SLIDES OF TALKS Akira Masuoka CIM AkiraMasuoka

Super Lie groups over a complete field

— From analytic to algebraic, and back —

Akira Masuoka (U. Tsukuba)

Joint work (in progress) with Mitsukazu Hoshi

CIM, Sep. 6, 2016

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66 One can easily learn algebraic groups, replacing AG with RT.

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66 One can easily learn algebraic groups, replacing AG with RT. Example GL(n) = {X = (x^ij), n × n | det X invertible}

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66 One can easily learn algebraic groups, replacing AG with RT. Example GL(n) = {X = (x^ij), n × n | det X invertible}

Group functor which associates to R, matrices withspecial entries x_{ij ∈ R.}

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66 One can easily learn algebraic groups, replacing AG with RT. Example GL(n) = {X = (x^ij), n × n | det X invertible}

Group functor which associates to R, matrices withspecial entries x_{ij ∈ R.}

Represented by the Hopf algebra k[xij, (det X )⁻¹] generated by the genericentries x_ij,

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66 One can easily learn algebraic groups, replacing AG with RT. Example GL(n) = {X = (x^ij), n × n | det X invertible}

Group functor which associates to R, matrices withspecial entries x_{ij ∈ R.}

Represented by the Hopf algebra k[xij, (det X )⁻¹] generated by the genericentries x_ij, with respect to the group law

∆(X ) = X ⊗ X , ϵ(X ) = I , S(X ) = X⁻¹.

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66 One can easily learn algebraic groups, replacing AG with RT. Example GL(n) = {X = (x^ij), n × n | det X invertible}

Group functor which associates to R, matrices withspecial entries x_{ij ∈ R.}

Represented by the Hopf algebra k[xij, (det X )⁻¹] generated by the genericentries x_ij, with respect to the group law

∆(X ) = X ⊗ X , ϵ(X ) = I , S(X ) = X⁻¹. The same in the super context, but that the domain of R is extended to super-com. super-algebras R = R_{0⊕ R1}.

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66 One can easily learn algebraic groups, replacing AG with RT. Example GL(n) = {X = (x^ij), n × n | det X invertible}

Group functor which associates to R, matrices withspecial entries x_{ij ∈ R.}

Represented by the Hopf algebra k[xij, (det X )⁻¹] generated by the genericentries x_ij, with respect to the group law

∆(X ) = X ⊗ X , ϵ(X ) = I , S(X ) = X⁻¹. The same in the super context, but that the domain of R is extended to super-com. super-algebras R = R_{0⊕ R1}. Example GL(m|n) = {

(X P

Q Y

)

| det X , det Y invertible} X m× m, Y n × n even entries; P m × n, Q n × m odd entries

0. Algebraic vs. Analytic

Waterhouse “Introduction to affine group schemes” GTM 66 One can easily learn algebraic groups, replacing AG with RT. Example GL(n) = {X = (x^ij), n × n | det X invertible}

Group functor which associates to R, matrices withspecial entries x_{ij ∈ R.}

Represented by the Hopf algebra k[xij, (det X )⁻¹] generated by the genericentries x_ij, with respect to the group law

∆(X ) = X ⊗ X , ϵ(X ) = I , S(X ) = X⁻¹. The same in the super context, but that the domain of R is extended to super-com. super-algebras R = R_{0⊕ R1}. Example GL(m|n) = {

(X P

Q Y

)

| det X , det Y invertible} X m× m, Y n × n even entries; P m × n, Q n × m odd entries Represented by k[xij, yst, (det X )⁻¹, (det Y )⁻¹_{] ⊗ ∧(p}it, qsj).

Def Ansuper affine group (scheme) is a representable functor G : (Super-algebras) → (Groups).

It is uniquely represented by a super Hopf algebra, k[G ].

Def Ansuper affine group (scheme) is a representable functor G : (Super-algebras) → (Groups).

It is uniquely represented by a super Hopf algebra, k[G ]. G is said to bealgebraic, if k[G ] is finitely generated.

Def Ansuper affine group (scheme) is a representable functor G : (Super-algebras) → (Groups).

It is uniquely represented by a super Hopf algebra, k[G ]. G is said to bealgebraic, if k[G ] is finitely generated.

To studysuper affine groupsthrough the correspondingsuper Hopf algebrasis more effective than in the non-super situation

Def Ansuper affine group (scheme) is a representable functor G : (Super-algebras) → (Groups).

It is uniquely represented by a super Hopf algebra, k[G ]. G is said to bealgebraic, if k[G ] is finitely generated.

To studysuper affine groupsthrough the correspondingsuper Hopf algebrasis more effective than in the non-super situation

because those super groups G are not determined by their functor points G (R) in a single R. [cf. A smooth algebraic group G over an algebraically closed field k is determined by G (k).]

Def Ansuper affine group (scheme) is a representable functor G : (Super-algebras) → (Groups).

It is uniquely represented by a super Hopf algebra, k[G ]. G is said to bealgebraic, if k[G ] is finitely generated.

To studysuper affine groupsthrough the correspondingsuper Hopf algebrasis more effective than in the non-super situation

because those super groups G are not determined by their functor points G (R) in a single R. [cf. A smooth algebraic group G over an algebraically closed field k is determined by G (k).]

Such study is active though only recently.

Def Ansuper affine group (scheme) is a representable functor G : (Super-algebras) → (Groups).

It is uniquely represented by a super Hopf algebra, k[G ]. G is said to bealgebraic, if k[G ] is finitely generated.

To studysuper affine groupsthrough the correspondingsuper Hopf algebrasis more effective than in the non-super situation

because those super groups G are not determined by their functor points G (R) in a single R. [cf. A smooth algebraic group G over an algebraically closed field k is determined by G (k).]

Such study is active though only recently.

Study ofsuper Lie groupshas a much longer history, though the definition is more involved.

