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Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ∼ = U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H.

We determine structure constants for the universal nonassociative enveloping algebra U(M) of the fourdimensional non-Lie Malcev algebra M by constructing a representation of U(M) by differential operators on the polynomial algebra P (M). These structure constants involve Stirling numbers of the second kind. This work is based on the recent theorem of Pérez-Izquierdo and Shestakov which generalizes the Poincaré-Birkhoff-Witt theorem from Lie algebras to Malcev algebras. We use our results for U(M) to determine structure constants for the universal alternative enveloping algebra A(M) = U(M)/I(M) where I(M) is the alternator ideal of U(M). The structure constants for A(M) were obtained earlier by Shestakov using different methods.Keywords Keywords structure constant, four-dimensional malcev algebra, differential operator, recent theorem, different method, malcev algebra, poincar birkhoff-witt theorem, polynomial algebra, second kind, alternator ideal, lie algebra, four-dimensional non-lie malcev algebra Disciplines Disciplines Algebra | Mathematics Comments Comments This article is published as Bremner, Murray R., Irvin R. Hentzel, Luiz A. Peresi, and Hamid Usefi. "Universal enveloping algebras of the four-dimensional Malcev algebra." Contemporary Mathematics 483 (2009): 73-89. Posted with permission.Abstract. We determine structure constants for the universal nonassociative enveloping algebra U (M) of the four-dimensional non-Lie Malcev algebra M by constructing a representation of U (M) by differential operators on the polynomial algebra P (M). The structure constants for U (M) involve the Stirling numbers of the second kind. This work is based on the recent theorem of Pérez-Izquierdo and Shestakov which generalizes the Poincaré-Birkhoff-Witt theorem from Lie algebras to Malcev algebras. We use our results for U (M) to determine structure constants for the universal alternative enveloping algebra A(M) = U (M)/I(M) where I(M) is the alternator ideal of U (M). The structure constants for A(M) were obtained earlier by Shestakov using different methods.

Ulcerative colitis (UC) is one of the most common forms of inflammatory bowel disease (IBD) characterized by inflammation of the mucosal layer of the colon. Diagnosis of UC is based on clinical symptoms, and then confirmed based on endoscopic, histologic and laboratory findings. Feature selection and machine learning have been previously used for creating models to facilitate the diagnosis of certain diseases. In this work, we used a recently developed feature selection algorithm (DRPT) combined with a support vector machine (SVM) classifier to generate a model to discriminate between healthy subjects and subjects with UC based on the expression values of 32 genes in colon samples. We validated our model with an independent gene expression dataset of colonic samples from subjects in active and inactive periods of UC. Our model perfectly detected all active cases and had an average precision of 0.62 in the inactive cases. Compared with results reported in previous studies and a model generated by a recently published software for biomarker discovery using machine learning (BioDiscML), our final model for detecting UC shows better performance in terms of average precision.

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