Abstract. In this paper we define pre-Malcev algebras and alternative quadrialgebras and prove that they generalize pre-Lie algebras and quadri-algebras respectively to the alternative setting. We use the results and techniques from [4,14] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota-Baxter operators.

In this paper we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with triality. This allows us to give a new construction of the universal enveloping algebras of Malcev algebras. Our work relies on the approach of Grishkov and Zavarnitsine to groups with triality.

We compute minimal sets of generators for the S n -modules (n ≤ 4) of multilinear polynomial identities of arity n satisfied by the Jordan product and the Jordan diproduct (resp. pre-Jordan product) in every triassociative (resp. tridendriform) algebra. These identities define Jordan trialgebras and post-Jordan algebras: Jordan analogues of the Lie trialgebras and post-Lie algebras introduced by Dotsenko et al., Pei et al., Vallette & Loday. We include an extensive review of analogous structures existing in the literature, and their interrelations, in order to identify the gaps filled by our two new varieties of algebras. We use computer algebra (linear algebra over finite fields, representation theory of symmetric groups), to verify in both cases that every polynomial identity of arity ≤ 6 is a consequence of those of arity ≤ 4. We conjecture that in both cases the next independent identities have arity 8, imitating the Glennie identities for Jordan algebras. We formulate our results as a commutative square of operad morphisms, which leads to the conjecture that the squares in a much more general class are also commutative. ASSOCIATIVE ALGEBRASSelf-dual Operation ab Relation (ab)c ≡ a(bc) Lie bracket [a, b] = ab − ba symmetry ab ≡ ba, relation (ab)c ≡ a(bc) Lie algebras, equation (1) Commutative associative algebras Jordan product a • b = ab + ba Jordan algebras, equation (2) No dual, operad is cubic not quadratic DIASSOCIATIVE ALGEBRAS DENDRIFORM ALGEBRAS Operations a b, a b, Definition 2.9 Operations a ≺ b, a b, Definition 2.9 Leibniz bracket {a, b} = a b − b a Leibniz algebras, Definition 2.12 Zinbiel algebras, Definition 2.12 Pre-Lie product {a, b} = a ≺ b − b a Perm algebras, Definition 2.16 Pre-Lie algebras, Definition 2.16 Jordan diproduct a • b = a b + b a Jordan dialgebras, Definition 2.18 No dual, operad is cubic not quadratic Pre-Jordan product a • b = a ≺ b + b a No dual, operad is cubic not quadratic Pre-Jordan algebras, Definition 2.18 TRIASSOCIATIVE ALGEBRAS TRIDENDRIFORM ALGEBRAS Operations a b, a ⊥ b, a b Operations a ≺ b, a b, a b Definition 3.1 Definition 3.1 Lie bracket [a, b] = a ⊥ b − b ⊥ a Leibniz bracket {a, b} = a b − b a Commutative tridendriform algebras, Lie trialgebras, Definition 3.4 Definition 3.4 Lie bracket [a, b] = a b − b a Commutative triassociative algebras, Pre-Lie product {a, b} = a ≺ b − b a Definition 3.8 Post-Lie algebras, Definition 3.8 Jordan product a • b = a ⊥ b + b ⊥ a Jordan diproduct a • b = a b + b a Jordan trialgebras, Section 4 No dual, operad is cubic not quadratic Jordan product a • b = a b + b a Pre-Jordan product a • b = a ≺ b + b a No dual, operad is cubic not quadratic Post-Jordan algebras, Section 6

Pre-Jordan algebras were introduced recently in analogy with pre-Lie algebras. A pre-Jordan algebra is a vector space A with a bilinear multiplication x · y such that the product x • y = x · y + y · x endows A with the structure of a Jordan algebra, and the left multiplications L·(x)

On the set Hn(K) of symmetric n × n matrices over the field K we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these nonassociative structures have the orthogonal Lie algebra so(n, K) as derivation algebra. This gives an embedding so(n, K) ⊂ so(N, K) for N = n+1 2 − 1. We obtain a sequence of reductive pairs (so(N, K), so(n, K)) that provides a family of irreducible Lie-Yamaguti algebras. In this paper we explain in detail the construction of these Lie-Yamaguti algebras. In the cases n ≤ 4, we use computer algebra to determine the polynomial identities of degree ≤ 6; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product.