Asuper Lie groupover a complete field is defined to be a group object in the category of super-manifolds.

Asuper Lie groupover a complete field is defined to be a group object in the category of super-manifolds.

Asuper-manifold is defined to be a topological space S given a struc. sheaf O^S of super-algebras, which satisfies some conditions.

Asuper Lie groupover a complete field is defined to be a group object in the category of super-manifolds.

Asuper-manifold is defined to be a topological space S given a struc. sheaf O^S of super-algebras, which satisfies some conditions. Example (see Deligne-Morgan) The analytic GL(m|n) is defined by its functor points in S = (S, O^{S) so as}

Asuper Lie groupover a complete field is defined to be a group object in the category of super-manifolds.

Asuper-manifold is defined to be a topological space S given a struc. sheaf O^S of super-algebras, which satisfies some conditions. Example (see Deligne-Morgan) The analytic GL(m|n) is defined by its functor points in S = (S, O^{S) so as}

GL(m|n) = the automorphisms of the super O^S-module O^m|nS ^.

Asuper Lie groupover a complete field is defined to be a group object in the category of super-manifolds.

Asuper-manifold is defined to be a topological space S given a struc. sheaf O^S of super-algebras, which satisfies some conditions. Example (see Deligne-Morgan) The analytic GL(m|n) is defined by its functor points in S = (S, O^{S) so as}

GL(m|n) = the automorphisms of the super O^S-module O^m|nS ^.

In essentially the same manner as in the algebraic situation, this is often presented as the group of matrices

(X P

Q Y

)

with entries in (OS⁾0 for X , Y , and entries in (OS⁾1 ^{for P, Q.}

Asuper Lie groupover a complete field is defined to be a group object in the category of super-manifolds.

Asuper-manifold is defined to be a topological space S given a struc. sheaf O^S of super-algebras, which satisfies some conditions. Example (see Deligne-Morgan) The analytic GL(m|n) is defined by its functor points in S = (S, O^{S) so as}

GL(m|n) = the automorphisms of the super O^S-module O^m|nS ^.

In essentially the same manner as in the algebraic situation, this is often presented as the group of matrices

(X P

Q Y

)

with entries in (OS⁾0 for X , Y , and entries in (OS⁾1 ^{for P, Q.}

Motivation I Do such matrix presentations, which we often encounter, and which are essentially the same as defining super algebraic groups, correctly define super Lie groups?

Hopf-algebraic study of super Lie groupsover a complete field k (e.g. R, C, Qp, Fp((X ))) of any characteristic

Hopf-algebraic study of super Lie groupsover a complete field k (e.g. R, C, Qp, Fp((X ))) of any characteristic

Aim I To give a comprehensive description of super Lie groups without referring to sheaves, proving a category equivalence

(Super Lie groups) ≈ (Harish-Chanda pairs).

Hopf-algebraic study of super Lie groupsover a complete field k (e.g. R, C, Qp, Fp((X ))) of any characteristic

Aim I To give a comprehensive description of super Lie groups without referring to sheaves, proving a category equivalence

(Super Lie groups) ≈ (Harish-Chanda pairs).

This was proved by Kostant, Koszul and Vishnyakova when k = R or C. The question to extend the result to p-adic fields was raised by Varadarajan, 2011.

Hopf-algebraic study of super Lie groupsover a complete field k (e.g. R, C, Qp, Fp((X ))) of any characteristic

Aim I To give a comprehensive description of super Lie groups without referring to sheaves, proving a category equivalence

(Super Lie groups) ≈ (Harish-Chanda pairs).

This was proved by Kostant, Koszul and Vishnyakova when k = R or C. The question to extend the result to p-adic fields was raised by Varadarajan, 2011.

Aim II To construct a natural functor

(Super algebraic groups) → (Super Lie groups).

This justifies defining super Lie groups, using matrix presentation.

Hopf-algebraic study of super Lie groupsover a complete field k (e.g. R, C, Qp, Fp((X ))) of any characteristic

Aim I To give a comprehensive description of super Lie groups without referring to sheaves, proving a category equivalence

(Super Lie groups) ≈ (Harish-Chanda pairs).

This was proved by Kostant, Koszul and Vishnyakova when k = R or C. The question to extend the result to p-adic fields was raised by Varadarajan, 2011.

Aim II To construct a natural functor

(Super algebraic groups) → (Super Lie groups).

This justifies defining super Lie groups, using matrix presentation. Aim III To describe Hopf-algebraically, analytic representations of super Lie groups, by means of the super Hopf algebras of

representative functions.

1. Complete fields, over which we will work

1. Complete fields, over which we will work

Anabsolute value| | : k → R≥0 on a field k is a multiplicative function with |1| = 1 which satisfies

1) x = 0 ⇔ |x| = 0 2) |x + y| ≤ |x| + |y|.

1. Complete fields, over which we will work

Anabsolute value| | : k → R≥0 on a field k is a multiplicative function with |1| = 1 which satisfies

1) x = 0 ⇔ |x| = 0 2) |x + y| ≤ |x| + |y|.

Def Acomplete fieldis a field given an absolute value, which is complete, but is not discrete.

1. Complete fields, over which we will work

Anabsolute value| | : k → R≥0 on a field k is a multiplicative function with |1| = 1 which satisfies

1) x = 0 ⇔ |x| = 0 2) |x + y| ≤ |x| + |y|.

Def Acomplete fieldis a field given an absolute value, which is complete, but is not discrete.

Thm (Ostrowski) If k is a complete field, then

◮ _k= R or C, or

◮ | | is non-Archimedian, i.e. 2^′) |x − y| ≤ Max(|x|, |y|), e.g., k = a finite ext. of Qp, or k = F ((T )), F a field. Ref J.-P. Serre, Lie algebras and Lie groups, SLN 1500.

