"Supersymmetry" is a hypothetical symmetry that relates the two kinds of particles in our universe: bosons and fermions. There's no experimental evidence for it.

But "superalgebra" is worth studying even if you don't care about supersymmetry.

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When you have two particles of the same kind, it's impossible to tell if you've switched them.

When you work out the consequences of this fact, using quantum mechanics and some math, it means every particle is either a boson or a fermion.

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Quantum mechanics says the state of any system - a particle, a bunch of particles, a cat - is described by a vector in a complex vector space.

But when you multiply this vector by a "phase" (a complex number of length 1), nothing you can measure changes.

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If you switch two identical particles that are "bosons", the vector describing their state gets multiplied by +1. It doesn't change at all.

But if you switch two identical particles that are "fermions", the vector describing their state gets multiplied by -1.

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This has huge consequences. The fundamental particles that describe "matter" are mostly fermions: electrons, neutrinos, quarks, etc. The particles that describe "forces" are bosons: photons, gluons, the W and Z, etc.

It's a lot of fun to see why. But not today....

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I want to explain superalgebra.

The first step is to replace vector spaces with supervector spaces. A "supervector space" is just a vector space V that's a direct sum of two subspaces V0 and V1.

In physics, V0 is the bosonic part, while V1 is the fermionic part.

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Mathematicians call V0 the "even" part and V1 the "odd" part.

The fun starts when we take the tensor product of two supervector spaces. In physics, this corresponds to *combining* two quantum systems - like two particles.

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When we combine two quantum systems into a bigger one it works like this:

boson + boson = boson
boson + fermion = fermion
fermion + boson = fermion
fermion + fermion = boson

It's just like adding even and odd numbers, with bosons as "even" and fermions as "odd"!

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For example a proton consists of 3 quarks, and quarks are fermions, so a proton is a fermion too:

fermion + fermion + fermion = fermion

This is the kind of fact that's built into superalgebra! It comes from the rule for tensor products of supervector spaces.

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If V and W are supervector spaces, we define their tensor product V⊗W by

(V⊗W)0 = V0⊗W0 ⊕ V1⊗W1

(V⊗W)1 = V0⊗W1 ⊕ V1⊗W0

This is just math for what I've already told you: 2 bosons make a boson, 2 fermions make a boson, but a fermion and a boson make a fermion.

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The fun *really* starts when we compare V⊗W and W⊗V. They're isomorphic, but in an interesting way: we stick in a minus sign when we switch two odd vectors.

This captures what happens when you switch two fermions in physics!

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So, if you're a mathematician, you say there's a "symmetric monoidal category" of supervector spaces.

This means you can take ALMOST EVERYTHING YOU DO WITH VECTOR SPACES and generalize it to supervector spaces.

You get a subject called SUPERALGEBRA.

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In fact, superalgebra was lurking in ordinary mathematics all along. It's very natural.

But there's also a lot of new math that was only discovered after we started studying superalgebra. Lots of fun stuff! The textbooks keep coming out....

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So you don't need to care about superstrings - or even physics at all - to care about superalgebra.

Superalgebra is built deeply into the fabric of mathematics. It just happens that it took the physical universe to teach us this fact.

Reality is the best teacher.

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To start learning about superalgebra, read this:

https://en.wikipedia.org/wiki/Super_vector_space

and then this:

https://en.wikipedia.org/wiki/Superalgebra

That'll get you started!

Then maybe try Varadarajan's "Supersymmetry for Mathematicians: An Introduction".

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