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We prove several positive results regarding representation of homotopy classes of spheres and algebraic groups by regular mappings. Most importantly we show that every mapping from a sphere to an orthogonal or a unitary group is homotopic to a regular one. Furthermore we prove that algebraic homotopy classes of spheres form a subgroup of the homotopy group, and that a similar result holds also for cohomotopy groups of arbitrary varieties. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating v and w , then there exists f ∈ S separating v and w. In this article we consider the action of G = SL 2 (C) × SL 2 (C) on the C -vector space M 2 , 2 n of n -tuples of 2 × 2 matrices by multiplication on the left and the right. Minimal generating sets S n of C [ M 2 , 2 n ] G are known, and | S n | = 1 24 (n 4 − 6 n 3 + 23 n 2 + 6 n). In recent work, Domokos [8] showed that for all n ≥ 1 , S n is a minimal separating set by inclusion, i.e. that no proper subset of S n is a separating set. This does not necessarily mean that S n has minimum cardinality among all separating sets for C [ M 2 , 2 n ] G. Our main result shows that any separating set for C [ M 2 , 2 n ] G has cardinality ≥ 5 n − 9. In particular, there is no separating set of size dim ⁡ (C [ M 2 n ] G) = 4 n − 6 for n ≥ 4. Further, S 4 has indeed minimum cardinality as a separating set, but for n ≥ 5 there may exist a smaller separating set than S n. We also consider the action of G = SL l (C) on M l , n by left multiplication. In that case the algebra of invariants has a minimum generating set of size ( n l ) (the l × l minors of a generic matrix) and dimension l n − l 2 + 1. We show that a separating set for C [ M l , n ] G must have size at least (2 l − 2) n − 2 (l 2 − l). In particular, C [ M l , n ] G does not contain a separating set of size dim ⁡ (C [ M l , n ] G) for l ≥ 3 and n ≥ l + 2. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem. [ABSTRACT FROM AUTHOR]

The category of linear algebraic groups admits non-surjective epimorphisms. For simple algebraic groups of rank 2 defined over algebraically closed fields, we show that the minimal dimension of a closed epimorphic subgroup is 3. [ABSTRACT FROM AUTHOR]

Let X be a real algebraic variety (resp. nonsingular real algebraic variety) and let Y be a homogeneous space for some linear real algebraic group. We prove that a continuous (resp. C ∞) map f : X → Y can be approximated by regular maps in the C 0 (resp. C ∞) topology if and only if it is homotopic to a regular map. Taking Y = S p , the unit p -dimensional sphere, we obtain solutions of several problems that have been open since the 1980's and which concern approximation of maps with values in the unit spheres. This has several consequences for approximation of maps between unit spheres. For example, we prove that for every positive integer n every C ∞ map from S n into S n can be approximated by regular maps in the C ∞ topology. Up to now such a result has only been known for five special values of n , namely, n = 1 , 2 , 3 , 4 or 7. [ABSTRACT FROM AUTHOR]

Let X and Y be nonsingular real algebraic varieties, dim ⁡ X ≥ dim ⁡ Y. Assume that the variety Y is malleable, compact and connected. Our main result implies that each regular map from X to Y is homotopic to a surjective regular map. The class of malleable varieties includes all homogeneous spaces for linear real algebraic groups. [ABSTRACT FROM AUTHOR]

We present some new analogs of Cassels–Tate dual exact sequence for connected reductive groups over global fields. We give also some extension of some important local-global exact sequences proved by Sansuc for connected linear algebraic groups over number fields, which are analogs of Cassels–Tate dual exact sequence, to the case of connected reductive groups over global function fields. As applications, we show that the Brauer-Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces under connected reductive groups over global function fields with connected reductive stabilizers are the only ones, extending some of Borovoi's results over number fields in this regard. [ABSTRACT FROM AUTHOR]

In this paper, we determine the modular invariants for some pseudo-reflection subgroups of the finite general linear group GL n (q) acting on the tensor product of the symmetric algebra S • (V) and the exterior algebra ∧ • (V) of the natural GL n (q) -module V. We are particularly interested in the case when the pseudo-reflection groups are subgroups of the parabolic subgroup of GL n (q) which are generalization of Weyl groups of Cartan type Lie algebra. [ABSTRACT FROM AUTHOR]

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field C (x) with derivation d d x and endomorphism f (x) ↦ f (x + 1). Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over C (x). [ABSTRACT FROM AUTHOR]