1. Complete fields, over which we will work

Anabsolute value| | : k → R≥0 on a field k is a multiplicative function with |1| = 1 which satisfies

1) x = 0 ⇔ |x| = 0 2) |x + y| ≤ |x| + |y|.

Def Acomplete fieldis a field given an absolute value, which is complete, but is not discrete.

Thm (Ostrowski) If k is a complete field, then

◮ _k= R or C, or

◮ | | is non-Archimedian, i.e. 2^′) |x − y| ≤ Max(|x|, |y|), e.g., k = a finite ext. of Qp, or k = F ((T )), F a field. Ref J.-P. Serre, Lie algebras and Lie groups, SLN 1500. In what follows we work over a complete field k.

2. Manifolds as ringed spaces

2. Manifolds as ringed spaces

Anopen sub-manifold of kⁿ is an open subset U ⊂ k^{n given the} sheaf, V 7→ F^U(V ), which associates to each open subset V ⊂ U the comm. algebra F^U(V ) = {all analytic functions V → k}.

2. Manifolds as ringed spaces

Anopen sub-manifold of kⁿ is an open subset U ⊂ k^{n given the} sheaf, V 7→ F^U(V ), which associates to each open subset V ⊂ U the comm. algebra F^U(V ) = {all analytic functions V → k}. Def. Amanifoldof dimension n is a topological space

T = (T , F) given a structure sheaf F of com. algebras, which is locally isomorphic to an open sub-manifold of kⁿ.

2. Manifolds as ringed spaces

Anopen sub-manifold of kⁿ is an open subset U ⊂ k^{n given the} sheaf, V 7→ F^U(V ), which associates to each open subset V ⊂ U the comm. algebra F^U(V ) = {all analytic functions V → k}. Def. Amanifoldof dimension n is a topological space

T = (T , F) given a structure sheaf F of com. algebras, which is locally isomorphic to an open sub-manifold of kⁿ.

At every pt. p ∈ T , the local algebra of germs and its completion are:

◮ _{Fp≃ k{X}

1, . . . , Xn}, the convergent power series algebra.

◮ _{Fbp≃ k[[X}

1, . . . , Xn]], the formal power series algebra.

2. Manifolds as ringed spaces

Anopen sub-manifold of kⁿ is an open subset U ⊂ k^{n given the} sheaf, V 7→ F^U(V ), which associates to each open subset V ⊂ U the comm. algebra F^U(V ) = {all analytic functions V → k}. Def. Amanifoldof dimension n is a topological space

T = (T , F) given a structure sheaf F of com. algebras, which is locally isomorphic to an open sub-manifold of kⁿ.

At every pt. p ∈ T , the local algebra of germs and its completion are:

◮ _{Fp≃ k{X}

1, . . . , Xn}, the convergent power series algebra.

◮ _{Fbp≃ k[[X}

1, . . . , Xn]], the formal power series algebra. Prop-Def The category of manifolds has finite products.

The group objects in the category are, therefore, defined. They are calledLie groups.

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

= To study G throughassociated (topological or discrete) super Hopf algebras

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

= To study G throughassociated (topological or discrete) super Hopf algebras

Let’s see first in the non-super situation.

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

= To study G throughassociated (topological or discrete) super Hopf algebras

Let’s see first in the non-super situation.

Given a Lie group G , the associated (three) Hopf algebras are divided into two groups according to their roles.

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

= To study G throughassociated (topological or discrete) super Hopf algebras

Let’s see first in the non-super situation.

Given a Lie group G , the associated (three) Hopf algebras are divided into two groups according to their roles.

Group I: For infinitesimal behavior of G

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

= To study G throughassociated (topological or discrete) super Hopf algebras

Let’s see first in the non-super situation.

Given a Lie group G , the associated (three) Hopf algebras are divided into two groups according to their roles.

Group I: For infinitesimal behavior of G

I-i. The com. complete local Hopf algebra_Fb_e, the completion of the local algebra F^e of germs at e.

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

= To study G throughassociated (topological or discrete) super Hopf algebras

Let’s see first in the non-super situation.

Given a Lie group G , the associated (three) Hopf algebras are divided into two groups according to their roles.

Group I: For infinitesimal behavior of G

I-i. The com. complete local Hopf algebra_Fb_e, the completion of the local algebra F^e of germs at e.

I-ii. The hyper-algebra hy(G )of G ; this is an irreducible cocom. Hopf algebra (Takeuchi).

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

= To study G throughassociated (topological or discrete) super Hopf algebras

Let’s see first in the non-super situation.

Given a Lie group G , the associated (three) Hopf algebras are divided into two groups according to their roles.

Group I: For infinitesimal behavior of G

I-i. The com. complete local Hopf algebra_Fb_e, the completion of the local algebra F^e of germs at e.

I-ii. The hyper-algebra hy(G )of G ; this is an irreducible cocom. Hopf algebra (Takeuchi).

We have an exact duality,_Fb_{e ↔ hy(G )}.

3. Associated Hopf-algebra objects

Hopf-algebraic study of super Lie groups G

= To study G throughassociated (topological or discrete) super Hopf algebras

Let’s see first in the non-super situation.

Given a Lie group G , the associated (three) Hopf algebras are divided into two groups according to their roles.

Group I: For infinitesimal behavior of G

I-i. The com. complete local Hopf algebra_Fb_e, the completion of the local algebra F^e of germs at e.

I-ii. The hyper-algebra hy(G )of G ; this is an irreducible cocom. Hopf algebra (Takeuchi).

We have an exact duality,_Fb_{e ↔ hy(G )}.

Group II: For representations of G (Mostow–Hochschild) II. The com. (not necessarily, finitely generated) Hopf algebra

R(G ) of representative functions.

