n-Örgü grubunun burav temsiline benzer temsilleri
Brau-like representation of the n-braid group
- Tez No: 46398
- Danışmanlar: PROF.DR. METİN ARIK
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Gruplar, Groups
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
/>>=/> +i >(9) e|n+->=e>+->(10) g'\n+l>=g'n+l\n + ±>(11) &'|n>=6'|n + l>(12) 1n *^> s g\n+±>=gn+l\n + l>(13) d\n>=dn\n>(14) where |n > and |n + | > belong to linear spaces V and W such that dimV - dimW and a,b,b',d:V->V f,g:W^V f',g':V-+W e:W^W. These definitions should satisfy the 17 equations arising from the relation (4). If öne applies (6-14) for 17 equations the relations among the elements of B are found as (l-O(l-O = (l-«M,)(l-<+1) = 3,(15) *n+en+dn^=2-q,(16) /n/: = (i-O(i-O,(17) gn+lg'n+l = (i - ^+1)(ı - O,(ıs) M'n = (l-<*n+1)(l-O,(19) f'nbngn+l = fnb'ng'n+l = («" - l)(en - 1)«+1 - 1).(20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hennitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (l - q)B + q are satisfied. For unitary B, this relation linearizes as B = (l - q) + qB^. Using this we show that there is no ünite dimensional unitary representation. However for the pseudo unitary case where pl Gj3i = G finite dimensional representations exits and we construct such a representation with a diagonal G with Gz = I. Example. For a real T = TO, let an > l be chosen. Then ali en 's will be viü
Özet (Çeviri)
/>>=/> + i > (9) e|n+->=e>+-> (10) g'\n+l>=g'n+l\n + ±> (11) b'\n>=b'n\n + l> (12) <7|n+->=gn+1|n + l> (13) d\n >= dn \n > (14) where \n > and \n + | > belong to linear spaces V and W such that dimV = dimW and a,b,b',d:V->V f,g:W-+V f',g':V-+W These definitions should satisfy the 17 equations arising from the relation (4). If one applies (6-14) for 17 equations the relations among the elements of B axe found as (i-0(i-0 = (i-«.+x)(i-<U) = 3, (15) an+^+dn+1=2-q, (16) /“/I = (l-0(l-en), (17) gn+lg'n+l = (i - <*n+1)(i - O, (is) M>(i-<*.+i)(i-0, (19) = («. ”l)(e. - 1)«+1 - 1). (20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hermitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (1 - q)B + q are satisfied. For unitary B, this relation linearizes as B - (1 - q) + qB^. Using this we show that there is no finite dimensional unitary representation. However for the pseudo unitary case where PjGfii = G finite dimensional representations exits and we construct such a representation with a diagonal G with G2 = I. Example. For a real r = r0, let an > 1 be chosen. Then all en 's will be viii/>>=/> +i >(9) e|n+->=e>+->(10) g'\n+l>=g'n+l\n + ±>(11) &'|n>=6'|n + l>(12) 1n *^> s g\n+±>=gn+l\n + l>(13) d\n>=dn\n>(14) where |n > and |n + | > belong to linear spaces V and W such that dimV - dimW and a,b,b',d:V->V f,g:W^V f',g':V-+W e:W^W. These definitions should satisfy the 17 equations arising from the relation (4). If öne applies (6-14) for 17 equations the relations among the elements of B are found as (l-O(l-O = (l-«M,)(l-<+1) = 3,(15) *n+en+dn^=2-q,(16) /n/: = (i-O(i-O,(17) gn+lg'n+l = (i - ^+1)(ı - O,(ıs) M'n = (l-<*n+1)(l-O,(19) f'nbngn+l = fnb'ng'n+l = («“ - l)(en - 1)«+1 - 1).