An unprecedented asymmetric acyl-carbamoylation of pendant alkenes tethered on aryl carbamic chlorides with both aliphatic and aromatic aldehydes has been developed via the cooperative catalysis of a chiral nickel-PHOX complex and tetrabutylammonium decatungstate. This reaction represents the first example of merging hydrogen-atom-transfer photochemistry and asymmetric transition metal catalysis in difunctionalization of alkenes. Using this protocol, a variety of oxindoles bearing a challenging quaternary stereogenic center are furnished under mild conditions in highly enantioselective manner.

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In this protocol aryl halides and α-bromo acetates are efficiently cross-coupled with an array of (hetero)­aromatic and aliphatic aldehydes under the cooperative catalysis of nickel and tetrabutylammonium decatungstate as a hydrogen-atom-transfer photocatalyst. This method provides a concise approach to a variety of ketones with high compatibility of various functional groups.

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Let (R, − ) be an arbitrary unital associative superalgebra with superinvolution over a commutative ring k with 2 invertible. The second homology of the generalized periplectic Lie superalgebra pm(R, − ) for m 3 has been completely determined via an explicit construction of its universal central extension. In particular, this second homology is identified with the first Z/2Z-graded dihedral homology of R with certain superinvolution whenever m 5.The super analogue of C. Kassel and J. L. Loday's work was obtained in [3,4]. The isomorphism between the second homology of the Lie superalgebra sl m|n (S) coordinated by a unital associative superalgebra S with m + n 5 and the first Z/2Z-graded cyclic homology HC 1 (S) was established. Recent investigation [2] further gave the identification between the second homology of the ortho-symplectic Lie superalgebra osp m|2n (R, − ) and the first Z/2Z-graded skew-dihedral homology of (R, − ) for (m, n) = (1, 1) or (2, 1), where (R, − ) is a unital associative superalgebra with superinvolution (see (2.1) for the definition). A series of deep investigations on the relationship between the homology theory of Lie algebras and the homology theory of associative algebras have been made in [12,13].Inspired by the above developments, we aim to establish an isomorphism that is analogous to C. Kassel and J. L. Loday's isomorphism for the generalized periplectic Lie superalgebra p m (R, − ) coordinatized by a unital associative superalgebra (R, − ) with superinvolution. As in Section 2, a generalized periplectic Lie superalgebra is defined as the derived sub-superalgebra of the Lie superalgebra of all skew-symmetric matrices with respect to the so-called periplectic superinvolution. It is a super analogue of a unitary Lie algebra introduced in [1]. This family of Lie superalgebras provides us with a realization of an arbitrary generalized root graded Lie superalgebra of type P (m − 1) for m = 4 up to central isogeny (cf.[5]), which is a complement to the realization of a root graded Lie superalgebra of type P (m − 1) given in [14].A primary result of this paper is Theorem 5.5 which states that the second homology of the Lie superalgebra p m (R, − ) with m 5 is isomorphic to the first Z/2Z-graded dihedral homology of (R, − • ρ), where − • ρ is the superinvolution on R obtained by twisting the superinvolution − with the sign map ρ (see (2.2) in Section 2). In the special case where R is super-commutative, the isomorphism indicates that the second homology of p m (k) ⊗ k R for a super-commutative superalgebra R is trivial, which was obtained by K. Iohara and Y. Koga in [7,8]. While the isomorphism also reveals that the second homology of p m (R, − ) is not necessarily trivial if R is not super-commutative.The methods used in this paper unsurprisingly involve an explicit construction of the universal central extension of p m (R, − ), which will be achieved via introducing the notion of the Steinberg periplectic Lie superalgebra stp m (R, − ) in Section 3.The isomorphism between the second ho...