(G , F) a Lie group over k

(G , F) a Lie group over k

I-i. Complete local Hopf algebra b_Fe:

(G , F) a Lie group over k

I-i. Complete local Hopf algebra b_Fe: The structure maps {e} ֒→ G ← G × G

on G induce maps of local algebras, k_{←− F}^ϵ e _{−→ F}^∆ G×G ,(e,e)^.

(G , F) a Lie group over k

I-i. Complete local Hopf algebra b_Fe: The structure maps {e} ֒→ G ← G × G

on G induce maps of local algebras, k_{←− F}^ϵ e _{−→ F}^∆ G×G ,(e,e)^.

These are completed to k_{←− b}^bϵ _{Fe −→ b}^∆b _{Fe ⊗ bb Fe}, with which b_Fe

is acomplete topological Hopf algebra.

(G , F) a Lie group over k

I-i. Complete local Hopf algebra b_Fe: The structure maps {e} ֒→ G ← G × G

on G induce maps of local algebras, k_{←− F}^ϵ e _{−→ F}^∆ G×G ,(e,e)^.

These are completed to k_{←− b}^bϵ _{Fe −→ b}^∆b _{Fe ⊗ bb Fe}, with which b_Fe

is acomplete topological Hopf algebra. Since Fbe ^{= k[[X ]],} F^be _{⊗ b}b _Fe= k[[X , Y ]]

as algebras, where X = (X₁, . . . , Xn), Y = (Y₁, . . . , Yn), giving b∆ is the same as giving f = (f₁, . . . , f_n), f_i ∈ k[[X , Y ]], satisfying the formal group law

f(f (X , Y ), Z ) = f (X , f (Y , Z )), f(X , 0) = 0 = f (0, Y ).

I-ii. Hyper-algebra hy(G ) of G :

I-ii. Hyper-algebra hy(G ) of G :Defined by hy(G ) := ^∪

r≥0

(F^e/mr)∗,

where m ⊂ F^e is the maximal ideal.

I-ii. Hyper-algebra hy(G ) of G :Defined by hy(G ) := ^∪

r≥0

(F^e/mr)∗,

where m ⊂ F^e is the maximal ideal.

◮ This is a cocom. irreducible Hopf algebra, in an exact duality with b_Fe so that

Fb^{e = (hy(G ))∗,} hy(G ) = the continuous dual of b_Fe.

I-ii. Hyper-algebra hy(G ) of G :Defined by hy(G ) := ^∪

r≥0

(F^e/mr)∗,

where m ⊂ F^e is the maximal ideal.

◮ This is a cocom. irreducible Hopf algebra, in an exact duality with b_Fe so that

Fb^{e = (hy(G ))∗,} hy(G ) = the continuous dual of b_Fe.

◮ _The Lie algebra of G is defined (Hopf-algebraically) by Lie(G ) := P(hy(G )).

I-ii. Hyper-algebra hy(G ) of G :Defined by hy(G ) := ^∪

r≥0

(F^e/mr)∗,

where m ⊂ F^e is the maximal ideal.

◮ This is a cocom. irreducible Hopf algebra, in an exact duality with b_Fe so that

Fb^{e = (hy(G ))∗,} hy(G ) = the continuous dual of b_Fe.

◮ _The Lie algebra of G is defined (Hopf-algebraically) by Lie(G ) := P(hy(G )).

If char k = 0, then hy(G ) = U(Lie(G )).

The duality b_Fe ↔ hy(G ) extends to their topological (co)modules.

The duality b_Fe ↔ hy(G ) extends to their topological (co)modules. Fact Given acomplete topological vector space V, there is a natural 1-1 correspondence between

The duality b_Fe ↔ hy(G ) extends to their topological (co)modules. Fact Given acomplete topological vector space V, there is a natural 1-1 correspondence between

◮ the topological b_Fe-comodule structures V → V b⊗ bFe^,

The duality b_Fe ↔ hy(G ) extends to their topological (co)modules. Fact Given acomplete topological vector space V, there is a natural 1-1 correspondence between

◮ the topological b_Fe-comodule structures V → V b⊗ bFe^{, and}

◮ the hy(G )-module structures hy(G ) ⊗ V → V s.t. every element of hy(G ) acts as a continuous linear endo. of V .

The duality b_Fe ↔ hy(G ) extends to their topological (co)modules. Fact Given acomplete topological vector space V, there is a natural 1-1 correspondence between

◮ the topological b_Fe-comodule structures V → V b⊗ bFe^{, and}

◮ the hy(G )-module structures hy(G ) ⊗ V → V s.t. every element of hy(G ) acts as a continuous linear endo. of V . Sum-up Hopf-algebraic treatment of Lie groups is thus done, introducing topological algebra. Then the relevant objects are often in a strong duality.

One can more easily learn Lie groups, replacing DC (=differential calculus) with HA (=Hopf algebra) (co-)actions(?)

II. Hopf alg. R(G ) of repres. functions(Mostow-Hochschild):

II. Hopf alg. R(G ) of repres. functions(Mostow-Hochschild): A finite-dim’nal G -module M isanalytic, if the corresp. matrix representation π : G → GL(M) = GL^r(k), composed with all projections of the r² entries GL_r(k) → k, is analytic.

II. Hopf alg. R(G ) of repres. functions(Mostow-Hochschild): A finite-dim’nal G -module M isanalytic, if the corresp. matrix representation π : G → GL(M) = GL^r(k), composed with all projections of the r² entries GL_r(k) → k, is analytic. Linearize π to kG _{→ Matr}(k), and dualize it to Matr(k)^∗ _{→ (kG )}^∗.

II. Hopf alg. R(G ) of repres. functions(Mostow-Hochschild): A finite-dim’nal G -module M isanalytic, if the corresp. matrix representation π : G → GL(M) = GL^r(k), composed with all projections of the r² entries GL_r(k) → k, is analytic. Linearize π to kG _{→ Matr}(k), and dualize it to Matr(k)^∗ _{→ (kG )}^∗. The image

C_M := Im(Matr(k)^∗_{→ (kG )}^∗) is a coalgebra,“the coefficient coalgebra of M”.

