of sine values. The Arabic mathematicians progressively made trigonometry into a science independent of its (astronomical) context. Applications of trigonometry analogous to those in the Indian siddhāntas are found in the astronomical works of al‐Khwārizmī. An analogous geometric construction for finding the azimuth according to the rule formulated in al‐Khwārizmī’s third treatise was provided by al‐Māhānī (ca. 825–888) in his Treatise on the Determination of the Azimuth at Any Time and in Any Place. The rules equivalent to the spherical sine and cosine theorems were also used by Thābit ibn Qurra in his Book on Horary Instruments Called Sundials. With Ḥabash, the applications of the tangent and cotangent functions went beyond the usual applications in the theory of sundials. The introduction of the tangent and cotangent and their application in astronomy was a novelty. The names zill (shadow) and zill maʿqus (reversed shadow) apparently are translations from Sanskrit. In the case of a vertical gnomon, al‐Ḥabas expressed the cosecant as the “diameter of the shadow” for a given height of the sun, i.e., as a hypotenuse. He computed a table for the cosecant with steps of 1 . For a long time, the chord was used along with the sine. A theory of these magnitudes is found in the work of al‐Battānī (ca. 858–929). In his astronomical work Islaḥ al‐Majisṭī (The Perfection of the Almagest), he systematically employed the trigonometric functions sine and versed sine with arguments between 0 and 180 . Since the cosine is defined as the sine of the complement of the angle and since no negative numbers are used, the versed sine is defined in the second quadrant as a sum of two quantities.

Geometry of Planes over Nonassociative Algebras

In this paper the geometric interpretation of the exceptional Lie groups F4, E6, E7, and E8 is given. These groups are groups of motions of elliptic hyperbolic planes over nonassociative algebras of octaves and split octaves and their tensor products with algebras of usual and split complex numbers, quaternions and octaves. The explicit expressions of motions of these planes and their figures of symmetry are presented.

Structures of Geometry

The objects of modern Geometry, spaces, like all objects of modern Mathematics, are sets of elements of arbitrary nature endowed with some mathematical structure. Such is the formulation in the best Encyclopaedia of Mathematics, the Elements of Mathematics by N. Bourbaki [Boul–9]. Many books in this encyclopaedic series, in particular Topological Vector Spaces [Bou5], and Lie Groups and Algebras [Bou8], are closely connected with modern Geometry, and the title Elements of Bourbaki emphasizes that this mathematical Encyclopaedia is a successor of the classical mathematical Encyclopaedia of Antiquity, the Elements of Euclid [Euc], the main part of which is devoted to geometry.

Quasielliptic, Quasihyperbolic, and Quasi-Euclidean Geometries

Since the Euclidean space R n and pseudo-Euclidean space R l n can be defined as the projective space P n with a distinguished hyperplane in which the absolute hyperquadrics of the spaces S n-1 or S l n-1 are given, there are spaces corresponding them by the duality principle of P n . These dual spaces are called co-Euclidean, respectively copseudo-Euclidean, spaces and are denoted by R n* , respectively R l n* .

Symplectic and Quasisymplectic Geometries

In 4.1.1 we have seen that the absolutes of the spaces S n , H n , S l n , and H l n are imaginary or real hyperquadrics, which, as we have seen in 2.8.3 are cosymmetry figures in the space P n . In 2.8.3 we have also seen that, besides hyperquadrics, in P 2n+1 there are cosymmetry figures of an other kind: linear complexes of lines. The space P 2n −1 in which a linear complex of lines is given is said to be a real quadratic symplectic space and is denoted by Sy 2n −1. The linear complex determining this space is called the absolute linear complex of Sy 2n −1. The collineations in P 2n −1 preserving the absolute linear complex of Sy 2n −1are called symplectic transformations in this space. The absolute linear complex of Sy 2n −1 can be defined by (2.109), where the (2n × 2n)-matrix (a ij ) can be reduced to the form (0.62); then (2.109) has the form

Euclidean, Pseudo-Euclidean, Conformal and Pseudoconformal Geometries

\sum\limits_{i = 0}^{n - 1} {{p^{2i,2i + 1}}} = 0.