Multidimensional Algebra

notation	meaning	basis	Alternative
	direct sum or Kronecker sum depending on context
	direct product or Kronecker product depending on context
^	exterior algebra
^k	exterior algebra kth blade
	scalar or real depending on context (see below)	e0
²	2 dimensional vector	e1,e2
³	3 dimensional vector	e1,e2,e3
n	n dimensional vector	e1…en
^²²	bivector based on 2D algebra	e12
^²³	bivector based on 3D algebra	e12,e31,e23
C	complex numbers
H	quaternions		³
O	octonions		7
aA	a is an element of the set A

Scalar

I have used the term 'scalar' interchangeably with the term 'real', that is, a continuous value that can be represented by a single number.

Strictly speaking the term 'scalar' should be reserved for a quantity that is used to scale a vector, that is change its magnitude without changing its direction, or in other words a scalar is the ratio of parallel vectors.

For instance I should not really call energy a scalar because there are no vectors involved.

I apologise for my lack of mathematical rigor here, its just that the word scalar seems to better express that it is not a vector and its less likely to cause confusion with the real part of a complex number. Also this (mis?)usage is quite common in the computer world.

Matrix

notation	meaning
[M]	matrix
[M]t	transpose of matrix (swap rows & columns)
[M]-1	inverse of matrix
[I]	identity matrix (ones on leading diagonal, otherwise zeros)

The individual elements of the matrix are numbered as follows,

m00	m01	m02	…	m0n
m10	m11	m12	…	m1n
m20	m21	m22	…	m2n
mp0	mp1	mp2	…	mpn

The first subscript represents the row, the second subscript represents the column

Tensor Notation

notation	example	meaning
subscript	e1	coordinate basis
superscript	x1	coordinate value
Einstein Summation Convention	eixi=e1x1+e2x2+e3x3=∑eixi	When the same index appears twice in an expression, once raised and once lowered, a sum is implied.
partial derivatives	∂a=∂/∂a

Tensors are discussed here.

Set Theory

notation	example	meaning
A	A = {a | aA}	set
a	aA	element of A
{a, b, c}		set containing a, b and c
	a H = {ah | hH}	left coset of a
(a,b)		cycle notation
<a,b>		all the elements of group generated by a and b
	<r,f | r³=1, f²=1, frf=r-1>	generators with constrains
| A |		grade (size) = number of elements in A
| G : H |		number of left cosets of H in G
×		Direct Product
		Semidirect Product (left)
	{n, h}φ {n',h'}	Semidirect Product (right)
		Bicrossed Product also known as Knit or Zappa-Szep product
	h = h'H such that a h' = h a	Normal subgroup a H = H a for all a in G
Ø		empty set
¬		not
∧		and (= intersection)
∨		or (U = union)
		is in (is an element of the set)
	AB	A is contained in B (or A is a subset of B)
→		if
<->		if and only if
		for all
		there exists
		is isomorphic to
(S)		The set of subsets of S (power set)
orb(s)		Orbit - set of elements that can be reached
stab(s)		Stabilizer - set of elements that don't move s

Sets are discussed on this page and groups here.

Category Theory

notation	example	meaning
AB		Map / Morphism / Function between mathematical structures that relates or preserves the structures in some way. May be injection, surjection or bijection (injection can be specifically indicated see below)
a b		A function in terms of elements of the structures aA, bB
		A function that is valid if all the functions drawn as solid lines are valid
		injective morphism (embedding)
		canonical map - map of G onto factor group G/H where H is a normal subgroup of G

Category Theory on this page and maps/morphisms here.

Greek Alphabet

	upper	lower		upper	lower		upper	lower
Alpha	A	α	Iota	I	ι	Rho	P	ρ
Beta	B	β	Kappa	K	κ	Sigma	Σ	σ
Gamma	Γ	γ	Lambda	Λ	λ	Tau	Τ	τ
Delta	Δ	δ	Mu	M	μ	Upsilon	Υ	υ
Epsilon	E	ε	Nu	N	ν	Phi	Φ	φ
Zeta	Z	ζ	Xi	Ξ	ξ	Chi	X	χ
Eta	H	η	Omicron	O	ο	Psi	Ψ	ψ
Theta	Θ	θ	Pi	Π	π	Omega	Ω	ω