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In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n).
In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).
We produce a rigid triple of classes in the algebraic group G(2) in characteristic 5, and use it to show that the finite groups G(2)(5(n)) are not (2, 5, 5)-generated.
De Gruyter2011
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For G a simple algebraic group over an algebraically closed field of characteristic 0, we determine the irreducible representations ρ:G→I(V), where I(V) denotes one of the classical groups SL(V), Sp(V
2015
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We continue our work, started in [9], on the program of classifying triples (X, Y, V), where X, Yare simple algebraic groups over an algebraically closed field of characteristic zero with X < Y, and V