From Leibniz homology to cyclic homology

For an algebra R over a commutative ring k, a natural homomorphism phi(*):HL*+1 (R) --> HH* (R) from Leibniz to Hochschild homology is constructed that is induced by an antisymmetrization map on the chain level. The map phi(*) is surjective when R = gl(A), A an algebra over a characteristic zero field. If f : A --> B is an algebra homomorophism, the relative groups HL*(gl f)) are studied, where gl(f) :gl(A) --> gl(B) is the induced map on matrices. Letting HC* denote cyclic homology, if f is surjective with nilpotent kernel, there is a natural surjection HL*+1 (gl(f)) --> HC*(f) in the characteristic zero setting.