Groupoidified Linear Algebra

Transcription

1 Christopher D. Walker () Department of Mathematics University of California, Riverside November 22, 2008

2 There is a systematic process (first described by James Dolan) called Degroupoidification which turns:

3 There is a systematic process (first described by James Dolan) called Degroupoidification which turns: Groupoids = Vector Spaces

4 There is a systematic process (first described by James Dolan) called Degroupoidification which turns: Groupoids = Vector Spaces Spans of Groupoids = Linear Operators

5 There is a systematic process (first described by James Dolan) called Degroupoidification which turns: Groupoids = Vector Spaces Spans of Groupoids = Linear Operators Groupoidification is an attempt to reverse this process. As with any categorification process, this reverse direction is not systematic.

6 Why groupoids and spans of groupoids?

7 Why groupoids and spans of groupoids? We use groupoids because they give us a way of categorifying the positive real numbers.

8 Why groupoids and spans of groupoids? We use groupoids because they give us a way of categorifying the positive real numbers. We use spans because the give us a way to describe the matrix of a linear operator.

9 A little bit of category theory.

10 A little bit of category theory. There are actually two categories we are working in:

11 A little bit of category theory. There are actually two categories we are working in: Grpd- the category of groupoids as objects and functors as morphisms

12 A little bit of category theory. There are actually two categories we are working in: Grpd- the category of groupoids as objects and functors as morphisms Span- the category of groupoids as objects and spans of groupoids as morphisms

13 What is a span?

14 What is a span? Definition Given groupoids X and Y, a span from X to Y is defined as Y q S p X where S is another groupoid and p : S X and q : S Y are functors

15 We can compose two spans using the weak pullback over the matching legs of the spans.

16 We can compose two spans using the weak pullback over the matching legs of the spans. Definition Given two functors f : S Y and g : T Y, we define the weak pullback as: TS π T π T T S g f Y where TS is the groupoid whose objects are triples (s, t, α) where α: f (s) g(t).

17 Where do the positive real numbers come from?

18 Where do the positive real numbers come from? Definition Given a groupoid X, we define the cardinality of X to be: X = [x] 1 Aut x where x ranges over all isomorphism classes in X

19 Where do the positive real numbers come from? Definition Given a groupoid X, we define the cardinality of X to be: X = [x] 1 Aut x where x ranges over all isomorphism classes in X When this sum converges, we call the groupoid tame, and we can obtain any positive real number. This gives us access to a field to define a vector space over.

20 Where do the positive real numbers come from? Definition Given a groupoid X, we define the cardinality of X to be: X = [x] 1 Aut x where x ranges over all isomorphism classes in X When this sum converges, we call the groupoid tame, and we can obtain any positive real number. This gives us access to a field to define a vector space over. Example - Let E be the groupoid of finite sets and bijections. Then:

21 Where do the positive real numbers come from? Definition Given a groupoid X, we define the cardinality of X to be: X = [x] 1 Aut x where x ranges over all isomorphism classes in X When this sum converges, we call the groupoid tame, and we can obtain any positive real number. This gives us access to a field to define a vector space over. Example - Let E be the groupoid of finite sets and bijections. Then: E

22 Where do the positive real numbers come from? Definition Given a groupoid X, we define the cardinality of X to be: X = [x] 1 Aut x where x ranges over all isomorphism classes in X When this sum converges, we call the groupoid tame, and we can obtain any positive real number. This gives us access to a field to define a vector space over. Example - Let E be the groupoid of finite sets and bijections. Then: E = 1 S n n

23 Where do the positive real numbers come from? Definition Given a groupoid X, we define the cardinality of X to be: X = [x] 1 Aut x where x ranges over all isomorphism classes in X When this sum converges, we call the groupoid tame, and we can obtain any positive real number. This gives us access to a field to define a vector space over. Example - Let E be the groupoid of finite sets and bijections. Then: E = 1 S n n = 1 n! = e n

24 Definition Given a groupoid X we define the degroupoidification of X as R X, where X is the set of isomorphism classes in X.

25 Definition Given a groupoid X we define the degroupoidification of X as R X, where X is the set of isomorphism classes in X. In order to produce a single vector (function) in R X, we consider a groupoid over X, p : Ψ X. We say Ψ is tame if p 1 (x) is tame for all x, where p 1 (x) is the essential preimage of x. We then define the function: Ψ ([x]) = p 1 (x)

26 Definition Given a groupoid X we define the degroupoidification of X as R X, where X is the set of isomorphism classes in X. In order to produce a single vector (function) in R X, we consider a groupoid over X, p : Ψ X. We say Ψ is tame if p 1 (x) is tame for all x, where p 1 (x) is the essential preimage of x. We then define the function: Ψ ([x]) = p 1 (x) Example - Again, consider E. Since E = N, R E = R[[x]].

