On the passage from finite braces to pre-Lie rings. (English) Zbl 1512.17055
Let \(p\) be a prime. The Lazard correspondence in group theory is between Lazard \(p\)-Lie rings and Lazard Lie \(p\)-groups. In [W. Rump, Note Mat. 34, 115–145 (2014; Zbl 1344.14029)], it was suggested that this correspondence could be extended to one between pre-Lie rings and braces. A formula of constructing a brace from a pre-Lie ring (using the Lazard correspondence) was also developed there.
A pre-Lie ring is a vector space \(A\) with a binary operation \(\cdot\) and a group operation \(+\) such that \((A,+)\) is abelian and \begin{align*} (x\cdot y)\cdot z - x\cdot (y\cdot z) & = (y\cdot x)\cdot z - y\cdot (x\cdot z)\\ (x + y) \cdot z & = x\cdot z + y\cdot z\\ x \cdot (y+z) & = x \cdot y + x\cdot z \end{align*} hold for all \(x,y,z\in A\).
A (left) brace is a set \(A\) equipped with two group operations \(+\) and \(\circ\) such that \((A,+)\) is abelian and \[ x \circ (y + z) + x = x\circ y + x \circ z \] holds for all \(x,y,z\in A\). Brace was introduced in [W. Rump, J. Algebra 307, 153–170 (2007; Zbl 1115.16022)] to describe all non-degenerate and involutive set-theoretic solutions of the Yang-Baxter equation. The definition stated here in from [F. Cedó et al., Commun. Math. Phys. 327, 101–116 (2014; Zbl 1287.81062)].
The paper under review shows that the aforementioned formula of Rump works for all left nilpotent pre-Lie rings of order \(p^n\) when \(n\leq p-2\). In particular, it was shown that for any \(n\leq p-2\), there is an injective mapping from left nilpotent pre-Lie rings of order \(p^n\) to left nilpotent braces of order \(p^n\) which preserves right nilpotency. Whether there is a passage from left nilpotent braces to left nilpotent pre-Lie rings remains an open question.
The paper under review also provides a method of constructing a pre-Lie ring from a brace. In particular, it was shown that for any \(k\leq p -1\) and \(n\leq p-2\), there is a one-to-one correspondence between strongly nilpotent braces of order \(p^n\) and nilpotent pre-Lie rings of order \(p^n\) which preserves nilpotency index.
A pre-Lie ring is a vector space \(A\) with a binary operation \(\cdot\) and a group operation \(+\) such that \((A,+)\) is abelian and \begin{align*} (x\cdot y)\cdot z - x\cdot (y\cdot z) & = (y\cdot x)\cdot z - y\cdot (x\cdot z)\\ (x + y) \cdot z & = x\cdot z + y\cdot z\\ x \cdot (y+z) & = x \cdot y + x\cdot z \end{align*} hold for all \(x,y,z\in A\).
A (left) brace is a set \(A\) equipped with two group operations \(+\) and \(\circ\) such that \((A,+)\) is abelian and \[ x \circ (y + z) + x = x\circ y + x \circ z \] holds for all \(x,y,z\in A\). Brace was introduced in [W. Rump, J. Algebra 307, 153–170 (2007; Zbl 1115.16022)] to describe all non-degenerate and involutive set-theoretic solutions of the Yang-Baxter equation. The definition stated here in from [F. Cedó et al., Commun. Math. Phys. 327, 101–116 (2014; Zbl 1287.81062)].
The paper under review shows that the aforementioned formula of Rump works for all left nilpotent pre-Lie rings of order \(p^n\) when \(n\leq p-2\). In particular, it was shown that for any \(n\leq p-2\), there is an injective mapping from left nilpotent pre-Lie rings of order \(p^n\) to left nilpotent braces of order \(p^n\) which preserves right nilpotency. Whether there is a passage from left nilpotent braces to left nilpotent pre-Lie rings remains an open question.
The paper under review also provides a method of constructing a pre-Lie ring from a brace. In particular, it was shown that for any \(k\leq p -1\) and \(n\leq p-2\), there is a one-to-one correspondence between strongly nilpotent braces of order \(p^n\) and nilpotent pre-Lie rings of order \(p^n\) which preserves nilpotency index.
Reviewer: Cindy Tsang (Tokyo)
MSC:
| 17D25 | Lie-admissible algebras |
| 16T25 | Yang-Baxter equations |
| 20E99 | Structure and classification of infinite or finite groups |
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