Second homology of generalized periplectic Lie superalgebras

Abstract

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...

Central extensions of generalized orthosymplectic Lie superalgebras

“…Proof. The proof is similar to [3,Proposition 5.7]. Recall that the Steinberg Lie superalgebra st m|2n (S) is the abstract Lie superalgebra generated by homogeneous elements e ij (a) of degree |i| + |j| + |a| for a ∈ R and 1 i = j m + 2n, subjecting to the relations:…”

“…Central extension theory is of vital importance in the study of Lie algebras and Lie superalgebras, which has attracted the attentions from mathematicians and physicist in recent decades (see [15] and [17] for a survey). In recent paper [3], the authors determined the universal central extensions of generalized periplectic Lie superalgebras and established a connection between the second homology group of a generalized periplectic Lie superalgebra and the dihedral group of its coordinate algebra with certain superinvolution. The current paper is devoted to study the universal central extensions of the generalized orthosymplectic Lie superalgebras coordinatized by unital associative superalgebras with superinvolution, which form another family of Lie superalgebras determined by superinvolutions.…”

“…Let k be a unital commutative ring with 2 invertible and R an associative k-superalgebra. We consider the associative superalgebra M m|2n (R) of (m + 2n) × (m + 2n)-matrices, in which the degree of the matrix unit e ij (a) is |e ij (a)| := |i| + |j| + |a| for a homogeneous element a ∈ R of degree |a| and 1 i, j m + 2n with the parity given by |i| := 0, if i m, 1, if i > m. (1.1) Analogous to the generalized periplectic Lie superalgebra discussed in [3], we further assume that the associative superalgebra R is equipped with a superinvolution, which is a k-linear map − : R → R preserving the Z/2Z-gradings and satisfying ab = (−1) |a||b|bā , andā = a, (1.2) for homogeneous a, b ∈ R. In this situation, the associative superalgebra M m|2n (R) possesses the generalized orthosymplectic superinvolution given by…”

“…Analogous to the generalized periplectic Lie superalgebra discussed in [3], we further assume that the associative superalgebra R is equipped with a superinvolution, which is a k-linear map − : R → R preserving the Z/2Z-gradings and satisfying ab = (−1) |a||b| bā, and ā = a, (…”

Remarks on the Second Homology Groups of Queer Lie Superalgebras

“…The universal central extensions of sl n (R), n 3 have been completely determined in [3], which has been further generalized to the Lie superalgebra sl m|n (R), m + n 3 in [4] and [7]. The authors addressed in [1] and [2] the case of an otho-symplectic Lie superalgebra and the case of a periplectic Lie superalgebra that are coordinatized by an associative superalgebra with a super anti-involution. These Lie superalgebras are Super analogy of unitary Lie algebra, thus it can help us to identify the second homology groups of these Lie superalgebras with the Z/2Z-graded version of the first dihedral and skew-dihedral homology group.…”