New knowls in the catalogue batch

Core idea

This batch creates 678 catalogue knowls and navigation pages, including this list. Existing canonical knowls are reused by many additional object records; the object catalogue includes both new and reused entries.

Open the category explorer · Catalogue overview

The separate orders and fractional ideals additions have their own list and are not counted here.

Algebras (104)

• Complex Albert algebra
• Complexification of C
• Complexification of H
• Complexification of O
• Full-matrix Jordan algebra M_1(C)^+
• Full-matrix Jordan algebra M_1(R)^+
• Full-matrix Jordan algebra M_2(C)^+
• Full-matrix Jordan algebra M_2(R)^+
• Full-matrix Jordan algebra M_3(C)^+
• Full-matrix Jordan algebra M_3(R)^+
• Full-matrix Jordan algebra M_n(C)^+
• Full-matrix Jordan algebra M_n(R)^+
• Herm_1(C) complexification
• Herm_1(C) Jordan algebra
• Herm_1(C_s) Jordan algebra
• Herm_1(H) complexification
• Herm_1(H) Jordan algebra
• Herm_1(H_s) Jordan algebra
• Herm_1(O) complexification
• Herm_1(O) Jordan algebra
• Herm_1(O_s) Jordan algebra
• Herm_1(R) Jordan algebra
• Herm_2(C) complexification
• Herm_2(C_s) Jordan algebra
• Herm_2(H) complexification
• Herm_2(H) Jordan algebra
• Herm_2(H_s) Jordan algebra
• Herm_2(O) complexification
• Herm_2(O_s) Jordan algebra
• Herm_2(R) Jordan algebra
• Herm_3(C) complexification
• Herm_3(C_s) Jordan algebra
• Herm_3(H) complexification
• Herm_3(H) Jordan algebra
• Herm_3(H_s) Jordan algebra
• Herm_3(R) Jordan algebra
• Herm_n(C) complexification
• Herm_n(C) Jordan algebra
• Herm_n(C_s) Jordan algebra
• Herm_n(H) complexification
• Herm_n(H) Jordan algebra
• Herm_n(H_s) Jordan algebra
• Herm_n(O) complexification
• Herm_n(O) Jordan algebra
• Herm_n(O_s) Jordan algebra
• Herm_n(R) Jordan algebra
• Hermitian Jordan embeddings from coefficient inclusions
• M_1(C) matrix algebra
• M_1(C_s) matrix algebra
• M_1(H) matrix algebra
• M_1(H_s) matrix algebra
• M_1(O) matrix algebra
• M_1(R) matrix algebra
• M_1(S) matrix algebra
• M_2(C) matrix algebra
• M_2(C_s) matrix algebra
• M_2(H) matrix algebra
• M_2(H_s) matrix algebra
• M_2(O) matrix algebra
• M_2(R) matrix algebra
• M_2(S) matrix algebra
• M_3(C) matrix algebra
• M_3(C_s) matrix algebra
• M_3(H) matrix algebra
• M_3(H_s) matrix algebra
• M_3(O) matrix algebra
• M_3(R) matrix algebra
• M_3(S) matrix algebra
• M_n(C) matrix algebra
• M_n(C_s) matrix algebra
• M_n(H) matrix algebra
• M_n(H_s) matrix algebra
• M_n(O) matrix algebra
• M_n(R) matrix algebra
• M_n(S) matrix algebra
• Nonunital Hermitian Jordan corner embedding
• Real sedenions S
• Spin factor J(C^1)
• Spin factor J(C^2)
• Spin factor J(C^3)
• Spin factor J(C^n)
• Spin factor J(R^1)
• Spin factor J(R^2)
• Spin factor J(R^3)
• Spin factor J(R^5)
• Spin factor J(R^9)
• Split complex numbers C_s
• Split octonions O_s
• Split quaternions H_s
• Split real Albert algebra
• Split-complex Hermitian matrices as full-matrix Jordan algebras
• Standard inclusions in the Cayley–Dickson tower
• Symmetric 1-by-1 complex Jordan algebra
• Symmetric 2-by-2 complex Jordan algebra
• Symmetric 3-by-3 complex Jordan algebra
• Symmetric n-by-n complex Jordan algebra
• Zero-product Jordan algebra C^1
• Zero-product Jordan algebra C^2
• Zero-product Jordan algebra C^3
• Zero-product Jordan algebra C^n
• Zero-product Jordan algebra R^1
• Zero-product Jordan algebra R^2
• Zero-product Jordan algebra R^3
• Zero-product Jordan algebra R^n