(20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hennitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (l - q)B + q are satisfied. For unitary B, this relation linearizes as B = (l - q) + qB^. Using this we show that there is no ünite dimensional unitary representation. However for the pseudo unitary case where pl Gj3i = G finite dimensional representations exits and we construct such a representation with a diagonal G with Gz = I. Example. For a real T = TO, let an > l be chosen. Then ali en 's will be viü/>>=/> + i > (9) e|n+->=e>+-> (10) g'\n+l>=g'n+l\n + ±> (11) b'\n>=b'n\n + l> (12) <7|n+->=gn+1|n + l> (13) d\n >= dn \n > (14) where \n > and \n + | > belong to linear spaces V and W such that dimV = dimW and a,b,b',d:V->V f,g:W-+V f',g':V-+W These definitions should satisfy the 17 equations arising from the relation (4). If one applies (6-14) for 17 equations the relations among the elements of B axe found as (i-0(i-0 = (i-«.+x)(i-<U) = 3, (15) an+^+dn+1=2-q, (16) /”/I = (l-0(l-en), (17) gn+lg'n+l = (i - <*n+1)(i - O, (is) M>(i-<*.+i)(i-0, (19) = («.“ l)(e. - 1)«+1 - 1). (20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hermitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (1 - q)B + q are satisfied. For unitary B, this relation linearizes as B - (1 - q) + qB^. Using this we show that there is no finite dimensional unitary representation. However for the pseudo unitary case where PjGfii = G finite dimensional representations exits and we construct such a representation with a diagonal G with G2 = I. Example. For a real r = r0, let an > 1 be chosen. Then all en 's will be viii/>>=/> +i >(9) e|n+->=e>+->(10) g'\n+l>=g'n+l\n + ±>(11) &'|n>=6'|n + l>(12) 1n *^> s g\n+±>=gn+l\n + l>(13) d\n>=dn\n>(14) where |n > and |n + | > belong to linear spaces V and W such that dimV - dimW and a,b,b',d:V->V f,g:W^V f',g':V-+W e:W^W. These definitions should satisfy the 17 equations arising from the relation (4). If öne applies (6-14) for 17 equations the relations among the elements of B are found as (l-O(l-O = (l-«M,)(l-<+1) = 3,(15) *n+en+dn^=2-q,(16) /n/: = (i-O(i-O,(17) gn+lg'n+l = (i - ^+1)(ı - O,(ıs) M'n = (l-<*n+1)(l-O,(19) f'nbngn+l = fnb'ng'n+l = («”- l)(en - 1)«+1 - 1).(20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hennitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (l - q)B + q are satisfied. For unitary B, this relation linearizes as B = (l - q) + qB^. Using this we show that there is no ünite dimensional unitary representation. However for the pseudo unitary case where pl Gj3i = G finite dimensional representations exits and we construct such a representation with a diagonal G with Gz = I. Example. For a real T = TO, let an > l be chosen. Then ali en 's will be viü/>>=/> + i > (9) e|n+->=e>+-> (10) g'\n+l>=g'n+l\n + ±> (11) b'\n>=b'n\n + l> (12) <7|n+->=gn+1|n + l> (13) d\n >= dn \n > (14) where \n > and \n + | > belong to linear spaces V and W such that dimV = dimW and a,b,b',d:V->V f,g:W-+V f',g':V-+W These definitions should satisfy the 17 equations arising from the relation (4). If one applies (6-14) for 17 equations the relations among the elements of B axe found as (i-0(i-0 = (i-«.+x)(i-<U) = 3, (15) an+^+dn+1=2-q, (16) /“/I = (l-0(l-en), (17) gn+lg'n+l = (i - <*n+1)(i - O, (is) M>(i-<*.+i)(i-0, (19) = («. ”l)(e. - 1)«+1 - 1). (20) Now, we try to find a representation in the case where B is unitary, i.e. BB^ = B^B = I where B^ is hermitean conjugate of B. Then (15-20) show that the Hecke algebra relation B2 = (1 - q)B + q are satisfied. For unitary B, this relation linearizes as B - (1 - q) + qB^. Using this we show that there is no finite dimensional unitary representation. However for the pseudo unitary case where PjGfii = G finite dimensional representations exits and we construct such a representation with a diagonal G with G2 = I. Example. For a real r = r0, let an > 1 be chosen. Then all en 's will be viii
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