II. Hopf alg. R(G ) of repres. functions(Mostow-Hochschild): A finite-dim’nal G -module M isanalytic, if the corresp. matrix representation π : G → GL(M) = GL^r(k), composed with all projections of the r² entries GL_r(k) → k, is analytic. Linearize π to kG _{→ Matr}(k), and dualize it to Matr(k)^∗ _{→ (kG )}^∗. The image

C_M := Im(Matr(k)^∗_{→ (kG )}^∗)

is a coalgebra,“the coefficient coalgebra of M”. The union R(G ) :=^∪

M

C_{M ⊂ (kG )}^∗

is a com. Hopf algebra,“the Hopf algebra of repres. functions”.

II. Hopf alg. R(G ) of repres. functions(Mostow-Hochschild): A finite-dim’nal G -module M isanalytic, if the corresp. matrix representation π : G → GL(M) = GL^r(k), composed with all projections of the r² entries GL_r(k) → k, is analytic. Linearize π to kG _{→ Matr}(k), and dualize it to Matr(k)^∗ _{→ (kG )}^∗. The image

C_M := Im(Matr(k)^∗_{→ (kG )}^∗)

is a coalgebra,“the coefficient coalgebra of M”. The union R(G ) :=^∪

M

C_{M ⊂ (kG )}^∗

is a com. Hopf algebra,“the Hopf algebra of repres. functions”. Fact (Loc. finite analytic G -modules) ≈ (R(G )-comodules)^⊗

Sum-up Representations of Lie groups can be thus treated Hopf-algebraically.

Sum-up Representations of Lie groups can be thus treated Hopf-algebraically.

Motivation II Want an analogous category equivalence, (L.f. analytic super G -modules) ≈ (super R(G )-comodules),^⊗ for super Lie groups G : Aim III.

4. Super-manifolds and super Lie groups

In what follows we assume char k ̸= 2.

4. Super-manifolds and super Lie groups

In what follows we assume char k ̸= 2.

Super-objects,e.g. super (Lie/Hopf) algebras, are the objects which are naturally defined in the symmetric tensor category of Z/2-graded vector spaces V = V_{0⊕ V1} w.r.t. the super-symmetry

V _{⊗ V}^′ _{→ V}^′⊗ V , v ⊗ v^′7→ (−1)^{|v ||v′|v′}⊗ v.

4. Super-manifolds and super Lie groups

In what follows we assume char k ̸= 2.

Super-objects,e.g. super (Lie/Hopf) algebras, are the objects which are naturally defined in the symmetric tensor category of Z/2-graded vector spaces V = V_{0⊕ V1} w.r.t. the super-symmetry

V _{⊗ V}^′ _{→ V}^′⊗ V , v ⊗ v^′7→ (−1)^{|v ||v′|v′}⊗ v. All super-algebras will be assumed to be super-commutative, ab_{= (−1)}^|a||b|ba.

4. Super-manifolds and super Lie groups

In what follows we assume char k ̸= 2.

Super-objects,e.g. super (Lie/Hopf) algebras, are the objects which are naturally defined in the symmetric tensor category of Z/2-graded vector spaces V = V_{0⊕ V1} w.r.t. the super-symmetry

V _{⊗ V}^′ _{→ V}^′⊗ V , v ⊗ v^′7→ (−1)^{|v ||v′|v′}⊗ v. All super-algebras will be assumed to be super-commutative, ab_{= (−1)}^|a||b|ba.

Example The exterior algebra ∧(W ) on a vector space W is the simplest example of super-algebras.

4. Super-manifolds and super Lie groups

In what follows we assume char k ̸= 2.

Super-objects,e.g. super (Lie/Hopf) algebras, are the objects which are naturally defined in the symmetric tensor category of Z/2-graded vector spaces V = V_{0⊕ V1} w.r.t. the super-symmetry

V _{⊗ V}^′ _{→ V}^′⊗ V , v ⊗ v^′7→ (−1)^{|v ||v′|v′}⊗ v. All super-algebras will be assumed to be super-commutative, ab_{= (−1)}^|a||b|ba.

Example The exterior algebra ∧(W ) on a vector space W is the simplest example of super-algebras.

Info Y. Shimadais developing a (dif’tial) Galois theory, believing the exterior algebras over fields= “fields” in the super-context.

Given a manifold (S, F) and a f.d. vector space W , then F ⊗ ∧(W ) : U 7→ F(U) ⊗ ∧(W ) is a sheaf of super-algebras on S.

Given a manifold (S, F) and a f.d. vector space W , then F ⊗ ∧(W ) : U 7→ F(U) ⊗ ∧(W )

is a sheaf of super-algebras on S. A super-ringed space of this form (S, F ⊗ ∧(W )) is called asplit super-manifoldof sdim. m|n, where m = dim S, n = dim W .

Given a manifold (S, F) and a f.d. vector space W , then F ⊗ ∧(W ) : U 7→ F(U) ⊗ ∧(W )

is a sheaf of super-algebras on S. A super-ringed space of this form (S, F ⊗ ∧(W )) is called asplit super-manifoldof sdim. m|n, where m = dim S, n = dim W .

Asuper-manifold is a super-ringed space (S, O) which is locally isomorphic to a split super-manifold.

Given a manifold (S, F) and a f.d. vector space W , then F ⊗ ∧(W ) : U 7→ F(U) ⊗ ∧(W )

is a sheaf of super-algebras on S. A super-ringed space of this form (S, F ⊗ ∧(W )) is called asplit super-manifoldof sdim. m|n, where m = dim S, n = dim W .

Asuper-manifold is a super-ringed space (S, O) which is locally isomorphic to a split super-manifold.

Lemma (1) Every manifold is a super-manifold.