Arithmetic (66)

• 2-adic integers
• 3-adic integers
• 5-adic integers
• Adeles of a global function field
• Adeles of a number field
• Adeles of F_q(t)
• Adeles of Q
• Eisenstein integers
• Field F_2
• Field F_27
• Field F_3
• Field F_4
• Field F_5
• Field F_8
• Field F_9
• Field F_p
• Field Q_2
• Field Q_3
• Field Q_5
• Field Q_p
• Finite adeles of a number field
• Finite adeles of Q
• Finite field F_(p^2)
• Finite field F_(p^3)
• Finite field F_(p^k)
• Finite ideles of a number field
• Finite ideles of Q
• Formal power series over F_2
• Formal power series over F_3
• Formal power series over F_5
• Formal power series over F_q
• Gaussian integers
• Gaussian rational field
• Hurwitz quaternion order
• Ideles of a global function field
• Ideles of a number field
• Ideles of F_q(t)
• Ideles of Q
• Imaginary quadratic field Q(sqrt(-3))
• Laurent-series field F_2((t))
• Laurent-series field F_3((t))
• Laurent-series field F_5((t))
• Laurent-series field F_q((t))
• Linear maps and field maps between finite fields
• Lipschitz quaternion order
• Polynomial ring F_2[t]
• Polynomial ring F_3[t]
• Polynomial ring F_5[t]
• Polynomial ring F_q[t]
• Quadratic number field Q(sqrt(d))
• Rational Hamilton quaternion algebra
• Rational matrix algebra M_1(Q)
• Rational matrix algebra M_2(Q)
• Rational matrix algebra M_3(Q)
• Rational matrix algebra M_n(Q)
• Rational-function field F_2(t)
• Rational-function field F_3(t)
• Rational-function field F_5(t)
• Rational-function field F_q(t)
• Real quadratic field Q(sqrt(2))
• Specified associative K-algebra A
• Specified base field F
• Specified extension field K/F
• Unramified degree 2 extension of Q_p
• Unramified degree 3 extension of Q_p
• Unramified degree n extension of Q_p

Categories (22)

• Category Jord of Jordan algebras
• Category Jord1 of unital Jordan algebras and unital maps
• Category of abelian groups
• Category of associative algebras with arbitrary homomorphisms
• Category of complex Lie groups
• Category of fields
• Category of finite groups
• Category of finite simple groups
• Category of groups
• Category of Lie algebras over a field
• Category of nonassociative algebras
• Category of real Lie groups
• Category of real Spin(8) representations
• Category of rings with arbitrary ring homomorphisms
• Category of sets
• Category of topological fields
• Category of topological groups
• Category of unital associative algebras
• Category of unital rings
• Category of unital topological rings
• Category of vector spaces over a field
• Category UJord of unital Jordan algebras with arbitrary maps

Finite groups (149)

