Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities a(bc) = a(cb), (ba)a 2 = (ba 2 )a, (b, a 2 , c) = 2(b, a, c)a.These identities are satisfied by the product ab = a ⊣ b + b ⊢ a in an associative dialgebra. We use computer algebra to show that every identity for this product in degree ≤ 7 is a consequence of the three identities in degree ≤ 4, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.where (a, b, c) = (ab)c − a(bc). The multilinear forms of the last two identities are J = (a(bc))d + (a(bd))c + (a(cd))b − (ab)(cd) − (ac)(bd) − (ad)(bc),Remark 9. Equivalent identities for the opposite product appear in the work of Kolesnikov. If we replace xy by yx in equations (26) and (27) of [17], replace x 1 , x 2 , x 3 , x 4 by a, b, c, d and apply right commutativity, then we obtain One easily verifies thatSee also the remarks on Jordan dialgebras in Pozhidaev [25], Section 3. Lemma 10. (Velásquez and Felipe [31]) The quasi-Jordan product in an associative dialgebra satisfies right commutativity and the quasi-Jordan identity.Lemma 11. (Bremner [3]) The quasi-Jordan product in an associative dialgebra satisfies the associator-derivation identity. The identities of Definition 8 imply every identity of degree ≤ 4 for the quasi-Jordan product in an associative dialgebra.Definition 12. A quasi-Jordan algebra is a nonassociative algebra over a field of characteristic = 2, 3 satisfying right commutativity and the quasi-Jordan identity.A semispecial quasi-Jordan algebra (also called a Jordan dialgebra) is a quasi-Jordan algebra satisfying the associator-derivation identity.Remark 13. Strictly speaking, a Jordan dialgebra as defined by Kolesnikov [17] and Pozhidaev [25] has two binary operations ⊣ and ⊢ which are in fact opposite as a result of the dialgebra version of commutativity, x ⊣ y = y ⊢ x. We simplify the notation by using only one operation and writing this operation as juxtaposition.Definition 14. A quasi-Jordan algebra is special if it is isomorphic to a subalgebra of D + for some associative dialgebra D. Every special algebra is semispecial.Glennie [7,8,9] (see also Hentzel [12]) discovered an identity satisfied by special Jordan algebras that is not satisfied by all Jordan algebras. In this paper we consider the analogous question for quasi-Jordan algebras. We use computer algebra to show that the identities in Definition 8 imply every identity of degree ≤ 7 for the quasi-Jordan product in an associative dialgebra. We demonstrate the existence of identities in degree 8 which do not follow from the identities of Definition 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity. These new identities are special identities in the following sense.

This paper classi es all identities of degree 7 satis ed by the ternary commutator in an associative ternary algebra. (Seven is the lowest degree for which non-trivial identities exist.) These identities are ternary generalizations of the Jacobi identity for Lie algebras.

Let 𝔤 be a simple complex (finite dimensional) Lie algebra, and let R be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed. Lie algebras of the form 𝔤⊗CR are considered; these generalize Kac–Moody loop algebras since for a curve of genus zero with two punctures R≂C[t,t−1]. The universal central extension of 𝔤⊗R is analogous to an untwisted affine Kac–Moody algebra. By Kassel’s theorem the kernel of the universal central extension is linearly isomorphic to the Kähler differentials of R modulo exact differentials. The dimension of the kernel for any R is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined. Restricting further to an elliptic curve with punctures at two points (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal central extension. The results involve Pollaczek polynomials, which are a genus-one generalization of ultraspherical (Gegenbauer) polynomials. © 1994 American Institute of Physics.

Abstract. We consider Lie algebras of the form g ® R where g is a simple complex Lie algebra and R = C[s, s~l, (s -1)_1, (s -a)-1] for a £ C -{0, 1} . After showing that R is isomorphic to a quadratic extension of the ring C[t, t~[] of Laurent polynomials, we prove that g ® R is a quasi-graded Lie algebra with a triangular decomposition. We determine the universal central extension of 0 ® R and show that the cocycles defining it are closely related to ultraspherical (Gegenbauer) polynomials.

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