Lie conformal algebras and the variational complex

Transcription

1 Le conformal algebras and the varatonal complex UMN Math Physcs Semnar February 10, 2014

2 Outlne Varatonal complex: a quck remnder Le conformal algebras and ther cohomology The varatonal complex as a LCA cohomology complex. References: 1 Alberto De Sole, Vctor Kac, Le conformal algebra cohomology and the varatonal complex 2 AS, VK, Alaa Barakat, Posson vertex algebras n the theory of Hamltonan equatons 3 AS, VK, The varatonal Posson cohomology 4 Bojko Bakalov, VK, Alexander Voronov, Cohomology of conformal algebras UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

3 Varatonal complex Recall: f u(x) = (u 1 (x), u 2 (x),..., u k (x)) s an extremum of a varatonal functonal L[u] = I f(x, u, u, u,... ) dx, then δf δu := n ( ) n f δf δu := = 0, = d dx δf δu 1... δf δu k s the Euler-Lagrange varatonal dervatve of f. ( = 1, 2,..., k) UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

4 Varatonal complex Recall: f u(x) = (u 1 (x), u 2 (x),..., u k (x)) s an extremum of a varatonal functonal L[u] = I f(x, u, u, u,... ) dx, then δf δu := n ( ) n f δf δu := = 0, = d dx δf δu 1... δf δu k s the Euler-Lagrange varatonal dervatve of f. ( = 1, 2,..., k) UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

5 Varatonal complex One easly checks that [ and, n partcular,, ] = δ (f) = 0. δu u (n 1) Example Takng f(u) = u2 2, ( δ δu (uu ) = u ) u +... (uu ) = u + (u) = 0 UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

6 Varatonal complex One easly checks that [ and, n partcular,, ] = δ (f) = 0. δu u (n 1) Example Takng f(u) = u2 2, ( δ δu (uu ) = u ) u +... (uu ) = u + (u) = 0 UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

7 Varatonal complex Q: Gven that δ δu (g) = 0, s g = f for some f? A: Yes, n some cases (for example, f f, g are polynomals n u, u, u,... ). δ Problem I: Descrbe the kernel of δu : V Vk for a more general space of functons V. δ Problem II: Descrbe the mage of δu n Vk. F 1 Theorem (Helmholtz): If F =... Im ( δ δu), then F k D F () = DF (), where the Frechet dervatve D F : V k V k s defned by (D F ()) j = n F u j n UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

8 Varatonal complex Q: Gven that δ δu (g) = 0, s g = f for some f? A: Yes, n some cases (for example, f f, g are polynomals n u, u, u,... ). δ Problem I: Descrbe the kernel of δu : V Vk for a more general space of functons V. δ Problem II: Descrbe the mage of δu n Vk. F 1 Theorem (Helmholtz): If F =... Im ( δ δu), then F k D F () = DF (), where the Frechet dervatve D F : V k V k s defned by (D F ()) j = n F u j n UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

9 Varatonal complex Q: Gven that δ δu (g) = 0, s g = f for some f? A: Yes, n some cases (for example, f f, g are polynomals n u, u, u,... ). δ Problem I: Descrbe the kernel of δu : V Vk for a more general space of functons V. δ Problem II: Descrbe the mage of δu n Vk. F 1 Theorem (Helmholtz): If F =... Im ( δ δu), then F k D F () = DF (), where the Frechet dervatve D F : V k V k s defned by (D F ()) j = n F u j n UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

10 Varatonal complex Puttng all that together, we obtan an exact sequence V/V u V k D() D () { Skew-symmetrc k-by-k dfferental matrces } These are the frst few terms of a larger varatonal complex. A geometrc approach (I.Anderson, W.Tulczyjew, A.Vnogradov,... ): thnk of the derham complex on the nfnte jet space of a bundle. A more algebrac approach (I.Gelfand, L.Dckey, I.Dorfman,... ): explot varous algebrac structures on the formal derham complex assocated wth V. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

