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mathematics, a unitary modular tensor category is a certain type of algebraic structure, defined by equipping a modular tensor category with additional
Unitary modular tensor category
Unitary_modular_tensor_category
Type of monoidal category
A modular tensor category (or modular fusion category) is a type of monoidal category that plays a role in the areas of topological quantum field theory
Modular_tensor_category
Particle
category, since φ = 2 cos ( π / 5 ) {\displaystyle \varphi =2\cos(\pi /5)} . The Fibonacci category is the unique unitary modular tensor category with
Fibonacci_anyons
asserts that the Drinfeld center of every spherical fusion category is a modular tensor category. Müger's theorem was introduced in 2003 by mathematician
Müger's_theorem
Algebraic theory
normalization is given below. Unitary modular tensor category (and Modular tensor category) Fibonacci anyons (and Fibonacci category) Kong, Liang; Zhang, Zhi-Hao
Algebraic theory of topological quantum information
Algebraic_theory_of_topological_quantum_information
Representation of a modular tensor category
In mathematics, the modular group representation (or simply modular representation) of a modular tensor category C {\displaystyle {\mathcal {C}}} is a
Modular_group_representation
Branch of mathematics that studies abstract algebraic structures
coalgebra. In general, the tensor product of irreducible representations is not irreducible; the process of decomposing a tensor product as a direct sum
Representation_theory
Algebra used in certain conformal field theories
under appropriate normalization the S-matrix of every modular tensor category can be made unitary, and the S-matrix entry S 0 σ {\displaystyle S_{0\sigma
Verlinde_algebra
Type of order at absolute zero
classified by unitary modular tensor categories. 2+1D bosonic topological orders with symmetry G are classified by G-crossed tensor categories. 2+1D bosonic/fermionic
Topological_order
Number in {..., –2, –1, 0, 1, 2, ...}
capital letter Z. Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008 LK Turner, FJ BUdden, D Knighton
Integer
Algebra used in 2D conformal field theories and string theory
vertex operator algebra admit a fusion tensor product operation, and form a braided tensor category. When the category of V-modules is semisimple with finitely
Vertex_operator_algebra
Algebraic construct of interest in theoretical physics
finite-dimensional. In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding
Quantum_group
representation tensor A tensor representation is roughly a representation obtained from tensor products (of certain representations). tensor product The tensor product
Glossary of representation theory
Glossary_of_representation_theory
Mathematical property
Texts in Mathematics, vol. 222 (2nd ed.), Springer MathOverflow:Are abelian non-degenerate tensor categories semisimple? Semisimple category at the nLab
Semi-simplicity
Elements taken to zero by a homomorphism
1,2,3,4,5\}} with modular addition, H {\displaystyle H} be the cyclic on 2 elements { 0 , 1 } {\displaystyle \{0,1\}} with modular addition, and f {\displaystyle
Kernel_(algebra)
paradox Category of groups Dimensional analysis Elliptic curve Galois group Gell-Mann matrices Group object Hilbert space Integer Lie group Matrix Modular arithmetic
List_of_group_theory_topics
Conformal field theory on a 2D spacetime
identified with (a component of) the energy–momentum tensor. In particular, the OPE of the energy–momentum tensor with a primary field is T ( y ) V Δ ( z ) = Δ
Two-dimensional conformal field theory
Two-dimensional_conformal_field_theory
History of maths
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: Categories of abstract algebraic structures
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Group that is a topological space with continuous group operations
representations, is to find the unitary dual (the space of all irreducible unitary representations) for the semisimple Lie groups. The unitary dual is known in many
Topological_group
Commutative group (mathematics)
nontrivial multiplication). Some important topics in this area of study are: Tensor product A.L.S. Corner's results on countable torsion-free groups Shelah's
Abelian_group
Mathematics timeline
geometric ideas, concepts from quantum field theory, and heavy use of category theory. Participants in the first phase of axiomatization were influenced
Timeline_of_manifolds
Representation of a group or algebra that is a direct sum of simple representations
irreducible representations occurring in the semisimple decomposition of a tensor product of irreducible representations. To be precise, the theorem concerns
Semisimple_representation
American computer scientist and engineer
software. Efficient simulation of quantum circuits with low tree-width using tensor-network contraction. Follow-up works extended this technique with approximations
Igor_L._Markov
Subgroup of the group of invertible n×n matrices
group G, together with the tensor product of representations, form a tannakian category RepG. In fact, tannakian categories with a "fiber functor" over
Linear_algebraic_group
U.S. presidential administration since 2025
expansion of presidential power under a maximalist interpretation of the unitary executive theory. Several of his actions ignored or violated federal laws
Second presidency of Donald Trump
Second_presidency_of_Donald_Trump
Type of quantum computer
similarly to that of two spin 1/2 particles. Particularly, we use the same tensor product ⊗ {\displaystyle \otimes } and direct sum ⊕ {\displaystyle \oplus
Topological_quantum_computer
Number with a real and an imaginary part
generalizes the transpose, hermitian matrices generalize symmetric matrices, and unitary matrices generalize orthogonal matrices. In control theory, systems are
Complex_number
Hungarian and American mathematician and physicist (1903–1957)
Schatten he initiated the study of nuclear operators on Hilbert spaces, tensor products of Banach spaces, introduced and studied trace class operators
John_von_Neumann
Theory of subatomic structure
wrapped into a small circle. Einstein introduced a non-symmetric metric tensor, while much later Brans and Dicke added a scalar component to gravity. These
String_theory
Representations of finite groups, particularly on vector spaces
the tensor product of the corresponding representation spaces. The second case is a representation of the group G {\displaystyle G} into the tensor product
Representation theory of finite groups
Representation_theory_of_finite_groups
Computer programming for quantum computers
partial trace/transpose, discrete Wigner transforms, stabilizer methods, and tensor-network utilities—as well as a library of named constructs (e.g., Bell/GHZ
Quantum_programming
Series of mathematics textbooks
Representation Theory, William Fulton, Joe Harris (1991, ISBN 978-3-5400-0539-1) Tensor Geometry — The Geometric Viewpoint and its Uses, Christopher T. J. Dodson
Graduate_Texts_in_Mathematics
248-dimensional exceptional simple Lie group
projective plane" because it can be built using an algebra that is the tensor product of the octonions with themselves, and is also known as a Rosenfeld
E8_(mathematics)
Series of mathematics books by Nicolas Bourbaki
topological vector spaces. Furthermore, two entirely new books (on category theory and modular forms) are stated to be under preparation. Springer Verlag became
Éléments_de_mathématique
Country in East Asia
advanced democracies in Asia. Under the 1987 constitution, it maintains a unitary presidential republic with a popularly elected unicameral legislature,
South_Korea
Special functions used to build correlation functions in 2D CFTs
classification of minimal models. These are related to the modular group representations of modular tensor categories. An arbitrary one-point block on the torus can
Virasoro_conformal_block
Country in West Asia
in 1991 during the dissolution of the Soviet Union. Modern Armenia is a unitary, multi-party, democratic nation-state. It is a developing country and ranks
Armenia
Grothendieck, Produits tensoriels topologiques et espaces nucléaires (Topological tensor products and nuclear spaces) Paul Jaffard, Les corps quasi-algébriquement
Séminaire Nicolas Bourbaki (1950–1959)
Séminaire_Nicolas_Bourbaki_(1950–1959)
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UNITARY MODULAR-TENSOR-CATEGORY
UNITARY MODULAR-TENSOR-CATEGORY
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