• Alternating group A_1
• Alternating group A_2
• Alternating group A_3
• Alternating group A_4
• Alternating group A_5
• Alternating group A_6
• Alternating group A_7
• Alternating group A_8
• Baby Monster group
• Conway group Co1
• Conway group Co2
• Conway group Co3
• Cyclic group C_1
• Cyclic group C_2
• Cyclic group C_3
• Cyclic group C_n
• Cyclic group C_p
• Dihedral group D_2
• Dihedral group D_2n
• Dihedral group D_4
• Dihedral group D_6
• Elementary abelian group C_p^1
• Elementary abelian group C_p^2
• Elementary abelian group C_p^3
• Elementary abelian group C_p^n
• Endomorphisms of a finite simple group
• E₆(2)
• E₆(q)
• E₇(2)
• E₇(q)
• E₈(2)
• E₈(q)
• Finite abelian group with specified invariant factors
• Finite Chevalley central quotient
• Finite general linear group GL_1(q)
• Finite general linear group GL_2(q)
• Finite general linear group GL_3(q)
• Finite general linear group GL_n(q)
• Finite Heisenberg group H_1(F_q)
• Finite Heisenberg group H_2(F_q)
• Finite Heisenberg group H_3(F_q)
• Finite Heisenberg group H_n(F_q)
• Finite matrix group GL_2(2)
• Finite matrix group GL_2(3)
• Finite matrix group SL_2(2)
• Finite matrix group SL_2(3)
• Finite projective orthogonal derived group
• Finite special linear group SL_1(q)
• Finite special linear group SL_2(q)
• Finite special linear group SL_3(q)
• Finite special linear group SL_n(q)
• First Janko group J1
• Fischer group Fi22
• Fischer group Fi23
• Fischer group Fi24′
• Fourth Janko group J4
• F₄(2)
• F₄(q)
• Generalized quaternion group Q_(2^n)
• Generalized quaternion group Q_16
• Generalized quaternion group Q_32
• G₂(2)
• G₂(2) derived subgroup
• G₂(3)
• G₂(q)
• Hall–Janko group J2
• Harada–Norton group
• Held group
• Higman–Sims group HS
• Hom, End, and Aut for finite cyclic groups
• Klein four-group
• Leech lattice
• Lyons group
• Mathieu group M11
• Mathieu group M12
• Mathieu group M22
• Mathieu group M23
• Mathieu group M24
• Mathieu point stabilizers
• McLaughlin group McL
• Monster group
• O'Nan group
• PSL(1,q)
• PSL(2,2)
• PSL(2,3)
• PSL(2,4)
• PSL(2,5)
• PSL(2,7)
• PSL(2,8)
• PSL(2,9)
• PSL(2,q)
• PSL(2,q²)
• PSL(3,2)
• PSL(3,4)
• PSL(3,q)
• PSL(4,2)
• PSL(4,q)
• PSL(n,q)
• PSp(2,q)
• PSp(2n,q)
• PSp(4,2)
• PSp(4,2) derived subgroup
• PSp(4,3)
• PSp(4,q)
• PSp(6,q)
• PSU(1,q)
• PSU(2,q)
• PSU(3,2)
• PSU(3,3)
• PSU(3,q)
• PSU(4,2)
• PSU(4,q)
• PSU(n,q)
• PΩ(2n+1,q)
• PΩ(3,q)
• PΩ(5,q)
• PΩ(7,q)
• PΩ+(2n,q)
• PΩ+(4,q)
• PΩ+(6,q)
• PΩ-(2n,q)
• PΩ-(4,q)
• PΩ-(6,q)
• Rudvalis group
• Sporadic Suzuki group Suz
• Steinberg fixed-point group
• Symmetric group S_1
• Symmetric group S_2
• Symmetric group S_3
• Symmetric group S_4
• Symmetric group S_6
• Sz(2)
• Sz(8)
• Sz(q)
• Third Janko group J3
• Thompson sporadic group
• Tits group
• Two sporadic subgroups of the Baby Monster
• ²E₆(2)
• ²E₆(q)
• ²F₄(2)
• ²F₄(8)
• ²F₄(q)
• ²G₂(27)
• ²G₂(3)
• ²G₂(3) derived subgroup
• ²G₂(q)
• ³D₄(2)
• ³D₄(q)

Lie algebras (138)

