DEV Community: The Mansions of Science

Definition of the Cartan Subalgebra

The Mansions of Science — Tue, 16 Jun 2026 09:55:22 +0000

Given an 8 dimensional Lie algebra with generators A B C D E F G H. If it is the case that A B C D all commute with each other, and if you add any of E F G H, you end up breaking commutativity, then the collection A B C D is called a Cartan subalgebra and it has rank 4. For a given Lie group, even this one, it can be proven that all Cartan subalgebras have the same rank. For example, another Cartan subalgebra could very well be E F G H, as long as these 4 commute and adding any of A B C D would break commutativity.

You can say this: The Cartan subalgebra picks out the largest set of directions where flows don’t interfere with each other.

Lie Algebra = Generators + Commutation Relations. That’s it.

The Mansions of Science — Tue, 16 Jun 2026 04:52:37 +0000

So imagine I have a Lie group G. By definition, a Lie group has infinite elements. Out of these infinite number of elements, there are 10 independent generators. Meaning, with these 10 elements, I can "get to" every other element of G. The set of these 10 elements, along with their commutation relations, [X,Y] = XY - YX, where X and Y are two of the generators, form the Lie algebra of this Lie group G. We say that the Lie algebra is 10 dimensional. For example, SO(3) is the group of all rotations in 3D space. It has 3 generators (infinitesimal rotations). So the Lie algebra of the Lie group SO(3) is a 3 dimensional algebra.

We also say that the tangent space at the identity element e is 10 dimensional. There are 10 different independent directions you can “go in”.

Bilinear Form = Function of 2 Vectors (That's it)

The Mansions of Science — Tue, 16 Jun 2026 04:48:44 +0000

Bilinear Form Confusion:

So, bilinear forms use to be confusing to me because I didn’t know what either bilinear meant or what forms meant. But, a form is, very often, just a function. Map is much more common among mathematicians but they’re just functions. They take in inputs and give you outputs. The same functions we’ve been using since middle school.

Now, they take in two vectors and give you back a number. That’s it. The most famous example is the dot product, which even high schoolers know. You take in 2 vectors and you output a number. That’s it.

Bilinear forms are a function that take in 2 vectors and give you back a number.

They have the word “linear” in them because if you plug in (a+b,c), it’s the same as plugging in (a,c) + (b,c). Same for (a,b+c).

If you have a symmetric bilinear form it means that (a,b) = (b,a). Skew-symmetric means what you think antisymmetric means (a,b) = -(b,a). So, like cross products.

Next, I’ll be using bilinear forms to define the symplectic group.

Skyscraper Sheaf = Dirac Delta Function

The Mansions of Science — Mon, 15 Jun 2026 01:17:48 +0000

You have a topological space, T, and a point in it, X.

The Skyscraper Sheaf is literally a function F that takes in a subset of T as input and if the subset contains X, you get a nonzero output. If the subset does not contain X, you get 0.

That's it.

The Picard Group and Picard Variety

The Mansions of Science — Mon, 15 Jun 2026 01:07:23 +0000

Given a topological space, T, you can define the Picard Group, Pic(T). The Picard Variety is also sometimes Pic(T).

You can build a fiber bundle by attaching 1D vector spaces at every point on T. You would get a line bundle.

Some of the line bundles will be isomorphic to each other. The set of all isomorphism classes of these line bundles forms the Picard Group. The group operation is the tensor product.

If you parametrize these isomorphism classes (13 params, 18 param, or 389213912, doesn't matter) into, say, 13 parameters, and somehow plotted them, you would get a 13D object. This is the Picard Variety.

Variety vs Manifold (2 Major Differences)

The Mansions of Science — Sun, 14 Jun 2026 20:20:10 +0000

There are 2 major differences between a variety and a manifold.

A manifold looks like R^n if you keep zooming in. A variety can have corners, cusps, crossings (X marks), etc. You cannot zoom in and get rid of them. If a variety does not have them we call it a smooth variety. All smooth varieties are manifolds.
An entire variety is described by a single polynomial. This can sometimes apply to a manifold, but often not the case. Maybe only some patches can be described by polynomials. But, in any case, often an entire manifold cannot be described by a single polynomial.

The Moduli Space of Elliptic Curves

The Mansions of Science — Sun, 14 Jun 2026 18:55:10 +0000

The Moduli Space of Elliptic Curves is parametrized by a single complex number, the j-invariant. There is a space, and every point in it has a single complex number called the j-invariant. Every point represents a family (infinite members) of elliptic curves that have the same j-invariant. You might notice that this moduli space must necessarily be isomorphic to the complex plane C.

Elliptic curves are one of the most studied objects in modern mathematics. They are famously related to modular forms, were used to prove Fermat's Last Theorem and are part of the Langlands Program.

What Is A Moduli Space?

The Mansions of Science — Sun, 14 Jun 2026 00:42:16 +0000

Moduli spaces study the structure of "families" of objects. For example, families of elliptic curves. You have a space where every point represents a "family" of some object: Triangles, curves, groups, whatever.

In modern mathematics, we often use the moduli stack instead. Related to schemes, spectrums of rings, fibers and principle G-bundles.

Learn Quantum Field Theory in 3.8 Minutes

The Mansions of Science — Sun, 14 Jun 2026 00:39:12 +0000

Second Quantization: Promote fields to operators and DEMOTE position

A Lie Group is a Group + a Manifold - That's It

The Mansions of Science — Sun, 14 Jun 2026 00:32:26 +0000

What is a Lie group?

The number line is R1. It is a 1-dimensional manifold; a 1-dimensional space. It is also a group, since all real numbers form an additive group. Therefore, the real number line R1 is a manifold and a group. A Lie group, after Norwegian mathematician Sophus Lie.

R2, R3, Rn are all Lie groups! :D

Visual Intuition for Differential Forms in 1.5 Minutes

The Mansions of Science — Sat, 13 Jun 2026 18:47:33 +0000

A one-form is a DIRECTED line segment (an arrow, a line with a direction). A two-form is a directed area segment. A three-form is a directed (oriented) volume element.

That's it.