Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: symplectic group Sp(2n,R) | Statement: [Heisenberg Lie algebra, hasAutomorphismGroupContaining, symplectic group Sp(2n,R)]

Target entity: symplectic group Sp(2n,R)
Context triple: [Heisenberg Lie algebra, hasAutomorphismGroupContaining, symplectic group Sp(2n,R)]

• A. SL(2,R)
  SL(2,R) is the Lie group of 2×2 real matrices with determinant 1, fundamental in representation theory, geometry, and the study of symmetries in mathematics and physics.
• B. special orthogonal group SO(n)
  The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
• C. special linear group SL(n,R)
  The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
• D. Spin(2,d)
  Spin(2,d) is the double-covering spin group of SO(2,d), serving as the relevant symmetry group for spinor fields in (d+1)-dimensional anti-de Sitter space.
• E. SL(2,C)
  SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: symplectic group Sp(2n,R)
Target entity description: The symplectic group Sp(2n,ℝ) is the Lie group of 2n×2n real matrices that preserve a nondegenerate skew-symmetric bilinear form, playing a central role in symplectic geometry and Hamiltonian mechanics.

• A. SL(2,R)
  SL(2,R) is the Lie group of 2×2 real matrices with determinant 1, fundamental in representation theory, geometry, and the study of symmetries in mathematics and physics.
• B. special orthogonal group SO(n)
  The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
• C. special linear group SL(n,R)
  The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
• D. Spin(2,d)
  Spin(2,d) is the double-covering spin group of SO(2,d), serving as the relevant symmetry group for spinor fields in (d+1)-dimensional anti-de Sitter space.
• E. SL(2,C)
  SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
• F. None of above. chosen

Target predicate: hasAutomorphismGroupContaining
Context triple: [Heisenberg Lie algebra, hasAutomorphismGroupContaining, symplectic group Sp(2n,R)]

• A. isAutomorphismGroupOf
  Indicates that a group is the full automorphism group consisting of all structure-preserving bijections (automorphisms) of a given mathematical object.
• B. hasAutomorphism
  Indicates that there exists a structure-preserving bijection from an entity to itself that maintains all relevant relations and operations.
• C. automorphismGroupIsomorphicTo
  Indicates that the automorphism group of one structure is isomorphic, as a group, to the automorphism group of another structure.
• D. hasOuterAutomorphismGroupOrder
  Indicates the relationship that specifies the order (size) of the outer automorphism group associated with a given mathematical structure.
• E. fullAutomorphismGroupOf
  Indicates that one entity is the complete group of all automorphisms (structure-preserving bijections from an object to itself) of the other entity.
• F. None of above. chosen