The Bogomolov multiplier of a multiplicative Lie algebra

TL;DR

The paper extends the Bogomolov multiplier to multiplicative Lie algebras by defining $B_0(G)$ as a subgroup of $H^2_{ML}(G,\mathbb{C}^*)$ and introduces the companion $\tilde{B}_0(G)$ via a Lie exterior framework. It establishes an isomorphism $B_0(G)\cong\tilde{B}_0(G)$ for finite $G$ by linking cohomology to the Schur multiplier $\tilde{M}(G)$ through a duality with $\mathrm{Hom}(\tilde{M}(G),\mathbb{C}^*)$, and derives a Hopf-type formula $\tilde{B_0}(G)\cong\frac{R\cap(F\star F)[F,F]}{\langle K(F)\cap R\rangle}$ along with a five-term exact sequence. The work introduces the curly Lie exterior square $G\curlywedge^L G$, providing a parallel mechanism to study $\tilde{B_0}(G)$ and defining a universal CTP extension framework; it further proves that Bogomolov multipliers are invariant under isoclinism of multiplicative Lie algebras. Collectively, these results connect central extensions, cohomology, and isoclinism in the multiplicative Lie algebra setting and extend classical group-theoretic Bogomolov theory to a broader algebraic context.

Abstract

In this paper, we develop the concept of the Bogomolov multiplier for a multiplicative Lie algebra and establish a Hopf-type formula. Consequently, we see that the Bogomolov multipliers of two isoclinic multiplicative Lie algebras are isomorphic.

Theorems & Definitions (35)

• Definition 2.1
• Proposition 2.1
• Definition 2.2
• Remark 2.2
• Definition 2.3
• Theorem 2.3
• Definition 2.4
• Definition 3.1
• Proposition 3.1
• proof