Some results on the quotient space by an algebraic group of automorphisms

1963 ◽  
Vol 149 (4) ◽  
pp. 286-301 ◽  
Author(s):  
C. S. Seshadri
Keyword(s):  
Algebraic Group ◽  
Quotient Space ◽  
Group Of Automorphisms

scholarly journals Algebraic singularities have maximal reductive automorphism groups

1989 ◽  
Vol 113 ◽  
pp. 181-186 ◽  
Author(s):  
Herwig Hauser ◽  
Gerd Müller
Keyword(s):  
Algebraic Group ◽  
Maximal Ideal ◽  
Analytic Functions ◽  
Automorphism Groups ◽  
Algebraic Subgroup ◽  
Group Of Automorphisms ◽  
Analytic Singularity ◽  
Cotangent Space ◽  
Algebraic Singularities

Let X = On/i be an analytic singularity where ṫ is an ideal of the C-algebra On of germs of analytic functions on (Cn, 0). Let denote the maximal ideal of X and A = Aut X its group of automorphisms. An abstract subgroup equipped with the structure of an algebraic group is called algebraic subgroup of A if the natural representations of G on all “higher cotangent spaces” are rational. Let π be the representation of A on the first cotangent space and A1 = π(A).

scholarly journals Squares and associative representations of two-dimensional evolution algebras

2020 ◽  
pp. 2150090 ◽  
Author(s):  
Maria Inez Cardoso Gonçalves ◽  
Daniel Gonçalves ◽  
Dolores Martín Barquero ◽  
Cándido Martín González ◽  
Mercedes Siles Molina
Keyword(s):  
Algebraic Group ◽  
Associative Algebra ◽  
Geometric Object ◽  
Two Dimensional ◽  
Group Of Automorphisms ◽  
Algebraic Invariant ◽  
Evolution Algebra

We associate a square to any two-dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behavior of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic group, getting in this form a new algebraic invariant. The study of associative representations of evolution algebras is also started and we get faithful representations for most two-dimensional evolution algebras. In some cases, we prove that faithful commutative and associative representations do not exist, giving rise to the class of what could be termed as “exceptional” evolution algebras (in the sense of not admitting a monomorphism to an associative algebra with deformed product).

Additive Groups in a Non-Symbolic ‘Artificial Algebra’

2020 ◽  
Author(s):  
Randolph C Grace ◽  
Nicola J. Morton ◽  
Matt Grice ◽  
Kate Stuart ◽  
Simon Kemp
Keyword(s):  
Algebraic Group ◽  
Explicit Instruction ◽  
Strong Support ◽  
Line Length

Grace et al. (2018) developed an ‘artificial algebra’ task in which participants learn to make an analogue response based on a combination of non-symbolic magnitudes by feedback and without explicit instruction. Here we tested if participants could learn to add stimulus magnitudes in this task in accord with the properties of an algebraic group. Three pairs of experiments tested the group properties of commutativity (Experiments 1a-b), identity and inverse existence (Experiments 2a-b) and associativity (Experiments 3a-b), with both line length and brightness modalities. Transfer designs were used in which participants responded on trials with feedback based on sums of magnitudes and later were tested with novel stimulus configurations. In all experiments, correlations of average responses with magnitude sums were high on trials with feedback, r = .97 and .96 for Experiments 1a-b, r = .97 and .96 for Experiments 2a-b, and ranged between r = .97 and .99 for Experiment 3a and between r = .82 and .95 for Experiment 3b. Responding on transfer trials was accurate and provided strong support for commutativity, identity and inverse existence, and associativity with line length, and for commutativity and identity and inverse existence with brightness. Deviations from associativity in Experiment 3b suggested that participants were averaging rather than adding brightness magnitudes. Our results confirm that the artificial algebra task can be used to study implicit computation and suggest that representations of magnitudes may have a structure similar to an algebraic group.

scholarly journals MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

2021 ◽  
Author(s):  
LUCAS FRESSE ◽  
IVAN PENKOV
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Interesting Case ◽  
Analogous Problem ◽  
Restrictive Condition ◽  
Dimensional Case ◽  
Essential Difference ◽  
Parabolic Subgroups ◽  
Finite Dimensional ◽  
Finite Dimensional Case

AbstractLet G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

scholarly journals Complete reducibility: variations on a theme of Serre

2021 ◽  
Author(s):  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Spherical Building ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
General Properties ◽  
Complete Reducibility ◽  
Special Cases ◽  
Variations On A Theme

AbstractIn this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

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