On Identical Relations in Free Polynilpotent Lie Algebras

Bounds in the restricted Burnside problem

“…Burnside proved that B(m, 3) is finite for all m, and also proved that 5(2,4) is finite. In 1940 Sanov [37] proved that B(m, 4) is finite for all m, and in 1958 Hall [14] proved that B(m, 6) is finite for all m. To date, no other Burnside groups (apart from the cyclic Burnside groups) are known to be finite, although a great deal of work has been done in an attempt to determine whether or not B (2,5) is finite. In the other direction, Novikov and Adjan [33][34][35] proved that B(m, n) is infinite if m > 1 and n is odd and n > 4381.…”

Normal enveloping algebras

“…On the other hand, if U (L) is normal then it satisfies the standard polynomial identity of degree 4 [Herstein 1976, Section 5]. Therefore, in view of a theorem of Latysěv [Bahturin 1987, Section 6.7,…”

The algebraic entropy of classical mechanics

“…If their conjecture is true, L is a PI-algebra [2,8], one which the identities which hold in the Lie algebra (such as Eq. (6)) are satisfied by all elements of the Lie algebra.…”

“…exp(a i τ A) exp(b i τ B) = exp(Z) (2) where Z ∈ L(A, B). Requiring Z = τ (A + B) + O(τ p+1 ) for some integer p > 1 gives a system of equations in the a i and b i which must be satisfied for the method to have order p. In the case of general A and B, then, at each order n = 1, .…”