11 Varatonal complex Puttng all that together, we obtan an exact sequence V/V u V k D() D () { Skew-symmetrc k-by-k dfferental matrces } These are the frst few terms of a larger varatonal complex. A geometrc approach (I.Anderson, W.Tulczyjew, A.Vnogradov,... ): thnk of the derham complex on the nfnte jet space of a bundle. A more algebrac approach (I.Gelfand, L.Dckey, I.Dorfman,... ): explot varous algebrac structures on the formal derham complex assocated wth V. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

12 Varatonal complex Puttng all that together, we obtan an exact sequence V/V u V k D() D () { Skew-symmetrc k-by-k dfferental matrces } These are the frst few terms of a larger varatonal complex. A geometrc approach (I.Anderson, W.Tulczyjew, A.Vnogradov,... ): thnk of the derham complex on the nfnte jet space of a bundle. A more algebrac approach (I.Gelfand, L.Dckey, I.Dorfman,... ): explot varous algebrac structures on the formal derham complex assocated wth V. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

13 Varatonal complex Defnton An algebra of dfferental functons V n the varables u 1,..., u k s a untal, commutatve, assocatve algebra equpped wth a dervaton : V V mutually commutng dervatons [ ] such that, and many s and n s. Example = u (n 1) f : V V, n Z + = 0 for all but fntely The polynomal algebra R k = C[u (n) ] =1,2,...,k;n Z+ wth := u (n+1) and usual dervatons. R k [x] wth (x) := 1 R k [log u, u 1 ] =1,2,...,k UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

14 Varatonal complex Defnton An algebra of dfferental functons V n the varables u 1,..., u k s a untal, commutatve, assocatve algebra equpped wth a dervaton : V V mutually commutng dervatons [ ] such that, and many s and n s. Example = u (n 1) f : V V, n Z + = 0 for all but fntely The polynomal algebra R k = C[u (n) ] =1,2,...,k;n Z+ wth := u (n+1) and usual dervatons. R k [x] wth (x) := 1 R k [log u, u 1 ] =1,2,...,k UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

15 Varatonal complex Defnton An algebra of dfferental functons V n the varables u 1,..., u k s a untal, commutatve, assocatve algebra equpped wth a dervaton : V V mutually commutng dervatons [ ] such that, and many s and n s. Example = u (n 1) f : V V, n Z + = 0 for all but fntely The polynomal algebra R k = C[u (n) ] =1,2,...,k;n Z+ wth := u (n+1) and usual dervatons. R k [x] wth (x) := 1 R k [log u, u 1 ] =1,2,...,k UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

16 Varatonal complex Defnton An algebra of dfferental functons V n the varables u 1,..., u k s a untal, commutatve, assocatve algebra equpped wth a dervaton : V V mutually commutng dervatons [ ] such that, and many s and n s. Example = u (n 1) f : V V, n Z + = 0 for all but fntely The polynomal algebra R k = C[u (n) ] =1,2,...,k;n Z+ wth := u (n+1) and usual dervatons. R k [x] wth (x) := 1 R k [log u, u 1 ] =1,2,...,k UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

17 Varatonal complex Defnton An algebra of dfferental functons V n the varables u 1,..., u k s a untal, commutatve, assocatve algebra equpped wth a dervaton : V V mutually commutng dervatons [ ] such that, and many s and n s. Example = u (n 1) f : V V, n Z + = 0 for all but fntely The polynomal algebra R k = C[u (n) ] =1,2,...,k;n Z+ wth := u (n+1) and usual dervatons. R k [x] wth (x) := 1 R k [log u, u 1 ] =1,2,...,k UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

18 Varatonal complex Defnton An algebra of dfferental functons V n the varables u 1,..., u k s a untal, commutatve, assocatve algebra equpped wth a dervaton : V V mutually commutng dervatons [ ] such that, and many s and n s. Example = u (n 1) f : V V, n Z + = 0 for all but fntely The polynomal algebra R k = C[u (n) ] =1,2,...,k;n Z+ wth := u (n+1) and usual dervatons. R k [x] wth (x) := 1 R k [log u, u 1 ] =1,2,...,k UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

