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Higherdimensional algebra VI: Lie 2algebras,
, 2004
"... The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We ..."
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Cited by 44 (12 self)
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The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We define a ‘semistrict Lie 2algebra ’ to be a 2vector space L equipped with a skewsymmetric 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 2algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang–Baxter equation. We construct a 2category of semistrict Lie 2algebras and prove that it is 2equivalent to the 2category of 2term L∞algebras in the sense of Stasheff. We also study strict and skeletal Lie 2algebras, obtaining the former from strict Lie 2groups and using the latter to classify Lie 2algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finitedimensional Lie algebra g a canonical 1parameter family of Lie 2algebras g � which reduces to g at � = 0. These are closely related to the 2groups G � constructed in a companion paper.
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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Nondegenerate invariant bilinear forms on non–associative algebras, Preprint Freiburg THEP 92/3, to appear
 Acta Math. Univ. Comenianae
"... 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. Finitedimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal st ..."
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Cited by 15 (0 self)
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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. Finitedimensional 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.
Multiplicative structures for Koszul algebras
, 2005
"... 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 ..."
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Cited by 7 (5 self)
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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.
Faculté des Sciences et Techniques
, 2008
"... 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 associ ..."
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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.