• Abelian complex Lie algebra of dimension 1
• Abelian complex Lie algebra of dimension 2
• Abelian complex Lie algebra of dimension 3
• Abelian complex Lie algebra of dimension n
• Abelian real Lie algebra of dimension 1
• Abelian real Lie algebra of dimension 2
• Abelian real Lie algebra of dimension 3
• Abelian real Lie algebra of dimension n
• Affine transformation Lie algebra aff(1,C)
• Affine transformation Lie algebra aff(1,R)
• Affine transformation Lie algebra aff(2,C)
• Affine transformation Lie algebra aff(2,R)
• Affine transformation Lie algebra aff(3,C)
• Affine transformation Lie algebra aff(3,R)
• Affine transformation Lie algebra aff(n,C)
• Affine transformation Lie algebra aff(n,R)
• e(1) — Euclidean motion Lie algebra
• e(2) — Euclidean motion Lie algebra
• e(3) — Euclidean motion Lie algebra
• e(n) — Euclidean motion Lie algebra
• e6 — compact exceptional Lie algebra
• e6 — split exceptional Lie algebra
• e6(-26) — noncompact exceptional real form
• e6(2) — noncompact exceptional real form
• e7 — compact exceptional Lie algebra
• e7 — split exceptional Lie algebra
• e7(-25) — noncompact exceptional real form
• e7(-5) — noncompact exceptional real form
• e8 — compact exceptional Lie algebra
• e8 — split exceptional Lie algebra
• e8(-24) — noncompact exceptional real form
• f4 — compact exceptional Lie algebra
• f4 — split exceptional Lie algebra
• g2 — compact exceptional Lie algebra
• g2 — split exceptional Lie algebra
• gl(1,C) — complex general linear Lie algebra
• gl(1,H) — quaternionic general linear Lie algebra
• gl(1,R) — real general linear Lie algebra
• gl(2,C) — complex general linear Lie algebra
• gl(2,H) — quaternionic general linear Lie algebra
• gl(2,R) — real general linear Lie algebra
• gl(3,C) — complex general linear Lie algebra
• gl(3,H) — quaternionic general linear Lie algebra
• gl(3,R) — real general linear Lie algebra
• gl(n,C) — complex general linear Lie algebra
• gl(n,H) — quaternionic general linear Lie algebra
• gl(n,R) — real general linear Lie algebra
• Heisenberg Lie algebra h2n+1(C)
• Heisenberg Lie algebra h2n+1(R)
• Heisenberg Lie algebra h3(C)
• Heisenberg Lie algebra h5(C)
• Heisenberg Lie algebra h5(R)
• Heisenberg Lie algebra h7(C)
• Heisenberg Lie algebra h7(R)
• Low-dimensional Lie algebra isomorphisms
• sl(1,C) — complex special linear Lie algebra
• sl(1,H) — quaternionic special linear Lie algebra
• sl(1,R) — real special linear Lie algebra
• sl(2,C) direct sum with itself
• sl(2,C) with scalars restricted to R
• sl(2,H) — quaternionic special linear Lie algebra
• sl(2,R) direct sum with itself
• sl(2,R) — real special linear Lie algebra
• sl(3,C) with scalars restricted to R
• sl(3,H) — quaternionic special linear Lie algebra
• sl(3,R) — real special linear Lie algebra
• sl(4,C) — complex special linear Lie algebra
• sl(n,C) — complex special linear Lie algebra
• sl(n,H) — quaternionic special linear Lie algebra
• sl(n,R) — real special linear Lie algebra
• so(1,1) — indefinite orthogonal real Lie algebra
• so(1,C) — complex orthogonal Lie algebra
