On the Dimension of Non-Abelian Tensor Squares of $n$-Lie Algebras
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Abstract
Let $L$ be an $n$-Lie algebra over a field $\mathbb{F}$. In this paper, we introduce the notion of non-abelian tensor square $L\otimes L$ of $L$ and define the central ideal $L\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\square L\cong L^{ab}\square L^{ab}$. Also, we establish the short exact sequence
\[ 0\to\mathcal{M}(L)\to\frac{L\otimes L}{L\square L}\to L^2\to 0 \]
and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.
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