19 Varatonal complex Defnton An algebra of dfferental functons V n the varables u 1,..., u k s a untal, commutatve, assocatve algebra equpped wth a dervaton : V V mutually commutng dervatons [ ] such that, and many s and n s. Example = u (n 1) f : V V, n Z + = 0 for all but fntely The polynomal algebra R k = C[u (n) ] =1,2,...,k;n Z+ wth := u (n+1) and usual dervatons. R k [x] wth (x) := 1 R k [log u, u 1 ] =1,2,...,k UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

20 Varatonal complex For an algebra of dfferental functons V n varables u 1,..., u k, we defne the varatonal dervatve δ δu : V Vk by δ δu (f) := n ( ) n f The derham complex over V s the free untal graded-commutatve assocatve algebra Ω (V) wth generators δu (n) of degree 1 ( = 1, 2,..., k, n Z + ). The dfferental δ on Ω (V) s defned by takng δ(f) := =1,2,...,k;n Z + f. δu (n) on f V and extendng further by the (graded) Lebnz rule. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

21 Varatonal complex For an algebra of dfferental functons V n varables u 1,..., u k, we defne the varatonal dervatve δ δu : V Vk by δ δu (f) := n ( ) n f The derham complex over V s the free untal graded-commutatve assocatve algebra Ω (V) wth generators δu (n) of degree 1 ( = 1, 2,..., k, n Z + ). The dfferental δ on Ω (V) s defned by takng δ(f) := =1,2,...,k;n Z + f. δu (n) on f V and extendng further by the (graded) Lebnz rule. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

22 Varatonal complex For an algebra of dfferental functons V n varables u 1,..., u k, we defne the varatonal dervatve δ δu : V Vk by δ δu (f) := n ( ) n f The derham complex over V s the free untal graded-commutatve assocatve algebra Ω (V) wth generators δu (n) of degree 1 ( = 1, 2,..., k, n Z + ). The dfferental δ on Ω (V) s defned by takng δ(f) := =1,2,...,k;n Z + f. δu (n) on f V and extendng further by the (graded) Lebnz rule. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

23 Varatonal complex Ω 0 (V)= V δ Ω 1 (V)=,n Vδu (n)... The C[]-acton on V = Ω 0 (V) can be extended to the entre complex Ω as an even dervaton and va (δu (n) ) = δu (n+1). Lemma commutes wth the dfferental δ. Defnton The varatonal complex over V s the quotent complex Ω (V) = Ω (V)/ Ω (V). UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

24 Varatonal complex Ω 0 (V)= V δ Ω 1 (V)=,n Vδu (n)... The C[]-acton on V = Ω 0 (V) can be extended to the entre complex Ω as an even dervaton and va (δu (n) ) = δu (n+1). Lemma commutes wth the dfferental δ. Defnton The varatonal complex over V s the quotent complex Ω (V) = Ω (V)/ Ω (V). UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

25 Varatonal complex Ω 0 (V)= V δ Ω 1 (V)=,n Vδu (n)... The C[]-acton on V = Ω 0 (V) can be extended to the entre complex Ω as an even dervaton and va (δu (n) ) = δu (n+1). Lemma commutes wth the dfferental δ. Defnton The varatonal complex over V s the quotent complex Ω (V) = Ω (V)/ Ω (V). UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

26 Varatonal complex Ω 0 (V)= V/V δ Ω 1 (V)= V k... 1 What s the cohomology of Ω (V)? 2 Do cochans and the dfferental operators of Ω (V) have a tangble nterpretaton? An algebra of dfferental functons V admts a fltraton by V n, = {f V f u (m) j = 0 for all(m, j) > (n, )} Theorem An algebra V s sad to be normal f If V s normal, then Ω (V) s exact. V n, = V n,. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

27 Varatonal complex Ω 0 (V)= V/V δ Ω 1 (V)= V k... 1 What s the cohomology of Ω (V)? 2 Do cochans and the dfferental operators of Ω (V) have a tangble nterpretaton? An algebra of dfferental functons V admts a fltraton by V n, = {f V f u (m) j = 0 for all(m, j) > (n, )} Theorem An algebra V s sad to be normal f If V s normal, then Ω (V) s exact. V n, = V n,. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

