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The notion of a group pervades mathematics and its applications to the physical sciences and beyond. A group is a mathematical structure: a collection of objects which can be added together like normal numbers. There must exist a unit : an object which when added to any other object does nothing, and each object must have an inverse : when an object and its inverse are added, we get the unit. However, unlike for usual addition, it is possible that the order of adding matters.Good examples arise from symmetry groups. Consider the rotations of a cube: this forms a group. The unit is the rotation by a zero angle, and addition is doing one rotation and then another: a bit of thought will show that the order you add rotations does matter! Notice that there are only finitely many rotations which leave the cube invariant. If we considered instead a sphere, then you can rotate by any angle, and so we get an infinite group. However, this group also have a notion of continuity : it makes sense to say that two rotations are close . We shall be interested in Locally Compact groups: these have this notion of continuity, but are also not too large in some technical sense.When the order of addition in a group doesn't matter, we say that our group is abelian. A locally compact abelian group has a dual group , which has an intimate relation to the Fourier transform. In everyday life you meet this when listing to music: the sound-wave is split into its component pure frequencies- we can think of these as building blocks. The group here is the real numbers under addition, and the pure frequencies are the periodic functions.When a group is not abelian, there are various notions of the dual , but none of these is an actual group. A way to deal with this problem is to give up on groups, and instead look at an algebra built out of the group (an algebra is a related mathematical structure with both addition and multiplication, interacting as you might expect). It turns out not to be so hard to define the algebra which should correspond to the dual group: even though the dual group doesn't actually exist! This is a slight mathematical fiction, but an incredibly productive one, which underpins what we call non-commutative mathematics . Often the term quantum is also used, as such objects were historically first considered in quantum mechanics.So, a quantum group is an algebra which looks like it is built from a group; a Locally Compact Quantum Group has some sort of continuity (technically, the algebra should be an Operator Algebra).While groups are interesting, in natural situations, we often only have a semigroup: this is when we lose the inverse axiom. An example arises from physical systems: classical mechanics, like the motion of a ball, are time-reversible, and so we get an action of the real numbers. But heat flow, for example, naturally only evolves forward in time, and so we are typically only interested in the semigroup of positive reals. We shall be interested in compact semigroups (which are, in some sense, small: the group of rotations of a sphere is compact, but the real numbers are not, as you can continue going out to infinity forever). We play this smallness off against weakening the continuity assumptions. Given a group, we can associate various universal compact semigroups to it: these have uses in combinatorics, and the abstract theory of group actions (topological dynamics) for example.Classically, one can build these compact semigroups out of various algebras associated to a group. By analogy, we get definitions for quantum groups. Alternatively, one can try to find an intrinsic definition of a quantum semigroup . Our project is to try to reconcile these two approaches (they are known to be different in some cases). Furthermore, we shall seek suitable definitions for a quantum semigroup which has rather weak continuity properties.
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