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The Kantor-Koecher-Tits construction for Jordan algebras. (English. Russian original) Zbl 0906.17026
Algebra Logic 35, No. 2, 96-104 (1996); translation from Algebra Logika 35, No. 2, 173-189 (1996).
By the Kantor-Koecher-Tits construction there is a Lie algebra \({\mathcal L}(B)\) associated to every Jordan algebra \(B\). The author shows that such a construction also exists for Jordan coalgebras. The key notion is a weak inner derivative of the dual algebra \(A^*\). If \(A\) is a Jordan coalgebra and \({\mathcal L}(A^*)\) the Lie algebra associated to the dual algebra, then (Theorem 1) the vector space \(L(A)\) dual to \({\mathcal L}(A^*)\) admits a comultiplication \(\Delta_L\) so that the associated coalgebra \(\langle L(A),\Delta_L\rangle\) is a Lie coalgebra with dual algebra \(L(A^*)\).

17C50 Jordan structures associated with other structures
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)