• so(1,R) — real orthogonal Lie algebra
• so(12,R) — real orthogonal Lie algebra
• so(2,1) — indefinite orthogonal real Lie algebra
• so(2,2) — indefinite orthogonal real Lie algebra
• so(2,C) — complex orthogonal Lie algebra
• so(2,R) — real orthogonal Lie algebra
• so(3,1) — indefinite orthogonal real Lie algebra
• so(3,3) — indefinite orthogonal real Lie algebra
• so(3,C) — complex orthogonal Lie algebra
• so(3,R) — real orthogonal Lie algebra
• so(4,1) — indefinite orthogonal real Lie algebra
• so(4,4) — indefinite orthogonal real Lie algebra
• so(4,C) — complex orthogonal Lie algebra
• so(4,R) — real orthogonal Lie algebra
• so(5,C) — complex orthogonal Lie algebra
• so(5,R) — real orthogonal Lie algebra
• so(6,C) — complex orthogonal Lie algebra
• so(6,R) — real orthogonal Lie algebra
• so(8,C) — complex orthogonal Lie algebra
• so(8,R) — real orthogonal Lie algebra
• so(n,C) — complex orthogonal Lie algebra
• so(n,n) — indefinite orthogonal real Lie algebra
• so(p,q) — indefinite orthogonal real Lie algebra
• so*(12) — quaternionic orthogonal real form
• sp(1) — compact symplectic Lie algebra
• sp(1,1) — indefinite quaternionic symplectic real Lie algebra
• sp(1,2) — indefinite quaternionic symplectic real Lie algebra
• sp(1,3) — indefinite quaternionic symplectic real Lie algebra
• sp(2) — compact symplectic Lie algebra
• sp(2,C) — complex symplectic Lie algebra
• sp(2,R) — real symplectic Lie algebra
• sp(2n,C) — complex symplectic Lie algebra
• sp(2n,R) — real symplectic Lie algebra
• sp(3) — compact symplectic Lie algebra
• sp(4,C) — complex symplectic Lie algebra
• sp(4,R) — real symplectic Lie algebra
• sp(6,C) — complex symplectic Lie algebra
• sp(6,R) — real symplectic Lie algebra
• sp(n) — compact symplectic Lie algebra
• sp(p,q) — indefinite quaternionic symplectic real Lie algebra
• Strictly upper triangular Lie algebra n1(C)
• Strictly upper triangular Lie algebra n1(R)
• Strictly upper triangular Lie algebra n2(C)
• Strictly upper triangular Lie algebra n2(R)
• Strictly upper triangular Lie algebra n3(C)
• Strictly upper triangular Lie algebra n3(R)
• Strictly upper triangular Lie algebra nn(C)
• Strictly upper triangular Lie algebra nn(R)
• su(1) — compact special unitary Lie algebra
• su(1,1) — indefinite special unitary real Lie algebra
• su(1,2) — indefinite special unitary real Lie algebra
• su(1,3) — indefinite special unitary real Lie algebra
• su(2) direct sum with itself
• su(2) — compact special unitary Lie algebra
• su(3) — compact special unitary Lie algebra
• su(3,3) — indefinite special unitary Lie algebra
• su(4) — compact special unitary Lie algebra
• su(6) — compact special unitary Lie algebra
• su(p,q) — indefinite special unitary real Lie algebra
• u(1) — compact unitary Lie algebra
• u(1,1) — indefinite unitary real Lie algebra
• u(1,2) — indefinite unitary real Lie algebra
• u(1,3) — indefinite unitary real Lie algebra
• u(2) — compact unitary Lie algebra
• u(3) — compact unitary Lie algebra
• u(p,q) — indefinite unitary real Lie algebra

