NN-point Virasoro algebras are multi-point Krichever--Novikov type algebras

We show how the recently again discussed $N$-point Witt, Virasoro, and affine Lie algebras are genus zero examples of the multi-point versions of Krichever--Novikov type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. The concept of almost-grading will yield information about triangular decompositions which are of importance in the theory of representations. As examples the algebra of functions, vector fields, differential operators, current algebras, affine Lie algebras, Lie superalgebras and their central extensions are studied. Very detailed calculations for the three-point case are given.Comment: 46 page

Universal geometric cluster algebras

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or reals. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B with universal geometric coefficients, or the universal geometric cluster algebra over B. Constructing universal coefficients is equivalent to finding an R-basis for B (a "mutation-linear" analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan F_B, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between F_B and g-vectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.Comment: Final version to appear in Math. Z. 49 pages, 5 figure

On geometrically equivalent S-acts

In this paper, considering the geometric equivalence for algebras of a variety $_{S}A$ of S-acts over a monoid S, we obtain representation theorems describing all types of the equivalence classes of geometrically equivalent S-acts of varieties $_{S}A$ over groups S.Comment: 13 pages, some statements were corrected and improve

Lie algebras in symmetric monoidal categories

We study algebras defined by identities in symmetric monoidal categories. Our focus is on Lie algebras. Besides usual Lie algebras, there are examples appearing in the study of knot invariants and Rozansky-Witten invariants. Our main result is a proof of Westbury's conjecture for K3-surface: there exists a Lie algebra homomorphism from Vogel's universal simple Lie algebra to the Lie algebra describing the Rozansky-Witten invariants of a K3-surface. Most of the paper involves setting up a proper language to discuss the problem and we formulate nine open questions as we proceed.Comment: Minor corrections all around. Now it has an answer to Question 7, due to J. Sawo