Given a manifold (S, F) and a f.d. vector space W , then F ⊗ ∧(W ) : U 7→ F(U) ⊗ ∧(W )

is a sheaf of super-algebras on S. A super-ringed space of this form (S, F ⊗ ∧(W )) is called asplit super-manifoldof sdim. m|n, where m = dim S, n = dim W .

Asuper-manifold is a super-ringed space (S, O) which is locally isomorphic to a split super-manifold.

Lemma (1) Every manifold is a super-manifold. (2) Given a super-manifold (S, O), then

S_{red= (S, Ored}),

where Ored(U) := (O(U))red= O(U)/^√0, is a manifold.

Prop-Def The category of super-manifolds has finite products.

Prop-Def The category of super-manifolds has finite products. The group objects in the category are, therefore, defined. They are calledsuper Lie groups.

Prop-Def The category of super-manifolds has finite products. The group objects in the category are, therefore, defined. They are calledsuper Lie groups.

To every super Lie group G = (G , O) associated are

Prop-Def The category of super-manifolds has finite products. The group objects in the category are, therefore, defined. They are calledsuper Lie groups.

To every super Lie group G = (G , O) associated are

· Lie group ^Gred= (G , Ored⁾

Prop-Def The category of super-manifolds has finite products. The group objects in the category are, therefore, defined. They are calledsuper Lie groups.

To every super Lie group G = (G , O) associated are

· Lie group ^Gred= (G , Ored⁾

· complete super Hopf algebra O^be

Prop-Def The category of super-manifolds has finite products. The group objects in the category are, therefore, defined. They are calledsuper Lie groups.

To every super Lie group G = (G , O) associated are

· Lie group ^Gred= (G , Ored⁾

· complete super Hopf algebra O^be

(≃ k[[X1, . . . , X_{m]] ⊗ ∧(Y1}, . . . , Y_n) as super-algebra)

Prop-Def The category of super-manifolds has finite products. The group objects in the category are, therefore, defined. They are calledsuper Lie groups.

To every super Lie group G = (G , O) associated are

· Lie group ^Gred= (G , Ored⁾

· complete super Hopf algebra O^be

(≃ k[[X1, . . . , X_{m]] ⊗ ∧(Y1}, . . . , Y_n) as super-algebra)

· super hyper-algebra hy(G ) = the continuous dual of b_Oe

Prop-Def The category of super-manifolds has finite products. The group objects in the category are, therefore, defined. They are calledsuper Lie groups.

To every super Lie group G = (G , O) associated are

· Lie group ^Gred= (G , Ored⁾

· complete super Hopf algebra O^be

(≃ k[[X1, . . . , X_{m]] ⊗ ∧(Y1}, . . . , Y_n) as super-algebra)

· super hyper-algebra hy(G ) = the continuous dual of b_Oe

· super Lie algebra Lie(G ) = P(hy(G ))

Prop-Def The category of super-manifolds has finite products. The group objects in the category are, therefore, defined. They are calledsuper Lie groups.

To every super Lie group G = (G , O) associated are

· Lie group ^Gred= (G , Ored⁾

· complete super Hopf algebra O^be

(≃ k[[X1, . . . , X_{m]] ⊗ ∧(Y1}, . . . , Y_n) as super-algebra)

· super hyper-algebra hy(G ) = the continuous dual of b_Oe

· super Lie algebra Lie(G ) = P(hy(G )) G recovers

◮ roughly fromG_red and Lie(G ),

◮ precisely from the Harish-Chandra pair.

5. Harish-Chandra pairs

All (super) module over Lie groups are assumed to be analytic.

5. Harish-Chandra pairs

All (super) module over Lie groups are assumed to be analytic. Def (Kostant) AHarish-Chandra pairis

a pair (F , V ) given a linear mapV _{⊗ V −→ Lie(F )}^{[ , ]} , where F is a Lie group, V is a f.d. right F -module, and [ , ] is F-equivariant and satisfies

[v , v^′] = [v^′, v ], v▹ [v , v ],

5. Harish-Chandra pairs

All (super) module over Lie groups are assumed to be analytic. Def (Kostant) AHarish-Chandra pairis

a pair (F , V ) given a linear mapV _{⊗ V −→ Lie(F )}^{[ , ]} , where F is a Lie group, V is a f.d. right F -module, and [ , ] is F-equivariant and satisfies

[v , v^′] = [v^′, v ], v▹ [v , v ],

where Lie(F ) x F by adjoint, and▹ indicates V x Lie(F ) induced from the given V x F .

Functor (Super Lie groups) → (Harish-Chandra pairs)

Functor (Super Lie groups) → (Harish-Chandra pairs) Let (G , O) ∈ (Super Lie groups).

Functor (Super Lie groups) → (Harish-Chandra pairs) Let (G , O) ∈ (Super Lie groups).

We have the assoc. Lie group Gred with struc. sheaf F := O^red.

Functor (Super Lie groups) → (Harish-Chandra pairs) Let (G , O) ∈ (Super Lie groups).

We have the assoc. Lie group Gred with struc. sheaf F := O^red. The right adjoint action g⁻¹hg ←p (h, g) induces

αg : b_Oe _{→ bO}e _{⊗ b}b _Og

id b⊗( )red

−→ O^be ⊗ b^b F^g.

Functor (Super Lie groups) → (Harish-Chandra pairs) Let (G , O) ∈ (Super Lie groups).

We have the assoc. Lie group Gred with struc. sheaf F := O^red. The right adjoint action g⁻¹hg ←p (h, g) induces

αg : b_Oe _{→ bO}e _{⊗ b}b _Og

id b⊗( )red

−→ O^be ⊗ b^b F^g.

Composed with the proj. b_Oe _{⊗ b}b _Fg _{→ bO}e _{⊗ k = b}b _Oe, there arise super-Hopf-algebra automorphisms βg : b_Oe _{→ bO}e_{, g ∈ G}red^,

Functor (Super Lie groups) → (Harish-Chandra pairs) Let (G , O) ∈ (Super Lie groups).

