Algebraic groups

Inner form

In mathematics, an inner form of an algebraic group over a field is another algebraic group such that there exists an isomorphism between and defined over (this means that is a -form of ) and in addition, for every Galois automorphism the automorphism is an inner automorphism of (i.e. conjugation by an element of ). Through the correspondence between -forms and the Galois cohomology this means that is associated to an element of the subset where is the subgroup of inner automorphisms of . Being inner forms of each other is an equivalence relation on the set of -forms of a given algebraic group. A form which is not inner is called an outer form. In practice, to check whether a group is an inner or outer form one looks at the action of the Galois group on the Dynkin diagram of (induced by its action on , which preserves any torus and hence acts on the roots). Two groups are inner forms of each other if and only if the actions they define are the same. For example, the -forms of are itself and the unitary groups and . The latter two are outer forms of , and they are inner forms of each other. (Wikipedia).

Video thumbnail

Inner products (video 3): Definition

Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning

From playlist Inner Products

Video thumbnail

Inner Products (video 4): Lengths and Distances, Part 1/2

Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning

From playlist Inner Products

Video thumbnail

Inner Products (video 7): Unconventional Inner Products

Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning

From playlist Inner Products

Video thumbnail

Inner products (video 8): Outro

Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning

From playlist Inner Products

Video thumbnail

How to find the composition of two areas by subtracting their areas

👉 Learn how to find the area and perimeter of composite shapes. A composite shape is a shape that is composed of different shapes. The area of a shape is the measure of the portion enclosed by the shape while the perimeter of a shape is the measure of the outline enclosing the shape. To f

From playlist Area and Perimeter

Video thumbnail

Interior and Exterior Angles

"Interior and exterior angles of regular and irregular polygons."

From playlist Shape: Angles

Video thumbnail

How to find the area of a figure using multiple area's

👉 Learn how to find the area and perimeter of composite shapes. A composite shape is a shape that is composed of different shapes. The area of a shape is the measure of the portion enclosed by the shape while the perimeter of a shape is the measure of the outline enclosing the shape. To f

From playlist Area and Perimeter

Video thumbnail

Learn how to find the composition area of a figure

👉 Learn how to find the area and perimeter of composite shapes. A composite shape is a shape that is composed of different shapes. The area of a shape is the measure of the portion enclosed by the shape while the perimeter of a shape is the measure of the outline enclosing the shape. To f

From playlist Area and Perimeter

Video thumbnail

Differential Forms | The Minkowski metric and the Hodge operator.

We explore the lifting of the Minkowski inner product to the space of 2 and 3 forms. Then we look at what effect this has on the corresponding Hodge operator. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-ma

From playlist Differential Forms

Video thumbnail

What is the different formulas for interior angles of a polygon

👉 Learn about the interior and the exterior angles of a polygon. A polygon is a plane shape bounded by a finite chain of straight lines. The interior angle of a polygon is the angle between two sides of the polygon. The sum of the interior angles of a regular polygon is given by the formul

From playlist Interior and Exterior Angles of Polygons

Video thumbnail

Differential Forms | The Hodge operator via an inner product.

We describe how to define a more generalized Hodge operator via an inner product of m-forms. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www.michael-penn.net Randolph College

From playlist Differential Forms

Video thumbnail

Lecture 7: Integration (Discrete Differential Geometry)

Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg

From playlist Discrete Differential Geometry - CMU 15-458/858

Video thumbnail

What is a Tensor? Lesson 23: Operations on p-forms. The Exterior Algebra.

What is a Tensor? Lesson 23: Operations on p-forms. The Exterior Algebra.

From playlist What is a Tensor?

Video thumbnail

Lie Groups and Lie Algebras: Lesson 10: The Classical Groups part VIII

Lie Groups and Lie Algebras: Lesson 10: The Classical Groups part VIII In this lecture we demonstrate the canonical form of a bilinear symmetric metric. This will help us appreciate that all of the most important types of metrics can be represented by matrices of a specific "canonical" ty

From playlist Lie Groups and Lie Algebras

Video thumbnail

General Inner Products in ℝⁿ. Matrix Representation

https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep

From playlist Part 4 Linear Algebra: Inner Products

Video thumbnail

Duality in Linear Algebra: Dual Spaces, Dual Maps, and All That

An exploration of duality in linear algebra, including dual spaces, dual maps, and dual bases, with connections to linear and bilinear forms, adjoints in real and complex inner product spaces, covariance and contravariance, and matrix rank. More videos on linear algebra: https://youtube.c

From playlist Summer of Math Exposition Youtube Videos

Video thumbnail

Lecture 15 - Kernel Methods

Kernel Methods - Extending SVM to infinite-dimensional spaces using the kernel trick, and to non-separable data using soft margins. Lecture 15 of 18 of Caltech's Machine Learning Course - CS 156 by Professor Yaser Abu-Mostafa. View course materials in iTunes U Course App - https://itunes.a

From playlist Machine Learning Course - CS 156

Video thumbnail

How to determine the sum of interior angles for any polygon

👉 Learn about the interior and the exterior angles of a polygon. A polygon is a plane shape bounded by a finite chain of straight lines. The interior angle of a polygon is the angle between two sides of the polygon. The sum of the interior angles of a regular polygon is given by the formul

From playlist Interior and Exterior Angles of Polygons

Video thumbnail

Lecture 2 | Modern Physics: Quantum Mechanics (Stanford)

Lecture 2 of Leonard Susskind's Modern Physics course concentrating on Quantum Mechanics. Recorded January 21, 2008 at Stanford University. This Stanford Continuing Studies course is the second of a six-quarter sequence of classes exploring the essential theoretical foundations of mode

From playlist Quantum Mechanics Prof. Susskind & Feynman

Related pages

Galois cohomology | Dynkin diagram | Algebraic group | Unitary group | Inner automorphism