On \((\alpha, \beta, \gamma)\)-structural algebras and Dynkin diagrams: beyond Lie algebras to triple systems (Q2874901)

The work introduces the notion of \((\alpha,\beta,\gamma)\)-structurable algebras, where \((\alpha,\beta,\gamma)\in\{-1,0,1\}\). If \((A,-)\) is an algebra with involution and we fix \((\alpha,\beta,\gamma)\in\{-1,0,1\}\) we can define \(V_{x,y}: A\to A\) by \(V_{x,y}=\alpha L_{L_x(\bar y)}+\beta R_x R_{\bar y}+\gamma R_y R_{\bar x}\). We can also define \(B_A(x,y,z):=V_{x,y}(z)\) and call this the \((\alpha,\beta,\gamma)\)-triple system obtained from the algebra \((A,-)\). We say that \((A,-)\) is an \((\alpha,\beta,\gamma)\)-structurable algebra if the following identity is satisfied: NEWLINE\[NEWLINE[V_{u,v},V_{x,y}]=V_{V_{u,v}(x),y}-V_{x,V_{v,u}(y)}NEWLINE\]NEWLINE for any \(u,v,x,y\in A\). If \((\alpha,\beta,\gamma)=(1,1,-1)\) then \((A,-)\) is a structurable algebra. Also \(A\) with the triple product \(B_A\) is a generalized Jordan triple system (GJTSs in the sequel). If \((\alpha,\beta,\gamma)=(1,-1,1)\), then \((A,-)\) is an antistructurable algebra. The paper contains also a classification of standard embedding Lie algebras of classical complex simple GJTS of second order. Then, it is obtained a classification of structurable algebra from GJTS of second order with a left unital element \(e\) such that \(eex=x\) for all \(x\). The paper also contains a discussion of almost complex structures and some final short comments on applications in \(M\)-theory and to other topics such as constructions of Lie algebras and superalgebras, Yang-Baxter equations, generalizations of generalized Jordan triple systems, Nambu identity and others.