The theory of Lie algebras can be categorified starting from a new notion of ‘2-vector space’, which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2-morphisms. We define a ‘semistrict Lie 2-algebra ’ to be a 2-vector space L equipped with a skew-symmetric bilinear functor [·, ·]: L × L → L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the ‘Jacobiator’, which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2-algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang–Baxter equation. We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L∞-algebras in the sense of Stasheff. We also study strict and skeletal Lie 2-algebras, obtaining the former from strict Lie 2-groups and using the latter to classify Lie 2-algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finite-dimensional Lie algebra g a canonical 1-parameter family of Lie 2-algebras g � which reduces to g at � = 0. These are closely related to the 2-groups G � constructed in a companion paper.

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Abstract. A bilinear form f on a nonassociative algebra A is said to be invariant iff f(ab, c) = f(a, bc) for all a, b, c ∈ A. Finite-dimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal structure of A if f is nondegenerate and introduce the notion of T ∗-extension of an arbitrary algebra B (i.e. by its dual space B ∗ ) where the natural pairing gives rise to a nondegenerate invariant symmetric bilinear form on A: = B ⊕ B ∗. The T ∗-extension involves the third scalar cohomology H3 (B, K) if B is Lie and the second cyclic cohomology HC 2 (B) if B is associative in a natural way. Moreover, we show that every nilpotent finitedimensional algebra A over an algebraically closed field carrying a nondegenerate invariant symmetric bilinear form is a suitable T ∗-extension. As a Corollary, we prove that every complex Lie algebra carrying a nondegenerate invariant symmetric bilinear form is always a special type of Manin pair in the sense of Drinfel’d but not always isomorphic to a Manin triple. Examples involving the Heisenberg and filiform Lie algebras (whose third scalar cohomology is computed) are discussed. 1.

Abstract. Let Λ = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over Λ is given in [9]. This resolution is shown to have a “comultiplicative ” structure in [7], and this is used to find a minimal projective resolution P of Λ over the enveloping algebra Λ e. Using these results we show that the multiplication in the Hochschild cohomology ring of Λ relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of Λ with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of Λ is shown to be surjective onto the graded centre of the Koszul dual.

Let A be a unital commutative associative algebra over a field of characteristic zero, k be a Lie algebra, and z a vector space, considered as a trivial module of the Lie algebra g: = A⊗k. In this paper, we give a description of the cohomology space H 2 (g, z) in terms of well accessible data associated to A and k. We also discuss the topological situation, where A and k are locally convex algebras.