We have the assoc. Lie group Gred with struc. sheaf F := O^red. The right adjoint action g⁻¹hg ←p (h, g) induces

αg : b_Oe _{→ bO}e _{⊗ b}b _Og

id b⊗( )red

−→ O^be ⊗ b^b F^g.

Composed with the proj. b_Oe _{⊗ b}b _Fg _{→ bO}e _{⊗ k = b}b _Oe, there arise super-Hopf-algebra automorphisms βg : b_Oe _{→ bO}e_{, g ∈ G}red^,

which are dualized to γg : hy(G ) → hy(G ), g ∈ Gred^.

Functor (Super Lie groups) → (Harish-Chandra pairs) Let (G , O) ∈ (Super Lie groups).

We have the assoc. Lie group Gred with struc. sheaf F := O^red. The right adjoint action g⁻¹hg ←p (h, g) induces

αg : b_Oe _{→ bO}e _{⊗ b}b _Og

id b⊗( )red

−→ O^be ⊗ b^b F^g.

Composed with the proj. b_Oe _{⊗ b}b _Fg _{→ bO}e _{⊗ k = b}b _Oe, there arise super-Hopf-algebra automorphisms βg : b_Oe _{→ bO}e_{, g ∈ G}red^,

which are dualized to γg : hy(G ) → hy(G ), g ∈ Gred^.

Prop (1) By γg_{, g ∈ G}red (their restrictions), hy(G )(Lie(G )) is a Hopf-algebra(Lie-algebra) object in (Right super G_red-modules).

Functor (Super Lie groups) → (Harish-Chandra pairs) Let (G , O) ∈ (Super Lie groups).

We have the assoc. Lie group Gred with struc. sheaf F := O^red. The right adjoint action g⁻¹hg ←p (h, g) induces

αg : b_Oe _{→ bO}e _{⊗ b}b _Og

id b⊗( )red

−→ O^be ⊗ b^b F^g.

Composed with the proj. b_Oe _{⊗ b}b _Fg _{→ bO}e _{⊗ k = b}b _Oe, there arise super-Hopf-algebra automorphisms βg : b_Oe _{→ bO}e_{, g ∈ G}red^,

which are dualized to γg : hy(G ) → hy(G ), g ∈ Gred^.

Prop (1) By γg_{, g ∈ G}red (their restrictions), hy(G )(Lie(G )) is a Hopf-algebra(Lie-algebra) object in (Right super G_red-modules). (2) The odd comp. Lie(G )₁ is thus a right G_red-module. The pair (G_red, Lie(G )₁), given the restricted bracket

[ , ] : Lie(G )_{1⊗ Lie(G )1→ Lie(G )0}= Lie(G_red), is a Harish-Chandra pair.

6. Main theorem

Thm G 7→ (Gred^{, Lie(G )}1) gives rise to a functor, (Super Lie groups) → (Harish-Chandra pairs), which is an equivalence.

Rem A main point of the proof is to construct a quasi-inverse, using thesmash-coproduct construction; Hopf-algebraic!

6. Main theorem

Thm G 7→ (Gred^{, Lie(G )}1) gives rise to a functor, (Super Lie groups) → (Harish-Chandra pairs), which is an equivalence.

Rem A main point of the proof is to construct a quasi-inverse, using thesmash-coproduct construction; Hopf-algebraic! Example G = GL(m|n), Gred= GL(m) × GL(n).

Lie(G )_{1= {A =}

(O A^′ A^′′ O

)

| A^{′ m}× n, A^{′′ n}× m} A^γ= γ⁻¹Aγ, γ =

(γ^′ O O γ^′′

)

∈ GL(m) × GL(n) [A, B] = AB + BA

7. The corresponding result in algebraic situation

7. The corresponding result in algebraic situation

Over an arbitrary field of char ̸= 2, an(algebraic) Harish-Chandra pairis defined to be a pair (F , V ) of an algebraic group (scheme) F and a f.d. right F -module V , given a bracket [ , ] : V ⊗ V → Lie(F ) as before.

7. The corresponding result in algebraic situation

Over an arbitrary field of char ̸= 2, an(algebraic) Harish-Chandra pairis defined to be a pair (F , V ) of an algebraic group (scheme) F and a f.d. right F -module V , given a bracket [ , ] : V ⊗ V → Lie(F ) as before.

The analogous category equivalence

(Super algebraic groups) → (Harish-Chandra pairs), was proved by myself, 2012.

7. The corresponding result in algebraic situation

Over an arbitrary field of char ̸= 2, an(algebraic) Harish-Chandra pairis defined to be a pair (F , V ) of an algebraic group (scheme) F and a f.d. right F -module V , given a bracket [ , ] : V ⊗ V → Lie(F ) as before.

The analogous category equivalence

(Super algebraic groups) → (Harish-Chandra pairs), was proved by myself, 2012. The converse construction is valid even for non-algebraic F (M-Shibata).

7. The corresponding result in algebraic situation

Over an arbitrary field of char ̸= 2, an(algebraic) Harish-Chandra pairis defined to be a pair (F , V ) of an algebraic group (scheme) F and a f.d. right F -module V , given a bracket [ , ] : V ⊗ V → Lie(F ) as before.

The analogous category equivalence

(Super algebraic groups) → (Harish-Chandra pairs), was proved by myself, 2012. The converse construction is valid even for non-algebraic F (M-Shibata).

Return to the situation: k is a complete field.

Prop If (F , V ) is an algebraic Harish-Chandra pair, (F (k), V ) is canonically regarded as an analytic Harish-Chandra pair, provided F is smooth (always if char k = 0).

