Atabey Kaygun

We construct a model categorical equivalence between the category of simplicial vector spaces and the category of representations of a crossed simplicial group \Delta G when each G_{n} is finite and the characteristic of the ground field is 0 .

This paper develops and tests a unified quantitative framework for designing electricity procurement portfolios in two major European power markets, EPEX and Nord Pool. We model day-ahead prices across multiple market areas and formulate procurement as a convex mean–variance portfolio problem in which a procurement plan corresponds to a vector of weights constrained by explicit feasibility sets. We study four feasibility regimes—ranging from long-only (simplex) allocations to bounded long–short portfolios and fully unconstrained short-allowed allocations—in order to isolate how implementability constraints reshape the efficient frontier. We then validate the deterministic quadratic solutions with a scenario-based Monte Carlo layer calibrated to match the empirical first and second moments, and we overlay minimum-variance futures hedges using the Nord Pool system price benchmark and German futures for the continental EPEX region.The evidence yields three main findings. First, the constraint regime induces a systematic cost–risk trade-off: greater allocation flexibility typically lowers variance, but it can raise expected procurement cost, whereas tighter feasibility sets often reduce expected cost while leaving higher residual risk. Second, Monte Carlo simulations reproduce the analytical mean–variance outputs closely, which confirms numerical stability and shows that the quadratic approximation captures the dominant drivers of realized procurement-cost dispersion in this application. Third, market structure governs hedge effectiveness: Nord Pool hedges deliver substantial variance reductions at economically meaningful hedge ratios with only moderate cost increases, while EPEX hedges deliver only marginal variance relief, consistent with higher basis risk and weaker spot–future alignment at the portfolio level.

We provide a unified geometric realization of the classical deformation complexes. We construct GL-equivariant bilinear incidence varieties whose diagonal slices recover the varieties of associative, commutative, Leibniz, and Lie algebra structures on a finite-dimensional vector space. We prove that the fiber of the incidence map at a given algebra law is canonically isomorphic to the space of 2-cocycles in the corresponding cohomology theory (Hochschild, Harrison, Leibniz, or Chevalley--Eilenberg). Furthermore, we introduce invariant bilinear forms to define open strata of separable and semisimple algebras, and demonstrate that these strata consist of open GL-orbits, establishing the rigidity of generic points in the coarse moduli spaces for all four geometries.

We calculate suitably localized Hochschild homologies of various quantum groups and Podleś spheres after realizing them as generalized Weyl algebras (GWAs). We use the fact that every GWA is birationally equivalent to a smash product with a 1-torus. We also address and solve the birational equivalence problem, and the birational smoothness problem for GWAs.

Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est isomorphism to ones between the Hopf-cyclic (co)homologies of quantized algebras of functions and quantized universal enveloping algebras, both in $h$-adic and $q$-deformation frameworks.

We show that every Kac-Moody algebra is birationally equivalent to a smash biproduct of two copies of a Weyl algebra together with a polynomial algebra. We also show that the same is true for quantized Kac-Moody algebras where one replaces Weyl algebras with their quantum analogues.