27 Definition Given a groupoid X we define the degroupoidification of X as R X, where X is the set of isomorphism classes in X. In order to produce a single vector (function) in R X, we consider a groupoid over X, p : Ψ X. We say Ψ is tame if p 1 (x) is tame for all x, where p 1 (x) is the essential preimage of x. We then define the function: Ψ ([x]) = p 1 (x) Example - Again, consider E. Since E = N, R E = R[[x]]. Also E is a tame groupoid over itself with the identity functor.

28 Definition Given a groupoid X we define the degroupoidification of X as R X, where X is the set of isomorphism classes in X. In order to produce a single vector (function) in R X, we consider a groupoid over X, p : Ψ X. We say Ψ is tame if p 1 (x) is tame for all x, where p 1 (x) is the essential preimage of x. We then define the function: Ψ ([x]) = p 1 (x) Example - Again, consider E. Since E = N, R E = R[[x]]. Also E is a tame groupoid over itself with the identity functor. Ẽ([n]) = 1 n!

29 Definition Given a groupoid X we define the degroupoidification of X as R X, where X is the set of isomorphism classes in X. In order to produce a single vector (function) in R X, we consider a groupoid over X, p : Ψ X. We say Ψ is tame if p 1 (x) is tame for all x, where p 1 (x) is the essential preimage of x. We then define the function: Ψ ([x]) = p 1 (x) Example - Again, consider E. Since E = N, R E = R[[x]]. Also E is a tame groupoid over itself with the identity functor. Ẽ([n]) = 1 n! Ẽ = e x.

30 We now produce a linear operator out of a span of groupoids.

31 We now produce a linear operator out of a span of groupoids. Definition Given a tame span of groupoids Y q S p X the linear operator S : R X R Y is given by S Ψ = SΨ, where Ψ is a groupoid over X, v : Ψ X, and SΨ is the weak pullback: Y π Ψ S q p X SΨ π S v Ψ

32 Considering the definition of weak pullback, we get a nice formula for the matrix entries of S :

33 Considering the definition of weak pullback, we get a nice formula for the matrix entries of S : S [x][y] = [s] p 1 (x) q 1 (y) Aut x Aut s

34 Addition We can add vectors and operators as follows:

35 Addition We can add vectors and operators as follows: Proposition Give two groupoids Φ and Ψ over X, the disjoint union Φ + Ψ forms a groupoid over X, and Φ + Ψ = Φ + Ψ

36 Addition We can add vectors and operators as follows: Proposition Give two groupoids Φ and Ψ over X, the disjoint union Φ + Ψ forms a groupoid over X, and Proposition Give two spans: Φ + Ψ = Φ + Ψ S T q S p S q T p T Y X Y X the disjoint union S + T forms a span from X to Y, and S + T = S + T

37 Scalar Multiplication We can multiply both vectors and linear operators by scalars as follows:

38 Scalar Multiplication We can multiply both vectors and linear operators by scalars as follows: Proposition Given a groupoid Λ and a groupoid Φ over X, the groupoid Λ Φ over X satisfies Λ Φ = Λ Φ.

39 Scalar Multiplication We can multiply both vectors and linear operators by scalars as follows: Proposition Given a groupoid Λ and a groupoid Φ over X, the groupoid Λ Φ over X satisfies Λ Φ = Λ Φ. Proposition Given a groupoid Λ and a span S Y X then Λ S = Λ S.

40 We also have a concept of inner product, which turns our vector space into a Hilbert space.

41 We also have a concept of inner product, which turns our vector space into a Hilbert space. Definition Given groupoids Φ and Ψ over X, we define the inner product Φ, Ψ to be this weak pullback: Φ, Ψ Φ Ψ X

42 This definition holds to the standard properties of inner products.

43 This definition holds to the standard properties of inner products. Proposition Given a groupoid Λ and groupoids Φ, Ψ, and Ψ over X, the following properties hold: Φ, Ψ Ψ, Φ. Φ, Ψ + Ψ Φ, Ψ + Φ, Ψ. Φ, Λ Ψ Λ Φ, Ψ.

44 Definition A groupoid Φ over X is called square-integrable if Φ, Φ is tame. We define L 2 (X ) to be the subspace of R X consisting of finite real linear combinations of vectors Φ where Φ is square-integrable.

45 Definition A groupoid Φ over X is called square-integrable if Φ, Φ is tame. We define L 2 (X ) to be the subspace of R X consisting of finite real linear combinations of vectors Φ where Φ is square-integrable. We then produce our Hilbert space. Proposition L 2 (X ) forms a Hilbert space with the inner product Ψ, Φ = Ψ, Φ.

46 Examples

47 Examples Hecke Algebras and Hecke Operators (Alex Hoffnung)

48 Examples Hecke Algebras and Hecke Operators (Alex Hoffnung) Quantum Harmonic Oscillators (Jeffrey Morton)

49 Examples Hecke Algebras and Hecke Operators (Alex Hoffnung) Quantum Harmonic Oscillators (Jeffrey Morton) Jordan-Schwinger Representations

50 Examples Hecke Algebras and Hecke Operators (Alex Hoffnung) Quantum Harmonic Oscillators (Jeffrey Morton) Jordan-Schwinger Representations Quiver Representations and Hall Algebras