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Linear Aspects of Quasigroup Triality

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Linear Aspects of Quasigroup Triality Linear aspects of quasigroup triality Alex W. Nowak Iowa State University Dissertation Defense April 13, 2020 A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Summary 1 Quasigroups and triality 2 Linear quasigroup theory 3 Modules over Mendelsohn triple systems 4 Abelian groups in MTS 5 Beyond set-theoretic triality A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Quasigroups and triality A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: a combinatorial perspective (Q; ·) is a quasigroup when T = f(x; y; x · y) j (x; y) 2 Q2g has the Latin square property: 8(x1; x2; x3); (y1; y2; y3) 2 T; jf1 ≤ i ≤ 3 j xi = yigj 6= 2. g For all g 2 S3, T = f(x1g; x2g; x3g) j (x1; x2; x3) 2 T g also has the Latin square property. If H is a subgroup of the kernel of this permutation action, then T is H-symmetric. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: a combinatorial perspective (Q; ·) is a quasigroup when T = f(x; y; x · y) j (x; y) 2 Q2g has the Latin square property: 8(x1; x2; x3); (y1; y2; y3) 2 T; jf1 ≤ i ≤ 3 j xi = yigj 6= 2. g For all g 2 S3, T = f(x1g; x2g; x3g) j (x1; x2; x3) 2 T g also has the Latin square property. If H is a subgroup of the kernel of this permutation action, then T is H-symmetric. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: a combinatorial perspective (Q; ·) is a quasigroup when T = f(x; y; x · y) j (x; y) 2 Q2g has the Latin square property: 8(x1; x2; x3); (y1; y2; y3) 2 T; jf1 ≤ i ≤ 3 j xi = yigj 6= 2. g For all g 2 S3, T = f(x1g; x2g; x3g) j (x1; x2; x3) 2 T g also has the Latin square property. If H is a subgroup of the kernel of this permutation action, then T is H-symmetric. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: an algebraic perspective (Q; ·; =; n) is a quasigroup when all the following hold: (IL) yn(y · x) = x; (IR) x = (x · y)=y; (SL) y · (ynx) = x; (SR) x = (x=y) · y: (12) (Q; ·; =; n) o / (Q; ◦; nn; ==) (23) 3; ck (23) s{ #+ (Q; n; ==; ·) (Q; nn; ◦; =) g 7 (12) (12) ' w (Q; ==; n; ◦) ks +3 (Q; =; ·; nn) (23) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Triality: an algebraic perspective (Q; ·; =; n) is a quasigroup when all the following hold: (IL) yn(y · x) = x; (IR) x = (x · y)=y; (SL) y · (ynx) = x; (SR) x = (x=y) · y: (12) (Q; ·; =; n) o / (Q; ◦; nn; ==) (23) 3; ck (23) s{ #+ (Q; n; ==; ·) (Q; nn; ◦; =) g 7 (12) (12) ' w (Q; ==; n; ◦) ks +3 (Q; =; ·; nn) (23) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Multiplication groups (Q; ·) is a quasigroup iff R(q): x 7! x · q and L(q): x 7! q · x are bijections. The group Mlt(Q) = hR(q);L(q)iSQ acts transitively on Q. ` Q is a subquasigroup of Q[X] = Q VhXiV and the subgroup of Mlt(Q[X]) generated by R(Q) [ L(Q) is the universal multiplication group of Q in V, U(Q; V). A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Multiplication groups (Q; ·) is a quasigroup iff R(q): x 7! x · q and L(q): x 7! q · x are bijections. The group Mlt(Q) = hR(q);L(q)iSQ acts transitively on Q. ` Q is a subquasigroup of Q[X] = Q VhXiV and the subgroup of Mlt(Q[X]) generated by R(Q) [ L(Q) is the universal multiplication group of Q in V, U(Q; V). A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Multiplication groups (Q; ·) is a quasigroup iff R(q): x 7! x · q and L(q): x 7! q · x are bijections. The group Mlt(Q) = hR(q);L(q)iSQ acts transitively on Q. ` Q is a subquasigroup of Q[X] = Q VhXiV and the subgroup of Mlt(Q[X]) generated by R(Q) [ L(Q) is the universal multiplication group of Q in V, U(Q; V). A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Universal stabilizers U(Q; V) also acts transitively on Q, so for any e 2 Q, define U(Q; V)e to be the universal stabilizer of Q in V. rq L(rq=e) 7 g L(r) L(q=e) e 3+ q R(enq) R(enqr) R(r) ' qr w −1 −1 Define Te(q) = R(enq)L(q=e) , Re(q; r) = R(enq)R(r)R(enqr) , −1 and Le(q; r) = L(q=e)L(r)L(rq=e) −1 U(Q; V)e will act on the fiber p feg in a split extension p : E ! Q. