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A class of axial decomposition algebras with Miyamoto group generated by two Miyamoto automorphisms and three eigenvalues $0,1$ and $eta$ is introduced and classified in the case with $eta otin{0,1,frac{1}{2}}$. This class includes specializations of 2-generated axial algebras of Majorana type $(xi,eta)$ to the case with $xi=eta$.
A class of axial algebras generated by two axes with eigenvalues 0, 1, $eta$ and $xi$ called axial algebras of Majorana type is introduced and classified when they are 2-generated, over fields of characteristics neither 2 nor 5 and there exists an au
The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler, is analyzed. Transitive irreducible graded Lie algebras $L=sum_{iin mathbb Z}L_i,$ over an algebraically closed field of characteristic $p>2,$ with clas
For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple restricted Lie algebras of types W(m;1) and S(m;1) (m>=2), in terms of numerical and group-theoretical invariants. Our main tool is automorphism group scheme
We give the description of homogeneous Rota-Baxter operators, Reynolds operators, Nijenhuis operators, Average operators and differential operator of weight 1 of null-filiform associative algebras of arbitrary dimension.
We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For $mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtch