For example, we still don’t know if it has the structure of an algebraic group of infinite dimension. In this paper, we will construct the Cremona group functor, calculate its Lie algebra and show that its Lie algebra is simple. | JOURNAL OF SCIENCE OF HNUE DOI Mathematical and Physical Sci. 2016 Vol. 61 No. 7 pp. 3-13 This paper is available online at http CREMONA GROUP FUNCTOR AND ITS LIE ALGEBRA Nguyen Dat Dang Faculty of Mathematics Hanoi National University of Education Abstract. Let Crn k Bir Pnk denote the set of all birational maps of the projective space Pnk over a field k. It is clear that Crn k is a group under composition of dominant rational maps called the Cremona group of order n. This group is not an algebraic group. It was studied for the first time by Luigi Cremona 1830 - 1903 an Italian mathematician. Although it has been studied since the 19th Century by many famous mathematicians it is still not well understood. For example we still don t know if it has the structure of an algebraic group of infinite dimension. In this paper we will construct the Cremona group functor calculate its Lie algebra and show that its Lie algebra is simple. Keywords Birational map cremona group group functor Lie algebra. 1. Lie algebra of a group functor . Extension of categories When we study the Cremona group Crn k we have the following question Is it an algebraic group a variety a scheme . . . If none of the above in what category does it belong In this paper we will present a category which is larger than the category of group schemes and contains Crn k as its object. In classic algebraic geometry we study the category of algebraic varieties which is the sets of solutions to systems of polynomial equations in An ou Pn . However because this category is not large enough we extend it in the category of schemes which is really larger. We can resume this extension stating the following theorem cf. 1 page 78 and page 104 . Theorem . Let k be an algebraically closed field. There is a natural fully faithful functor t Var k Sch k from the category Var k of varieties over k to Sch k schemes over k. For any variety V its topological space is .
