# A History of Geometrical Methods by Julian Lowell Coolidge By Julian Lowell Coolidge

Full, authoritative background of the innovations for facing geometric equations covers improvement of projective geometry from historic to fashionable instances, explaining the unique works, commenting at the correctness and directness of proofs, and exhibiting the relationships among arithmetic and different highbrow advancements. 1940 edition.

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Let R be a bialgebra or Ropf algebra and let HM be the category of algebra representations as before. Then 0, defined using module notation by h 1> (v 0 w) = 2: h 1 1> v 0 h2 1> w (via the coproduct) • makes H M into a monoidal category. Since the constructions have dual counterparts, if A is a bialgebra or a Ropf algebra, then the category MA of right A-comodules is a monoidal cateory. The tensor product comodule V 0 W is defined via the coaction (A88) ,8Vl8>w(v 0 w) = 2:v 1 0 w 1 0 v 2w 2 in terms of the coactions on V, W and the product of A.

H -+ H 0 H. Conversely, gi ven a coaction V -+ H 0 V one defines >, v, w (v 0 w) = 2: vI c> W 0 v 2 and checks that it is a natural transformation. • Next let H be a bialgebra and C = HM; then CO over C (without an invertibility condition on >,) can be identified with the category ~M of crossed modules and comodules (or D(H)-module - D(H) will be discussed later as needed). In particular a left D(H)-module is a vector space V which is both a left H-module by c> and a left H-comodule by (3( v) = 2: vI 0 v 2 satisfying (A90) 2: h 1v 10h 2C>V 2 = 2:(h 1c>v)1 0 (h 1c>v)2.

For G semisimple and H reductive there exists an indefinite metric on M such that these ftows are geodesics. The metric is induced by the Killing form and is Riemannian if G is compact. Instead of working with the points of M one could equally well work with functions in COO(M) (classical point observables). The trajectories, ar geodesics, are labeled by elements of 9 and these are called classical momentum observables. An equivalent set up is to work in the phase space X = T* Mor rather with functions on X.