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1. Braided Frobenius algebras from certain Hopf algebras

   https://doi.org/10.1142/S0219498823500123  

   Saito, Masahico; Zappala, Emanuele (September 2021, Journal of Algebra and Its Applications)

   A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations. 

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2. Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras

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   Baudier, Florent P; Braga, Bruno M; Farah, Ilijas; Vignati, Alessandro; Willett, Rufus (January 2024, Journal of Functional Analysis)

   We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated with a uniformly locally finite metric space X. Under weak assumptions, these C*-algebras contain embedded copies of certain matrix algebras. We aim to show they cannot contain any other von Neumann algebras. One of our main results shows that the only embedded von Neumann algebras are the “obvious” ones. 

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3. Classification of algebras of level two in the variety of nilpotent algebras and Leibniz algebras

   https://doi.org/10.1016/j.geomphys.2018.08.012  

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   Kleshchev, Alexander; Muth, Robert (July 2019, Transactions of the American Mathematical Society)
5. Unimodular graded Poisson Hopf algebras: GRADED POISSON HOPF ALGEBRAS

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   Brown, Kenneth A.; Zhang, James J. (October 2018, Bulletin of the London Mathematical Society)