28 Varatonal complex Ω 0 (V)= V/V δ Ω 1 (V)= V k... 1 What s the cohomology of Ω (V)? 2 Do cochans and the dfferental operators of Ω (V) have a tangble nterpretaton? An algebra of dfferental functons V admts a fltraton by V n, = {f V f u (m) j = 0 for all(m, j) > (n, )} Theorem An algebra V s sad to be normal f If V s normal, then Ω (V) s exact. V n, = V n,. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

29 Varatonal complex Ω 0 (V)= V/V δ Ω 1 (V)= V k... 1 What s the cohomology of Ω (V)? 2 Do cochans and the dfferental operators of Ω (V) have a tangble nterpretaton? An algebra of dfferental functons V admts a fltraton by V n, = {f V f u (m) j = 0 for all(m, j) > (n, )} Theorem An algebra V s sad to be normal f If V s normal, then Ω (V) s exact. V n, = V n,. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

30 Varatonal complex Theorem Let V be an algebra of dfferental functons on k varables u 1,..., u k. Then for p 2, Ω r (V) = { r V k V/V}. The dfferental δ acts on S Ω r (V) va r+1 δs(p 1,..., P r+1 ) = ( 1) j+1 (X P js)(p 1,..., ˆP j,..., P r+1 ) j=0 where X P s a dervaton (an evolutonary vector feld ) defned as X P := ( n P ).,n UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

31 Varatonal complex As a corollary, f an algebra V s normal, then δf δu = 0 ff f = g + const F Im ( δ δu) ff DF () DF () = 0. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

32 Posson vertex and Le conformal algebras Let V be a C[]-module, where acts as a dervaton. A λ-bracket on V s a C-lnear map V V C[λ] V denoted by f g {f λ g} subject to sesqulnearty: {f λ g} = λ{f λ g}, {f λ g} = ( + λ)λ{f λ g}; left and rght Lebnz rules (f V has a structure of a untal assocatve algebra); Jacob dentty: {{f λ g} µ h} + {g µ {f λ h}} = {{f λ g} µ+λ h} Defnton A Posson vertex algebra W s a dfferental algebra wth a skew-symmetrc λ-bracket. A Le conformal algebra V s a C[]-module wth a wth a skew-symmetrc λ-bracket. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

33 Posson vertex and Le conformal algebras Let V be a C[]-module, where acts as a dervaton. A λ-bracket on V s a C-lnear map V V C[λ] V denoted by f g {f λ g} subject to sesqulnearty: {f λ g} = λ{f λ g}, {f λ g} = ( + λ)λ{f λ g}; left and rght Lebnz rules (f V has a structure of a untal assocatve algebra); Jacob dentty: {{f λ g} µ h} + {g µ {f λ h}} = {{f λ g} µ+λ h} Defnton A Posson vertex algebra W s a dfferental algebra wth a skew-symmetrc λ-bracket. A Le conformal algebra V s a C[]-module wth a wth a skew-symmetrc λ-bracket. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

34 Posson vertex and Le conformal algebras Let V be a C[]-module, where acts as a dervaton. A λ-bracket on V s a C-lnear map V V C[λ] V denoted by f g {f λ g} subject to sesqulnearty: {f λ g} = λ{f λ g}, {f λ g} = ( + λ)λ{f λ g}; left and rght Lebnz rules (f V has a structure of a untal assocatve algebra); Jacob dentty: {{f λ g} µ h} + {g µ {f λ h}} = {{f λ g} µ+λ h} Defnton A Posson vertex algebra W s a dfferental algebra wth a skew-symmetrc λ-bracket. A Le conformal algebra V s a C[]-module wth a wth a skew-symmetrc λ-bracket. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

35 Posson vertex and Le conformal algebras Let V be a C[]-module, where acts as a dervaton. A λ-bracket on V s a C-lnear map V V C[λ] V denoted by f g {f λ g} subject to sesqulnearty: {f λ g} = λ{f λ g}, {f λ g} = ( + λ)λ{f λ g}; left and rght Lebnz rules (f V has a structure of a untal assocatve algebra); Jacob dentty: {{f λ g} µ h} + {g µ {f λ h}} = {{f λ g} µ+λ h} Defnton A Posson vertex algebra W s a dfferental algebra wth a skew-symmetrc λ-bracket. A Le conformal algebra V s a C[]-module wth a wth a skew-symmetrc λ-bracket. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

