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In this paper, first, we study linear deformations of a Lie–Yamaguti algebra and introduce the notion of a Nijenhuis operator. Then we introduce the notion of a product structure on a Lie–Yamaguti algebra, which is a Nijenhuis operator [Formula: see text] satisfying [Formula: see text]. There is a product structure on a Lie–Yamaguti algebra if and only if the Lie–Yamaguti algebra is the direct sum of two subalgebras (as vector spaces). There are some special product structures, each of which corresponds to a special decomposition of the original Lie–Yamaguti algebra. In the same way, we introduce the notion of a complex structure on a Lie–Yamaguti algebra. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on a Lie–Yamaguti algebra.

In this study, high-throughput technology was used to reveal the core microbial community in sorghum-based fermented grains during different fermentation periods and to quantify the impacts of physicochemical parameters and microbial interspecies relationships on the volatile flavors. Headspace solid-phase microextraction, coupled with gas chromatography-mass spectrometry, was used to select 14 major volatile products with relative content greater than 1% in at least one sample, including three alcohols, one acid, eight esters, and two alkanes. The relative content of alkanes was only high on the first day and continued to decrease during the later fermentation stage. As fermentation progressed, the relative content of ethanol, ethyl acetate (aroma), and isoamyl alcohol (pungent, spicy) first increased and then decreased. In addition, the relative content of other ethyl esters continued to increase. In the early stage of fermentation (1-7 days), the temperature, moisture, and alcohol content showed an upward trend, while the content of reducing sugar decreased. As the temperature decreased in the middle and later stages (7-28 days), the physicochemical parameters tended to stabilize. In community composition, the dominant bacterial genera were Lactobacillus, Streptomyces, and Acetobacter, and the fungal genera were mainly Issatchenkia, Torulaspora, and Pichia. Network analysis identified a total of 10 core microbiota as the main contributors of esters and alkane metabolites. Moreover, total acidity and reducing sugar played important roles in promoting the formation of core microbiota and succession of dominant taxa.

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