8. Application I — From algebraic to analytic

8. Application I — From algebraic to analytic

Prop There exists a canonical functor

(Smooth super algebraic groups) → (Super Lie groups) which sends the algebraic GL(m|n) to the analytic GL(m|n), and which preserves super hyper-algebras and super Lie algebras.

8. Application I — From algebraic to analytic

Prop There exists a canonical functor

(Smooth super algebraic groups) → (Super Lie groups) which sends the algebraic GL(m|n) to the analytic GL(m|n), and which preserves super hyper-algebras and super Lie algebras. Faithful on the connected ones. Not full. Not onto.

8. Application I — From algebraic to analytic

Prop There exists a canonical functor

(Smooth super algebraic groups) → (Super Lie groups) which sends the algebraic GL(m|n) to the analytic GL(m|n), and which preserves super hyper-algebras and super Lie algebras. Faithful on the connected ones. Not full. Not onto.

Conclusion A matrix presentation to define a super Lie group is understood to indicate the super Lie group which arises from the smooth super algebraic group defined by the matrix presentation. Remark We have supposed “affine” above. But even in

non-affinesituation one can construct an extended functor (Smooth locally algebraic super-groups) → (Super Lie groups), which arises from

(Smooth locally algebraic super-schemes) → (Super-manifolds).

9. Application II — From analytic to algebraic

9. Application II — From analytic to algebraic

Let G ∈ (Super Lie groups).

9. Application II — From analytic to algebraic

Let G ∈ (Super Lie groups).

Def Asuper G-module structureon a super vector-space M is a morphism G → GL(M) = GL(m|n) in (Super Lie groups), where m|n = sdim M.

9. Application II — From analytic to algebraic

Let G ∈ (Super Lie groups).

Def Asuper G-module structureon a super vector-space M is a morphism G → GL(M) = GL(m|n) in (Super Lie groups), where m|n = sdim M.

Analytic interpretation Suppose (F , V ) ∈ (Harish-Chandra pairs) corresponds to G . Giving a super G -module structure as above is the same as giving

9. Application II — From analytic to algebraic

Let G ∈ (Super Lie groups).

Def Asuper G-module structureon a super vector-space M is a morphism G → GL(M) = GL(m|n) in (Super Lie groups), where m|n = sdim M.

Analytic interpretation Suppose (F , V ) ∈ (Harish-Chandra pairs) corresponds to G . Giving a super G -module structure as above is the same as giving

(1) a super F -module structure on M,

9. Application II — From analytic to algebraic

Let G ∈ (Super Lie groups).

Def Asuper G-module structureon a super vector-space M is a morphism G → GL(M) = GL(m|n) in (Super Lie groups), where m|n = sdim M.

Analytic interpretation Suppose (F , V ) ∈ (Harish-Chandra pairs) corresponds to G . Giving a super G -module structure as above is the same as giving

(1) a super F -module structure on M, and (2) a super-linear map V ⊗ M → M, (v, m) 7→ vm

9. Application II — From analytic to algebraic

Let G ∈ (Super Lie groups).

Def Asuper G-module structureon a super vector-space M is a morphism G → GL(M) = GL(m|n) in (Super Lie groups), where m|n = sdim M.

Analytic interpretation Suppose (F , V ) ∈ (Harish-Chandra pairs) corresponds to G . Giving a super G -module structure as above is the same as giving

(1) a super F -module structure on M, and (2) a super-linear map V ⊗ M → M, (v, m) 7→ vm s.t. f(v^fm) = v (fm), v(v^′m) + v^′(vm) = [v , v^′]m.

9. Application II — From analytic to algebraic

Let G ∈ (Super Lie groups).

Def Asuper G-module structureon a super vector-space M is a morphism G → GL(M) = GL(m|n) in (Super Lie groups), where m|n = sdim M.

Analytic interpretation Suppose (F , V ) ∈ (Harish-Chandra pairs) corresponds to G . Giving a super G -module structure as above is the same as giving

(1) a super F -module structure on M, and (2) a super-linear map V ⊗ M → M, (v, m) 7→ vm s.t. f(v^fm) = v (fm), v(v^′m) + v^′(vm) = [v , v^′]m. Algebraic interpretation Let F denote the affine (may not algebraic) group corresponding to the Hopf algebra R(F ) of repres. functions.

9. Application II — From analytic to algebraic

Let G ∈ (Super Lie groups).

Def Asuper G-module structureon a super vector-space M is a morphism G → GL(M) = GL(m|n) in (Super Lie groups), where m|n = sdim M.

Analytic interpretation Suppose (F , V ) ∈ (Harish-Chandra pairs) corresponds to G . Giving a super G -module structure as above is the same as giving

(1) a super F -module structure on M, and (2) a super-linear map V ⊗ M → M, (v, m) 7→ vm s.t. f(v^fm) = v (fm), v(v^′m) + v^′(vm) = [v , v^′]m. Algebraic interpretation Let F denote the affine (may not algebraic) group corresponding to the Hopf algebra R(F ) of repres. functions. Then (F, V ) forms an algebraic Harish-Chandra pair. Giving a super G -mod. structure as above is the same as giving the analogous data (1^′), (2^′) with F replaced by F.

Let G be the super affine group scheme constructed from (F, V ).

Let G be the super affine group scheme constructed from (F, V ). It is the “algebraic closure” of the given super Lie group G .

Let G be the super affine group scheme constructed from (F, V ). It is the “algebraic closure” of the given super Lie group G . Prop (L.f. analytic super G -modules)≈ (Super G-modules)^⊗

(= (Super k[G]-comodules))

Let G be the super affine group scheme constructed from (F, V ). It is the “algebraic closure” of the given super Lie group G . Prop (L.f. analytic super G -modules)≈ (Super G-modules)^⊗

(= (Super k[G]-comodules))

Conclusion The super Hopf algebra k[G] which represents G is the super Hopf algebra R(G ) of representative functions on G .