Structures of G(2) Type and Nonintegrable Distributions in Characteristic p

Abstract

ABSTRACT. Lately we observe: (1) an upsurge of interest (in particular, triggered by a paper by Atiyah and Witten) to manifolds with G(2)-type structure; (2) classifications are obtained of simple (finite dimensional and graded vectorial) Lie superalgebras over fields of complex and real numbers and of simple finite dimensional Lie algebras over algebraically closed fields of characteristic p greater than 3; (3) importance of nonintegrable distributions in observations (1) and (2).We add to interrelation of (1)- (3) an explicit description of several exceptional simple Lie algebras for p=2, 3 (Brown, Ermolaev, Frank, and Skryabin algebras, and analogs of Melikyan algebras) as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by g (2), o(7), sp(4), sp(10) and the Brown algebra br(3). The description is performed in terms of Cartan-Tanaka-Shchepochkina prolongs and is similar to descriptions of simple Lie superalgebras of vector fields with polynomial coefficients. Our results illustrate usefulness of Shchepochkina's algorithm and SuperLie package; at least two families of simple Lie algebras found in the process are new.

New simple modular Lie superalgebras as generalized prolongs

“…For p < 5, the above KSh-procedure and Melikyan's examples do not produce all simple finite-dimensional Lie algebras; there appear other examples and several old ones disappear. In [13], we returned tó E. Cartan's description of Z-graded Lie algebras as subalgebras of vectorial Lie algebras preserving certain nonintegrable distributions; we thus interpreted the "mysterious" exceptional examples of simple Lie algebras for p = 3 (the Brown, Frank, Ermolaev and Skryabin algebras), further elucidated Kuznetsov's interpretation [16] of Melikyan's algebras (as prolongs of the nonpositive part of the exceptional Lie algebra g(2) in one of its Z-gradings) and discovered three new series of simple Lie algebras. In [1], the same approach yielded bj, a simple super version of g(2), and Bj(1; N |7), a simple super version of the Melikyan algebra.…”

“…We use the standard notation of [10] and [22]; for the precise definition (algorithm) of Cartan-Tanaka-Shchepochkina (CTS) complete and partial prolongations, see [21]. Let mg(A) denote the realization of the Lie (super)algebra g(A) with the help of the mth Cartan matrix A; for their numbering and list, see [13], [2], and [3]. The grading deg e ± i = ±s i of Chevalley generators of g(A) is said to be simplest if all but one coordinates of the vector s = (s 1 , .…”

“…It is based on an idea entirely different from that of the KSh-conjecture. In it, as well as in [13] and [1], CTS-prolongs play the main role. (In the same way, Cartan obtained all simple Z-graded Lie algebras of polynomial growth and finite depth-the Lie algebras of type (1)-at the time when the root technique was not developed yet.…”

Деформации алгебры Ли $\mathfrak o(5)$ в характеристиках $3$ и $2$

“…Немного раньше Кос-трикина, используя явное описание 3-параметрического семейства T( , , ) неприво-димых 3-мерных sl(2)-модулей в характеристике 3, полученное Рудаковым и Шафа-ревичем [11], Рудаков построил картановское продолжение 2 ( , , ) := (T( , , ), gl(2)) * , являющееся Z-градуированным 3-параметрическим семейством деформаций алгеб-ры Ли o(5) (в [9] конструкция Рудакова упомянута). Эта конструкция прозрачна, а явные формулы описаны в [13]: а именно,…”

“…Отметим, что Грозман и Лейтес [13] нашли исключительные значения параметров ( , , ), при которых картановское продолжение…”