Special and exceptional Jordan dialgebras (Q2885390)

Over a field of characteristic different from 2 and 3 Jordan algebras are defined by the multilinear polynomial identities \(x_1x_2=x_2x_1\) and NEWLINE\[NEWLINE J(x_1,x_2,x_3,x_4)=x_1(x_2(x_3x_4))+(x_2(x_1x_3)x_4+x_3(x_2(x_1x_4)) NEWLINE\]NEWLINE NEWLINE\[NEWLINE -(x_1x_2)(x_3x_4)-(x_1x_3)(x_2x_4)-(x_3x_2)(x_1x_4). NEWLINE\]NEWLINE Jordan dialgebras form the class of all dialgebras satisfying the identities NEWLINE\[NEWLINE (x\dashv y)\vdash z=(x\vdash y)\vdash z,\, x\dashv(y\vdash z)=x\dashv(y\dashv z) NEWLINE\]NEWLINE and the dialgebraic analogues of the Jordan identities NEWLINE\[NEWLINE x_1\vdash x_2=x_2\dashv x_1,\, J(x_1,\ldots,\dot{x}_i,\ldots,x_4)=0,\,i=1,2,3,4, NEWLINE\]NEWLINE where \((a_1\cdots \dot{a}_k\cdots a_n)=(a_1\vdash\cdots\vdash a_k\dashv\cdots\dashv a_n)\) for any nonassociative word \((a_1\cdots a_k\cdots a_n)\), i.e., the element \(a_k\) is central for the monomial. The Jordan dialgebra is special if it can be embedded into an associative dialgebra considered as a Jordan dialgebra with operations NEWLINE\[NEWLINE a_{(\vdash)}b={1\over 2}(a\vdash b+b\dashv a),\, a_{(\dashv)}b={1\over 2}(a\dashv b+b\vdash a). NEWLINE\]NEWLINE In the paper under review the author develops an approach for reducing problems on Jordan dialgebras to the case of ordinary Jordan algebras. He shows that straightforward generalizations of the classical Cohn, Shirshov, and Macdonald theorems do not hold for Jordan dialgebras. He finds the exact analogues of these statements. (1) The free special Jordan algebra with one or two generators coincides with the set of symmetric elements of the free associative dialgebra. The inclusion is strict for the case of three generators. (The theorem of Cohn includes also the case of free algebras with three generators.) (2) There is a two-generated Jordan dialgebra which is a homomorphic image of the free special Jordan dialgebra but is not special. Hence the class of special Jordan dialgebras is not a variety. (This is an analogue of an example of Cohn for Jordan algebras.) (3) Any one-generated Jordan dialgebra is special. The two-generated free Jordan dialgebra is also special. (By the Shirshov theorem any two-generated Jordan algebra is special.) (4) Any polynomial in three variables \(f(x,y,\dot{z})\) which is linear in the central variable \(z\) and vanishes on the special Jordan dialgebras vanishes on all Jordan dialgebras. (This is an analogue of the Macdonald theorem for polynomial identities in three variables which are linear in one of the variables and vanish on the special Jordan algebras. A counterexample of Bremner and Peresi shows that there is no straightforward analogue for dialgebras.) Finally, the author studies multilinear special identities which hold in all special Jordan algebras and do not hold in all Jordan algebras. He finds a natural correspondence between special identities for ordinary algebras and dialgebras.