Abstract Algebra
Table of Contents
1. Group
See group theory.
2. Ring
See ring theory.org.
2.1. Hypercomplex Number
2.1.1. Complex Number
2.1.2. Split Complex Number
2.1.3. Dual Number
2.1.4. Clifford Algebra
2.1.5. Quotient Ring
3. Module
4. Vector Space
See vector space.
5. Algebra
5.1. Free Algebra
- Noncommutative polynomial ring
5.1.1. Definition
- For a commutative ring \(R\), the free (associative unital) \(R\)-algebra on \(n\) indeterminates \(X = \{X_i\}_{i=1}^n\) is the free \(R\)-module with all words over the alphabet \(X\) as the basis, equipped with concatenation as multiplication.
- \[ R\langle X\rangle := \bigotimes_{w\in X^*}Rw \] where \(X^*\) denotes the free monoid on \(X\).
5.2. Weyl Algebra
5.2.1. Definition
- Noncommutative ring \(R[\Delta]\) of differential operators with
polynomial coefficients on a partial differential ring \((R, \Delta)\) satisfying
\(\forall r\in R, \partial_ir = r\partial_i + \partial_i(r)\).
- \[ R[\Delta]/\langle \partial_i r - r\partial_i - \partial_i(r)\rangle \]
- Using commutator: \([\partial_i, x_i] = 1\).
5.2.1.1. n-th Weyl Algebra
- \(A_n\)
- Infinite family of algebras, also called Weyl algebras.
- Ring of differential operators with polynomial coefficients in \(n\) variables.
5.2.1.2. Abstract Construction
- \[
W(V) := T(V)/\langle v\otimes u - u\otimes v - \omega(v,u)\rangle
\]
- where \(\omega\) is an symplectic form on a vector space \(V\).
5.2.2. Properties
- Example of a simple ring that is not a matrix ring over a division ring.
- Noncommutative example of a domain
- Example of an Ore extension
- Isomorphic to the quotient of the free algebra on two generators: \[ A_1 \cong R\langle X, Y\rangle/(YX-XY-1). \]
5.3. Witt Algebra
5.3.1. Definition
- Lie algebra of meromorphic vector fields defined on the Riemann sphere.
- Complexification of the Lie algebra of polynomial vector fields on a circle.
- Lie algebra of the derivations of the ring \(\mathbb{C}[z, z^{-1}]\) that satisfies two properties:
- Linearity
- Leibniz Rule
- This conditions restrict the space to correspond to the ring of Laurant polynomials.
5.3.1.1. Basis
- The basis for the Laurant polynomial is mapped to the basis for the Witt algebra.
- The basis is vector fields \(L_n\), for any \(n\in \mathbb{Z}\): \[ L_n = -z^{n+1}\frac{\partial}{\partial z}. \]
5.3.1.2. Lie Bracket
- \[ [L_m, L_n] := L_mL_n - L_nL_m = (m-n)L_{m+n} \]
5.3.2. Properties
- The first order derivatives with Laurant polynomial coefficients.
- Central extension of Witt algebra is called the Virasoro algebra.