Bracket Notation #
This file provides notation which can be used for the Lie bracket, for the commutator of two subgroups, and for other similar operations.
Main Definitions #
Bracket L Mfor a binary operation that takes something in
Land something in
Mand produces something in
M. Defining an instance of this structure gives access to the notation
We introduce the notation
⁅x, y⁆ for the
bracket of any
Bracket structure. Note that
these are the Unicode "square with quill" brackets rather than the usual square brackets.
⁅x, y⁆is the result of a bracket operation on elements
y. It is supported by the
Brackettypeclass.bracket : L → M → M
Bracket class has three intended uses:
- for certain binary operations on structures, like the product
⁅x, y⁆of two elements
yin a Lie algebra or the commutator of two elements
yin a group.
- for certain actions of one structure on another, like the action
⁅x, m⁆of an element
xof a Lie algebra on an element
min one of its modules (analogous to
SMulin the associative setting).
- for binary operations on substructures, like the commutator
⁅H, K⁆of two subgroups
Kof a group.
⁅x, y⁆ is the result of a bracket operation on elements
It is supported by the
- One or more equations did not get rendered due to their size.