36 Cohomology of Le conformal algebras Let V be a Le conformal algebra and M be an V-module. The space of k-cochans of V wth coeffcents n M s the space Γ(V, M) of C-lnear maps such that γ : A k C[λ 1,..., λ k ] M γ(a 1,..., a,..., a k ) = λ γ(a 1,..., a,..., a k ) γ s skew-symmetrc w.r.t. smultaneous permutatons of a s and λ s. The dfferental Γk (V, M) Γ k+1 (V, M) s defned by k+1 δ γ(a 1,..., a k+1 ) = ( 1) +1 (... ) + }{{} =1 Exercse k+1,j=1 <j ( 1) k++j+1 (... ) }{{} Exercse UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

37 Cohomology of Le conformal algebras Let V be a Le conformal algebra and M be an V-module. The space of k-cochans of V wth coeffcents n M s the space Γ(V, M) of C-lnear maps such that γ : A k C[λ 1,..., λ k ] M γ(a 1,..., a,..., a k ) = λ γ(a 1,..., a,..., a k ) γ s skew-symmetrc w.r.t. smultaneous permutatons of a s and λ s. The dfferental Γk (V, M) Γ k+1 (V, M) s defned by k+1 δ γ(a 1,..., a k+1 ) = ( 1) +1 (... ) + }{{} =1 Exercse k+1,j=1 <j ( 1) k++j+1 (... ) }{{} Exercse UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

38 Cohomology of Le conformal algebras Let V be a Le conformal algebra and M be an V-module. The space of k-cochans of V wth coeffcents n M s the space Γ(V, M) of C-lnear maps such that γ : A k C[λ 1,..., λ k ] M γ(a 1,..., a,..., a k ) = λ γ(a 1,..., a,..., a k ) γ s skew-symmetrc w.r.t. smultaneous permutatons of a s and λ s. The dfferental Γk (V, M) Γ k+1 (V, M) s defned by k+1 δ γ(a 1,..., a k+1 ) = ( 1) +1 (... ) + }{{} =1 Exercse k+1,j=1 <j ( 1) k++j+1 (... ) }{{} Exercse UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

39 Cohomology of Le conformal algebras The complex Γ(V, M) can be gven a C[]-acton va ( γ)(a 1,..., a k ) = ( M + λ λ k )( γ(a 1,..., a k )) Lemma and δ on Γ(V, M) commute. Hence, the reduced cohomology complex Γ(V, M) := Γ(V, M)/ Γ(V, M) s well-defned. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

40 Cohomology of Le conformal algebras The complex Γ(V, M) can be gven a C[]-acton va ( γ)(a 1,..., a k ) = ( M + λ λ k )( γ(a 1,..., a k )) Lemma and δ on Γ(V, M) commute. Hence, the reduced cohomology complex Γ(V, M) := Γ(V, M)/ Γ(V, M) s well-defned. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

41 Cohomology of Le conformal algebras Lemma Let V be an algebra of dfferental functons n varables (u ). Consder the Le conformal algebra A = C[]u wth the zero bracket. Then V s a module over A wth the acton gven by u λ f := n λ n f. Theorem The varatonal complex Ω (V) s somorphc to the reduced cohomology complex Γ (A, V). The cochans of Γ (A, V) can be descrbed rather explctly n terms of the poly-λ-brackets. Ths descrpton passed to Ω (V) gves us the structure theorem for Ω (V) dscussed above. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex

42 Cohomology of Le conformal algebras Lemma Let V be an algebra of dfferental functons n varables (u ). Consder the Le conformal algebra A = C[]u wth the zero bracket. Then V s a module over A wth the acton gven by u λ f := n λ n f. Theorem The varatonal complex Ω (V) s somorphc to the reduced cohomology complex Γ (A, V). The cochans of Γ (A, V) can be descrbed rather explctly n terms of the poly-λ-brackets. Ths descrpton passed to Ω (V) gves us the structure theorem for Ω (V) dscussed above. UMN Math Physcs Semnar Le conformal algebras and the varatonal complex