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The paper studies generic commutative and anticommutative algebras of a fixed dimension, their invariants, covariants and algebraic properties (e.g., the structure of subalgebras).


We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants.


Using related sequences of Lucas numbers, other 3manifolds are constructed, their geometric structures determined, and a curious relationship between the homology and the invariant tracefield examined.


Multivalued groups, their representations and Hopf algebras


This paper introduces the concept ofnvalued groups and studies their algebraic and topological properties.


Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms.


We also study the structure of the exceptional?graded transitive Lie superalgebras and give their geometric realization.


A characterization of linearly reductive groups by their invariants


Results of this kind were first obtained by Moore and Seiberg, but their paper contains serious gaps.


We study their general properties and apply these results to Schubert varieties.


These are analogous to "fusion rules" in tensor product decomposition and their derivation obtains from an analysis of theRmatrix.


Secon, we are interested in dominant polynomial mapsF:?n→?n1 whose connected components of their generic fibers are contractible.


Quantum Symmetric Pairs and Their Zonal Spherical Functions


We investigate the sets C(S) of such x for various orbits S and their relations with each other.


Finally, we study their reducibility of the action of the Casimirs on the zeroweight spaces of selfdual gmodules and obtain complete classification results for g = sln and g2.


Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k.


Here we show that this statement remains true for extensions of finite complex reflection groups by elements in their normalizer.


Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations


We consider free affine actions of unipotent complex algebraic groups on Cn and prove that such actions admit an analytic geometric quotient if their degree is at most 2.


A notion of a mixed representation of a quiver can be derived from ordinary quiver representation by considering the dual action of groups on "vertex" vector spaces together with their usual action.

