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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 $$L(A^*)$$.

##### MSC:
 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.)