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Universal stabilizers U(Q; V) also acts transitively on Q, so for any e 2 Q, define U(Q; V)e to be the universal stabilizer of Q in V. rq L(rq=e) 7 g L(r) L(q=e) e 3+ q R(enq) R(enqr) R(r) ' qr w −1 −1 Define Te(q) = R(enq)L(q=e) , Re(q; r) = R(enq)R(r)R(enqr) , −1 and Le(q; r) = L(q=e)L(r)L(rq=e) −1 U(Q; V)e will act on the fiber p feg in a split extension p : E ! Q. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Universal stabilizers U(Q; V) also acts transitively on Q, so for any e 2 Q, define U(Q; V)e to be the universal stabilizer of Q in V. rq L(rq=e) 7 g L(r) L(q=e) e 3+ q R(enq) R(enqr) R(r) ' qr w −1 −1 Define Te(q) = R(enq)L(q=e) , Re(q; r) = R(enq)R(r)R(enqr) , −1 and Le(q; r) = L(q=e)L(r)L(rq=e) −1 U(Q; V)e will act on the fiber p feg in a split extension p : E ! Q. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality Universal stabilizers U(Q; V) also acts transitively on Q, so for any e 2 Q, define U(Q; V)e to be the universal stabilizer of Q in V. rq L(rq=e) 7 g L(r) L(q=e) e 3+ q R(enq) R(enqr) R(r) ' qr w −1 −1 Define Te(q) = R(enq)L(q=e) , Re(q; r) = R(enq)R(r)R(enqr) , −1 and Le(q; r) = L(q=e)L(r)L(rq=e) −1 U(Q; V)e will act on the fiber p feg in a split extension p : E ! Q. A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Quasigroups and triality H-symmetry classes Q LS RS C P TS A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory Linear quasigroup theory A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory H-symmetry: modules as models Q: ZhR; Li-modules −1 −1 −1 −1 x · y = xR + yL, x=y = xR − yLR , xny = yL − xRL ±1 C(xy = yx): Z[R ]-modules −1 −1 x · y = (x + y)R, x=y = xR − y, xny = yR − x ±1 LS(y · yx = x): Z[R ]-modules −1 x · y = xR − y = xny, x=y = (x + y)R , 3 P(y · xy = x): Z[R]=(R + 1)-modules −1 −1 x · y = xR + yR , x=y = xR + yR = xny TS: Z-modules x · y = x=y = xny = −(x + y) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory H-symmetry: modules as models Q: ZhR; Li-modules −1 −1 −1 −1 x · y = xR + yL, x=y = xR − yLR , xny = yL − xRL ±1 C(xy = yx): Z[R ]-modules −1 −1 x · y = (x + y)R, x=y = xR − y, xny = yR − x ±1 LS(y · yx = x): Z[R ]-modules −1 x · y = xR − y = xny, x=y = (x + y)R , 3 P(y · xy = x): Z[R]=(R + 1)-modules −1 −1 x · y = xR + yR , x=y = xR + yR = xny TS: Z-modules x · y = x=y = xny = −(x + y) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory H-symmetry: modules as models Q: ZhR; Li-modules −1 −1 −1 −1 x · y = xR + yL, x=y = xR − yLR , xny = yL − xRL ±1 C(xy = yx): Z[R ]-modules −1 −1 x · y = (x + y)R, x=y = xR − y, xny = yR − x ±1 LS(y · yx = x): Z[R ]-modules −1 x · y = xR − y = xny, x=y = (x + y)R , 3 P(y · xy = x): Z[R]=(R + 1)-modules −1 −1 x · y = xR + yR , x=y = xR + yR = xny TS: Z-modules x · y = x=y = xny = −(x + y) A. W. Nowak (ISU) Quasigroup Triality April 13, 2020 Linear quasigroup theory H-symmetry: modules as models Q: ZhR; Li-modules −1 −1 −1 −1 x · y = xR + yL, x=y = xR − yLR , xny = yL − xRL ±1 C(xy = yx): Z[R ]-modules −1 −1 x · y = (x + y)R, x=y = xR − y, xny = yR − x ±1 LS(y · yx = x): Z[R ]-modules −1 x · y = xR − y = xny, x=y = (x + y)R , 3 P(y · xy = x): Z[R]=(R + 1)-modules −1 −1 x · y = xR + yR , x=y = xR + yR = xny TS: Z-modules x · y = x=y = xny = −(x + y) A.
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