Lie groups (162)

• Additive C^1
• Additive C^2
• Additive C^3
• Additive C^n
• Additive R^1
• Additive R^2
• Additive R^3
• Additive R^n
• Aff(1,C)
• Aff(1,R)
• Aff(2,C)
• Aff(2,R)
• Aff(3,C)
• Aff(3,R)
• Aff(n,C)
• Aff(n,R)
• E(1)
• E(2)
• E(3)
• E(n)
• E6 (complex, simply connected)
• E6 (split, connected adjoint)
• E7 (complex, simply connected)
• E7 (split, connected adjoint)
• E8 (complex, simply connected)
• E8 (split, connected adjoint)
• F4 (complex, simply connected)
• F4 (split, connected adjoint)
• G2 (complex, simply connected)
• G2 (split, connected adjoint)
• GL(1,C)
• GL(1,H)
• GL(1,R)
• GL(2,C)
• GL(2,H)
• GL(2,R)
• GL(3,C)
• GL(3,H)
• GL(3,R)
• GL(n,C)
• GL(n,H)
• GL(n,R)
• Heisenberg H_2n+1(C)
• Heisenberg H_3(C)
• Heisenberg H_3(R)
• Heisenberg H_5(C)
• Heisenberg H_5(R)
• Heisenberg H_7(C)
• Heisenberg H_7(R)
• Identity component SO0(p,q)
• O(1,1)
• O(1,2)
• O(1,3)
• O(1,C)
• O(1,R)
• O(2,C)
• O(2,R)
• O(3,C)
• O(3,R)
• O(n,C)
• O(n,R)
• O(p,q)
• PGL(1,C)
• PGL(1,R)
• PGL(2,C)
• PGL(2,R)
• PGL(3,C)
• PGL(3,R)
• PGL(n,C)
• PGL(n,R)
• Positive real multiplicative group
• PSL(1,C)
• PSL(1,R)
• PSL(2,R)
• PSL(3,C)
• PSL(3,R)
• PSL(n,C)
• PSL(n,R)
• SE(1)
• SE(2)
• SE(3)
• SE(n)
• SL(1,C)
• SL(1,H)
• SL(1,R)
• SL(2,C) × SL(2,C)
• SL(2,H)
• SL(2,R)
• SL(3,C)
• SL(3,H)
• SL(3,R)
• SL(4,C)
• SL(n,C)
• SL(n,H)
• SL(n,R)
• SO(1,1)
• SO(1,2)
• SO(1,3)
• SO(1,C)
• SO(1,R)
• SO(2,C)
• SO(2,R)
• SO(3,C)
• SO(4,C)
• SO(4,R)
• SO(5,C)
• SO(5,R)
• SO(6,C)
• SO(6,R)
• SO(8,C)
• SO(8,R)
• SO(n,C)
• SO(p,q)
• Sp(1)
• Sp(1,1)
• Sp(1,2)
• Sp(1,3)
• Sp(2)
• Sp(2,C)
• Sp(2,R)
• Sp(2n,C)
• Sp(2n,R)
• Sp(3)
• Sp(4,C)
• Sp(4,R)
• Sp(6,C)
• Sp(6,R)
• Sp(p,q)
• Spin(1,C)
• Spin(1,R)
• Spin(2,C)
• Spin(2,R)
• Spin(3,C)
• Spin(3,R)
• Spin(4,C)
• Spin(4,R)
• Spin(5,C)
• Spin(5,R)
• Spin(6,C)
• Spin(6,R)
• Spin(8,C)
• Spin(8,R)
• Spin(n,C)
• Spin(n,R)
• SU(1)
• SU(1,1)
• SU(1,2)
• SU(1,3)
• SU(2) × SU(2)
• SU(3)
• SU(4)
• SU(p,q)
• Torus T^1
• Torus T^2
• Torus T^3
• Torus T^n
• U(1,1)
• U(1,2)
• U(1,3)
• U(2)
• U(3)
• U(p,q)

Morphisms (9)

• Comparing K-linear and F-linear endomorphisms of a K-algebra
• Conjugation on SL(2,C) distinguishes real and complex group maps
• Endomorphisms of C as a vector space and as an algebra
• Endomorphisms of Q(i) as a rational vector space and algebra
• Finite-field Hom-sets under field and ring conventions
• Hom, End, and Aut depend on the category
• Jordan Hom-sets under the three unit conventions
• Jordan maps from the scalar algebra are idempotents
• Linear endomorphisms of sl(2,C) after forgetting its bracket

Navigation (11)

• A table of finite groups
• Elementary finite groups and families catalogue
• Fields, local objects, and orders catalogue
• Finite groups of Lie type catalogue
• Lie algebras catalogue
• Lie groups catalogue
• Magic-square outputs and triality representations catalogue
• Mathematical object catalogue
• Reading the Lie-group table
• Scalar, associative, and Jordan algebras catalogue
• Sporadic finite simple groups catalogue

Relationships (16)

• Compact Freudenthal magic square
• Derivation algebras of cubic Hermitian Jordan algebras
• Intertwiners between the three Spin(8) modules
• Real-form conventions for magic squares
• The Lie algebra su(3) direct sum su(3)
• The negative half-spin representation of Spin(8)
• The positive half-spin representation of Spin(8)
• The triple direct sum of sp(1)
• The vector representation of Spin(8)
• Tits decomposition of the magic square
• Transposition symmetry of the magic square
• Triality algebra of a composition algebra
• Triality and central quotients of Spin(8)
• Triality decomposition of compact f4
• Triality decomposition of the magic square
• Vinberg magic-square construction