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 construction In the case of 4-dimensional anticommutative algebras a construction is given that links the associated cubic surface and the 27 lines on it with the structure of subalgebras of the algebra. The aim of this paper is to discuss a construction of a class of linear isomorphisms σ:S(g)→U(g) which commute with the adjoint representation. However, there are many examples that do not arise from this construction. Using the properties ofn-Hopf algebras we show that certain spaces do not admit the structure of ann-valued group and that certain commutativen-valued groups do not arise by applying then-coset construction to any commutative group. We give complete proofs of the K-theoretic construction of the quantized enveloping algebra of affine gl(n) sketched in [GV]. The first part of this paper describes the construction of pseudo-Riemannian homogeneous spaces with special curvature properties such as Einstein spaces, using corresponding known compact Riemannian ones. This construction is based on the notion of a certain duality between compact and non-compact homogeneous spaces. The main goal of this paper is to show that this construction produces many new Gelfand pairs associated with nilpotent Lie groups. Geometric construction of the global base of the quantum modified algebra of A geometric construction of the modified quantum algebra ofgln was given in [BLM]. It was then observed independentely by Lusztig and Ginzburg-Vasserot (see [L1], [GV]) that this construction admits an affine analogue in terms of periodic flags of lattices. Such central extension gives rise to a goup scheme that leads to a geometric construction of unrestricted representations. We generalize to the case of a symmetric variety the construction of the enveloping semigroup of a semisimple algebraic group due to E. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the R-matrix construction of quantum immanants. A Universal Construction for Moduli Spaces of Decorated Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature. We also establish a connection between the composition of the functors, and the "centralizer construction" of the Yangian ${\rm Y}(\mathfrak{gl}_n)$ discovered by G. This includes the polynomiality of the invariant subalgebra of the symmetric algebra of a (truncated) parabolic subalgebra, the existence of a slice to the regular coadjoint orbits and the construction of maximal Poisson Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?

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